Final 17 0016 · 2019-09-26 · Lee, Jihoon (Chung-Ang University, Korea) Li, Zhilin (North Carolina State University, USA) Liu, Hongyu (Hong Kong Baptist University, Hong Kong )

VOLUME 21 NUMBER 2 / June 2017

The Korean Society for Industrial and Applied Mathematics

ContentsVOL. 21 No. 2 June 2017

A MULTIPHASE LEVEL SET FRAMEWORK FOR IMAGE SEGMENTATION

USING GLOBAL AND LOCAL IMAGE FITTING ENERGY

DULTUYA TERBISH, ENKHBOLOR ADIYA, MYUNGJOO KANG ············ 63

MATHEMATICAL UNDERSTANDING OF CONSCIOUSNESS AND

UNCONCIOUSNESS

NAMI LEE, EUN YOUNG KIM, CHANGSOO SHIN ······························ 75

THE CONE PROPERTY FOR A CLASS OF PARABOLIC EQUATIONS

MINKYU KWAK, BATAA LKHAGVASUREN ····································· 81

NUMERICAL SOLUTIONS OF AN UNSTEADY 2-D INCOMPRESSIBLE

FLOW WITH HEAT AND MASS TRANSFER AT LOW, MODERATE,

AND HIGH REYNOLDS NUMBERS

V. AMBETHKAR, D. KUSHAWAHA ················································· 89

ISSN 1226-9433(print)ISSN 1229-0645(electronic)

Vol. 21 No. 2 June 2017 The K

orean Society for Industrial and Applied M

athematics K

SIAM

Journal of the Korean Society for Industrial and Applied Mathematics

제 21권 2호

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ISSN 1226-9433(print) ISSN 1229-0645(electronic)

Journal of the KSIAMEditor-in-Chief

Seo, Jin Keun (Yonsei University, Korea)

Associate Editor-In-Chief Managing EditorsChen, Zhiming (Chinese Academy of Sciences, China)Lee, June-Yub (Ewha w. University, Korea)Tang, Tao (Hong Kong Baptist University, Hong Kong)

Cho, Jin-Yeon (Inha University, Korea)Kim, Junseok (Korea University, Korea)

Editorial BoardAhn, Hyung Taek (University of Ulsan, Korea)Ahn, Jaemyung (KAIST, Korea)Campi, Marco C. (University of Brescia, Italy)Ha, Tae Young (NIMS, Korea)Han, Pigong (Chinese Academy of Sciences, P.R.China)Hodges, Dewey (Georgia Tech., USA)Hwang, Gang Uk (KAIST, Korea)Jung, Il Hyo (Pusan Natl University, Korea)Kim, Hyea Hyun (Kyung Hee University, Korea)Kim, Hyoun Jin (Seoul Natl University, Korea)Kim, Jeong-ho (Inha University, Korea)Kim, Kyu Hong (Seoul Natl University, Korea)

Kim, Yunho (UNIST, Korea)Lee, Eunjung (Yonsei University, Korea)Lee, Jihoon (Chung-Ang University, Korea)Li, Zhilin (North Carolina State University, USA)Liu, Hongyu (Hong Kong Baptist University, Hong Kong)Min, Chohong (Ewha w. University, Korea)Park, Soo Hyung (Konkuk University, Korea)Shim, Gyoocheol (Ajou University, Korea)Shin, Eui Seop (Chobuk Natl University, Korea)Shin, Sang Joon (Seoul Natl University, Korea)Sohn, Sung-Ik (Gangneung-Wonju Natl University, Korea)Wang, Zhi Jian (North Carolina State University, USA)

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Contact info.THE KOREAN SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICSRoom 405, Industry-University Research Center, Yonsei Univ.50, Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of KoreaJung, Eunok (e-mail : [email protected]) Tel : +82-2-2123-8078

Journal of KSIAM was launched in 1997, is published four times a year by the Korean Society for Industrial and Applied Mathematics. Total or part of the articles in this journal are abstracted in KSCI, Mathematical Reviews, CrossRef, Korean Science, and NDSL . Full text is available at http://www.ksiam.org/archive.Copyright © 2017, the Korean Society for Industrial and Applied Mathematics


Volume 21, Number 2, June 2017

Contents



DULTUYA TERBISH, ENKHBOLOR ADIYA, MYUNGJOO KANG ············· 63


UNCONCIOUSNESS

NAMI LEE, EUN YOUNG KIM, CHANGSOO SHIN ······························· 75


MINKYU KWAK, BATAA LKHAGVASUREN ······································ 81




V. AMBETHKAR, D. KUSHAWAHA ·················································· 89

J. KSIAM Vol.21, No.2, 63–73, 2017 http://dx.doi.org/10.12941/jksiam.2017.21.063

A MULTIPHASE LEVEL SET FRAMEWORK FOR IMAGE SEGMENTATIONUSING GLOBAL AND LOCAL IMAGE FITTING ENERGY

DULTUYA TERBISH1, ENKHBOLOR ADIYA1, AND MYUNGJOO KANG2

1NATIONAL UNIVERSITY OF MONGOLIA, MONGOLIA

E-mail address: [email protected], [email protected] NATIONAL UNIVERSITY, KOREA

E-mail address: [email protected]

ABSTRACT. Segmenting the image into multiple regions is at the core ofimage processing.Many segmentation formulations of an images with multiple regions have been suggested overthe years. We consider segmentation algorithm based on the multi-phase level set method inthis work. Proposed method gives the best result upon other methods found in the references.Moreover it can segment images with intensity inhomogeneity and have multiple junction. Weextend our method (GLIF) in [T. Dultuya, and M. Kang,Segmentation with shape prior usingglobal and local image fitting energy, J.KSIAM Vol.18, No.3, 225–244, 2014.] using a multi-phase level set formulation to segment images with multipleregions and junction. We test ourmethod on different images and compare the method to other existing methods.

1. INTRODUCTION

Segmentation of images with multiple regions is an important research in image segmenta-tion field. Various works of segmentation methods have developed. Main goals of the segmen-tation methods are to partition an image into reasonable number of regions, to detect the objectswith different intensity from background of the image and torepresent the multiple junctionin the image. Region based model of Mumford-Shah (MS) functional [1] is fundamental ap-proach of most segmentation methods. The nonconvexity of the (MS) functional makes itdifficult to be minimized. To simplify the (MS) model, Chan and Vese (Chan-Vese)[2] re-formulate the (MS) functional using the level set method that introduced in [3]. However,the (Chan-Vese) method is suitable for two-region image segmentation. To segment multipleregions, Vese-Chan extended their method in [4]; however, multi-phase level set makes themethod computationally expensive because of the re-initialization process of the level set func-tions. The piecewise constant (PC) and the piecewise smooth(PS) models were proposed inthis extension. The (PC) model works well on the images with intensity homogeneity whereasthe (PS) model can segment images with intensity inhomogeneity. Nevertheless, these methodsare difficult to implement and also increases the computational cost.

Received by the editors May 2 2017; Accepted May 29 2017; Published online June 10 2017.2000Mathematics Subject Classification.93B05.Key words and phrases.Segmentation, GLIF model, multiple junction, multi-phaselevel set method.

63

64 T.DULTUYA, A.ENKHBOLOR, AND M.KANG

Another common approach is the active contour model , which is an edge based model.Using the active contour models [10, 11, 12], the local imagefitting (LIF) approach was pro-posed to segment images with intensity inhomogeneity in [5]. Similar approaches have beenproposed in [6, 8]. They regularize the level set function using Gaussian filtering for variationallevel set; thus, the (LIF) eliminates the re-initialization process. The (LIF) method gives bettersegmentation results and is more computationally efficientthan the (PS) model. We formulatedanother efficient approach (GLIF) in [17] for images with intensity inhomogeneity. Howeverthis model cannot segment multiple regions and cannot represent the multiple junction. We willextend the (GLIF) method using multi-phase level set formulation to partition multiple regionswith junction in this work.

Organization of this paper is followed by: Section 2 introduces previous works. We re-view the segmentation models of images with multiple regions. Also, significant segmentationmethods for images with inhomogeneity are introduced. The main contribution of this work ispresented in Section 3. In Section 3, we extend our method to segment images with multipleregions using multi-phase level set method and local image fitting energy. We set a local imagefitting term to cope with the intensity inhomogeneity of the image. To overcome sensitivity ofinitialization, a global image fitting term with multi-phase level set is considered. Numericalexperiments are discussed in Section 4. At last, we concludeour work in Section 5.

2. RELATED WORKS

The first fundamental model for image segmentation was proposed by Mumford and Shahin [1]. Their main idea is to approximate an image by a simplified image as a combination ofregions of constant intensities and the smoothness of the contours was disregarded. These ideaswere incorporated into a variational framework; an initialimageu0, find pair(u,C), whereuis a nearly piecewise smooth approximation ofu0 andC is the set of edges. Mumford andShah proposed to find(u,C) by minimizing the following functional:

FMS(u,C) =

∫

Ω−C

(u− u0)2dx+ µ

∫

Ω−C

|∇u|2dx+ ν

∫

C

dσ (2.1)

whereΩ is a bounded open set ofR2, µ andν are nonnegative constants and∫

Cdσ is the

length ofC. Methods of solving the general (MS) model are complicated and computationallyexpensive, even though, (2.1) is a natural method of segmentation.

To overcome the disadvantage of (MS) functional, a region-based segmentation method withlevel sets was proposed by Chan and Vese [2, 13]. The level setformulation of the (Chan-Vese)model can be written as

EChan−V ese(c1, c2, φ) =

∫

Ω

(

(u0 − c1)2H(φ) + (u0 − c2)

2(1−H(φ)))

dx

+ µ

∫

Ω

|∇H(φ)|dx + ν

∫

Ω

H(φ)dx

whereu0 is a given image on the bounded open subsetΩ in R2.

MULTIPLE REGION SEGMENTATION 65

Vese and Chan extended their model using a multi-phase levelset formulation [16] to par-tition multiple regions. Piecewise constant (PC) and Piecewise smooth(PS) models were pro-posed in [4]. The (PC) model has the advantage that it can represent multiple regions andjunction.

In the (PC) model, level set functionsφi : Ω → R, i = 1, ...,m were considered. The unionof the zero-level sets ofφi will represent the contours in the segmented image. The segments orphases in the domainΩ can be defined by the following way: Two pixels(x1, y1) and(x2, y2)in Ω will belong to the same phase or class if and only ifH(Φ(x1, y1)) = H(Φ(x2, y2)). HereΦ = (φ1, ..., φm) is the vector of level set functions andH(Φ) = (H(φ1), ...,H(φm)) is thevector of Heaviside functions whose components are only 1 or0.

Up ton = 2m phases or classes can be defined in the domain of definitionΩ. The classesdefined in this way form a disjoint decomposition and covering of Ω. Therefore, each pixel(x, y) ∈ Ω will belong to only one class, thus there is no vacuum or overlap among the phases.The set of curves C is represented by the union of the zero level sets of the functionsφi. Asshown in Figure 1, we need two level set functions(m = 2) to represent four phases(n = 4)in the (PC) model. Therefore, the energy of 4 phase (PC) modelfor level set representation is

FIGURE 1. 4 phases of two level set functions

written by:

EPC4 (c,Φ) =

n∑

i

m∑

j

∫

Ω

(u0 − ci,j)2Hi,jdx+

∫

Ω

|∇H(φ1)|+

∫

Ω

|∇H(φ2)|

where

H11 = H(φ1)H(φ2), H12 = H(φ1)(1 −H(φ2))

H21 = (1−H(φ1))H(φ2), H22 = (1−H(φ1))(1−H(φ2))

andci,j = mean(u0), i = 1, 2, j = 1, 2 in each region/phase.Figure 2 shows the segmentation result of a noisy synthetic image with a multiple junction.

Using only one level set function in Chan-Vese model (Figure2(a)), the triple junction cannot


(a) (b) (c) (d)

FIGURE 2. Segmentation result of (PC) model for image with triple junction

be represented (Figure 2(b)). If we utilize two level set functions (Figure 2(c)) in (PC) modelwith n = 4, the triple junction can be represented and 4 phases are extracted as shown in Figure2(d). The (PC) model has advantage that it can represent the triple junction and more than 2phases.

However, the authors Vese and Chan extended their model to multi-phase level set frame-work, it requiresm level set functions and2m phases to detect2m regions. To overcome thisdifficulty or to reduce this storage, Lie [18] proposed Piecewise Constant Level Set Method(PCLSM). In this model, they introduced using only one levelset function to detect all regions.The authors assumed that a piecewise constant level set function φ = i in reach region. Thesegmentation can be formulated as a minimization of the following functional:

F (c, φ) =1

2

∫

Ω

|u− u0|2dx+ β

n∑

i=1

∫

Ω

|∇ψi|dx

Where:u is formed byu =∑n

i=1ciψi, hereψi are basis functions and these functions defined

by using polynomial approach that is formed by

ψi =1

αi

n∏

j=1,j 6=i

(φ− j) andαi =

n∏

k=1,k 6=i

(i− k)

and a piecewise constant level set functionφ satisfiesφ= i in Ωi for i=1, 2, · · ·, n.For uniquely classifying each point or intensity in the image, they introduced a polynomial

of degreen that isK(φ) =∏n

i=1(φ − i) for the constraint. Therefore they used a constraint

K(φ) = 0 and solved the following constrained minimization problem:

minc,φ

F (c, φ) subject toK(φ) = 0.

If a given functionφ : Ω → R satisfiesK(φ) = 0, there exists a uniquei ∈ 1, ..., n for everyx ∈ Ω such thatφ(x) = i. Thus, each pointx ∈ Ω can belong to one and only one phase ifK(φ) = 0. It can be solved by the augmented Lagrangian method.

These methods work well for images with intensity homogeneity (or roughly constant ineach phases) but do not work for the images with intensity inhomogeneity. So, we will discuss


segmentation methods for images with intensity inhomogeneity. Most images with intensityinhomogeneity occur at medical image processing. In particular, inhomogeneities in magneticresonance images (MRI) arise from nonuniform magnetic fields produced by ratio-frequencycoils, as well as from variations in object susceptibility.Therefore, many segmentation ap-proaches have been developed for these images.

The first approach is the (PS) model proposed by Vese and Chan [4]. The (PS) model isformulated as minimizing the following energy whenn = 2 (two phase case):

EPS2 (u+, u−,Φ) =

∫

Ω

(

(u0 − u+)2H(φ) + (u0 − u−)2(1−H(φ)))

dxdy

+ µ

∫

Ω

(

|∇u+|2H(φ) + |∇u−|2(1−H(φ)))

dxdy + ν

∫

Ω

|∇H(φ)|

Here,u = u+H(φ) + u−(1−H(φ)) andφ can be expressed by introducing two functionsu+

andu− such that

u(x, y) =

u+(x, y), if φ(x, y) ≥ 0

u−(x, y), if φ(x, y) < 0.

This model can be extended to segment an image with intensityinhomogeneity and includetwo or more phases. Figure 3 shows the segmentation result ofa noisy image with intensityinhomogeneity. By the second term of the functional, a noisecan be removed as shown inFigure 3. Even though the (PS) model can segment an image by reducing the influence ofintensity inhomogeneity, it is computationally expensiveand inefficient in practice.

FIGURE 3. Segmentation result of (PS) model for noisy image

Compared to the (PS) model, a more inexpensive and accurate models were proposed in[5, 6, 7, 8, 17]. These models are based on a kernel functionK(x) with a localization propertythat K(x) decreases and approaches zero as|x| increases. All methods choose the kernelfunctionK(x) as a Gaussian kernel

Kσ(x) =1

(2π)n\2σnexp

(

−|x|2

2σ2

)

,


with a scale parameterσ > 0. This scale parameter is a standard deviation of the kerneland it plays an important role. By suitable chosen, it can control the region-scalability fromsmall neighborhoods to the entire image domain. The scale parameter should be properlychosen according to the contents of a given image. In particular, when an image is too noisyor has low contrast, a large value ofσ should be chosen. Unfortunately, this may cause a highcomputational cost. Small values ofσ can cause undesirable result as well.

We proposed global and local image fitting (GLIF) method in [17] by taking the advantage ofthe Chan-Vese and (LIF) models. The (GLIF) method defined theenergy functional as follows:

EGLIF (φ, c1, c2,m1,m2) = (1− α)ELIF (φ,m1,m2) + αEGIF (φ, c1, c2) (2.2)

whereα is a positive constant such that(0 ≤ α ≤ 1). The value ofα should be small when theintensity inhomogeneity in an image is severe. The local intensity fitting energyELIF is equalto the (LIF) model in [8], and the global intensity fitting (GIF) energyEGIF is the term of theChan-Vese model without regularizing term. Andm1 andm2 are the optimal fitting functionsgiven by following equation that is introduced in (LIF) model:

m1 = mean(u0 ∈ (x ∈ Ω|φ(x) < 0 ∩Wk(x)))

m2 = mean(u0 ∈ (x ∈ Ω|φ(x) > 0 ∩Wk(x)))(2.3)

Here, the rectangular window function is denoted byWk(x). They defined the local fittedimage as

uLFI = m1H(φ) +m2(1−H(φ)). (2.4)

Then proposed local intensity fitting (LIF) energy formulation is given by

ELIF =1

2

∫

Ω

∣

∣u0(x)− uLFI(x)∣

∣

2dx , x ∈ Ω. (2.5)

In this model, a Gaussian filtering is applied to regularize the level set function, i.e.,φ = Gγ ∗φ, whereγ is the standard deviation. Our method can segment an images with intensity inho-mogeneity or multiple objects with different intensities.However, it cannot segment multipleregions. Also the proposed method cannot represent multiple junction. Therefore we willextend our method in section 3.

3. PROPOSED MODEL

We extend the energy (2.2) using multi-phase level set framework. By idea of Vese-Chanmodel, we note the regions byi, with 1 ≤ i ≤ 2m = n. Then the extension of our model canbe written as

Fn(c,m,Φ) = α∑

1≤i≤n

∫

Ω

|u0 − ci|2Hidx+ (1− α)

∫

Ω

∣

∣

∣u0 −

∑

1≤i≤n

miHi

∣

∣

∣

2

dx

Where,ci = mean(u0) in each regioni,mi are optimal fitting functions andHi are the Heavi-side functions whose components are only 1 or 0. In this functional, the first term encouragesto drive the motion of the contour far away from object boundaries and to represent multiple


junction. And the second term enables to cope with intensityinhomogeneity. For the purposeof illustration, let us write the above energy forn = 4 phases or regions as follows.

The fitting term of the Vese-Chan model, excluding regularization terms, is given by:

EG4 =

∫

Ω

(

|u0 − c1|2H1 + |u0 − c2|

2H2 + |u0 − c3|2H3 + |u0 − c4|

2H4

)

dx

=

4∑

i=1

∫

Ω

|u0 − ci|2Hidx

(3.1)

where:H1 = H(φ1)H(φ2), H2 = H(φ1)(1 − H(φ2)), H3 = (1 − H(φ1))H(φ2), H4 =(1 −H(φ1))(1 −H(φ2)) andci = mean(u0), i = 1, 2, 3, 4 in each region. We call this termby global image fitting (GIF) term. This term encourages to improve the convergence speed byeliminating the segmentation process’ sensitivity to initialization and to represent the multiplejunction.

We extend (LIF) model on two level set functions by following:

EL4 =

∫

Ω

∣

∣u0 −m1H1 −m2H2 −m3H3 −m4H4

∣

∣

2dx (3.2)

where

mi =Kσ(x) ∗

(

u0(x) ·Hi

)

Kσ(x) ∗Hi, i = 1, 2, 3, 4.

Here,Kσ(x) is a Gaussian Kernel with scale parameterσ. The scale parameter should beproperly chosen depending on the contents of an image. In general,σ should be chosen frominterval of [1; 3].

The proposed energy functional consists of a local image fitting term and global image fittingterm. Specifically,

F4 = (1− α)EL4 + αEG

4 (3.3)

whereα is a positive constant such that(0 ≤ α ≤ 1). The value ofα should be smallfor images with severe intensity inhomogeneity. The local image fitting termEL

4 enables themodel to cope with intensity inhomogeneity. Furthermore, it includes a local force to attractthe contours and stop it at object boundaries. The global image fitting termEG

4 allows flexibleinitialization of the contours. Also, it includes a global force to drive the motion of the contourfar away from object boundaries.

We now discuss the numerical approximation for minimizing theF4 functional. Constantsci that minimize the energy in (3.3) are given by

ci =

∫

u0(x)Hi(x)dx∫

Hi(x)dx, i = 1, 2, 3, 4 (3.4)

By theory of calculus of variations, differentiating with respect toφ1, φ2 for fixed ci and weobtain the following gradient descent flow

∂φ1

∂t= −δ(φ1)

(

H(φ2)(a1 − a3) + (1−H(φ2))(a2 − a4)

)

(3.5)


∂φ2

∂t= −δ(φ2)

(

H(φ1)(a1 − a2) + (1−H(φ1))(a3 − a4)

)

(3.6)

whereai = (1− α)(u0(x)−mi) + α(u0(x)− ci)2, i = 1, 2, 3, 4.

The algorithm for solvingF4 is as follows:

Step 1: Initialize the level set functionφ1 andφ2.Step 2: Computeci according to (3.4).Step 3: Evolve the level set functionφ1 andφ2 according to (3.5), (3.6).Step 4: Regularize the level set functionφj using a Gaussian kernel,

i.e.,φj = Gγ ∗ φj , j = 1, 2 whereγ is the standard deviation.Step 5: Check whether the evolution is stationary. If not, return tostep 3.

4. EXPERIMENTAL RESULTS

Using gradient descent flows (3.5), (3.6) and the above algorithm, segmentation results areproduced faster and require fewer iteration than the Vese-Chan and (PCLSM) models. Exper-imental results are illustrated in Figure 4, Figure 5 and Figure 6. Our algorithm works wellon images with intensity inhomogeneity and it can segment multiple regions and can representmultiple junction (Figure 4(b,c), Figure 5(b,c) and Figure6(b,c)). The scale parameterσ isequal to3 for these images and the regularizing parameterγ is properly chosen0.8. Theseresults are better to the results of the Vese-Chan and (PCLSM) models.

In Figure 4(d), Figure 5(d) and Figure 6(d), the computed averages by (PCLSM) are il-lustrated. Notice that better results are produced by our method. Furthermore, computationaltimes are relatively high using the proposed method. In Table1, we compare the number of iter-ations and computational times for the (PCLSM), Vese-Chan models to our proposed method.

TABLE 1. Computation time results.

Methods Junction Brain lung

Iterations(time(s)) Iterations(time(s)) Iterations(time(s))

Vese-Chan 420 (528.85) 200 (184.16) 400 (469.09)PCLSM 492 (148.55) 679 (153.22) 596 (291.84)Proposed 12 (25.24) 4 (15.23) 2 (6.54)


(a) (b) (c) (d)

FIGURE 4. Segmentation result of proposed method: (a) is the given imagewith the initial level set; (b) is the result of proposed method; (c) is the com-puted average of proposed method; (d) is the computed average of the PCLSM.

(a) (b) (c) (d)


(a) (b) (c) (d)



5. CONCLUSION

We proposed the multi-phase level set method of image segmentation driven by global andlocal image fitting energy. Our method worked well for imageswith multiple regions andintensity inhomogeneity. Moreover, it can allowed flexibleinitialization of the contours. Inorder to cope with the intensity inhomogeneity of the image,we set a local image fitting term.To overcome initialization sensitivity and to represent multiple junction, a global image fittingterm was considered. Our segmentation results were obtained faster, requiring fewer iterationsthan the Vese-Chan and (PCLSM) models.

ACKNOWLEDGEMENTS

Myungjoo Kang was supported by NRF(2014R1A2A1A10050531, 2015R1A5A1009350)and IITP-MSIP(No. B0717-16-0107).

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of computational physics,127 (1996), 179–195.


[17] T. Dultuya, and M. Kang,Segmentation with shape prior using global and local image fitting energy, Journalof the Korean Society for Industrial and Applied Mathematics,18 (2014), 225–244.

[18] J. Lei, M. Lysaker and X.C. Tai,Piecewise constant level set methods and image segmentation, Scale-spaceand PDE methods in Computer vision: 5th International Conference, Scale-Space 2005, R.Kimmel, N.Sochen,and J. Weickerts, Eds. Heidelberg, Germany: Springer-Verlag,3459 (2005), 573–583


MATHEMATICAL UNDERSTANDING OF CONSCIOUSNESS ANDUNCONCIOUSNESS

NAMI LEE1†, EUN YOUNG KIM2, AND CHANGSOO SHIN3

1DR. LEE’ S PSYCHOLANALYSIS CLINIC , KOREA


2DEPARTMENT OF NEUROPSYCHIATRY, SEOUL NATIONAL UNIVERSITY HOSPITAL, MENTAL HEALTH

CENTER, SEOUL NATIONAL UNIVERSITY HEALTH CARE CENTER, KOREA


3DEPARTMENT OFENERGY RESOURCESENGINEERING, SEOUL NATIONAL UNIVERSITY, KOREA


ABSTRACT. This paper approaches the subject of consciousness and unconsciousness froma mathematical point of view. It sets up a hypothesis that when unconscious state becomesconscious state, high density energy is released. We argue that the process of transformation ofunconsciousness into consciousness can be expressed usingthe infinite recursive Heaviside stepfunction. We claim that differentiation of the potential ofunconsciousness with respect to timeis the process of being conscious in a world where only time exists, since the thinking processnever have any concrete space. We try to attribute our unconsciousness to a special solutionof the multi-dimensional advection partial differential equation which can be represented bythe finite recursive Heaviside step function. Mathematicallanguage explains how the infinitiveneural process is perceived and understood by consciousness in a definitive time.

1. INTRODUCTION

Human beings have tried to understand the world and themselves with numbers and im-ages since the antiquity. Verbal language cannot embrace the whole scope of abstract world.Mathematics have been considered to represent different dimension of the world and mental-ity beyond language. The Pythagoreans believed that the principles of mathematics were theprinciples of all things [1]. Rene Descartes claimed that scientific knowledge should be builtup with the fundamentals of mathematics and geometry [1]. Scientism tries to make believethat the evidence of real existence should be tested and validated only by numerical changesor visual and auditory proofs. Abstract or spiritual mentalfunctions such as thinking, affect,and belief, which cannot be numerically positioned or physically tested, have been consideredas something not-scientific and non-existent. Meanwhile, Nobel Laureate and atheist Peter

Received by the editors April 12 2017; Accepted May 29 2017; Published online June 14 2017.2000Mathematics Subject Classification.00A71, 62P15.Key words and phrases.Recursive Heaviside step function, Consciousness, Unconsciousness.† Corresponding author.

75

76 N. LEE, E.Y. KIM, AND C. SHIN

Mcdawar said; That there is indeed a limit upon science is made very likely by the existenceof questions that science cannot answer and that no conceivable advances of science wouldempower it to answer [2]. The classical physics and mathematics did not have any clue torelate brain mental function to numerical formula because mental function does not have anyspatial dimension, but only having time dimension. Modern physicists solved this puzzle byconfessing that the consciousness (and unconsciousness) of the person acts in the world but isnot spatially extent [3]. While Freud and Jung empirically and intuitively inferred the existenceof unconsciousness, contemporary modern physics made possible clarify the complex systemof consciousness and unconsciousness after the collaboration of Wolfgang Pauli and Jung [4].Despite the development of various human brain research, westill do not have a suitable set ofmathematical languages to understand the consciousness [5].

We begin by introducing the recursive Heaviside step functions that the human psyche hasno space to locate but exists only with time. The process of being conscious can be explainedas constant process of differentiation of recursive Heaviside step function representing uncon-scious of human. Following Shin and Cha’s approach of which our unconsciousness to a spe-cial solution of the multi-dimensional advection-like partial differential equation, we show thathumankind is totally dependent upon unconscious that is represented by the infinite recursiveHeaviside step function.

2. THE ADVECTION-LIKE EQUATION AND RECURSIVE HEAVISIDE STEP FUNCTION

The advection equation∂U

∂t= −c

∂U

∂x(2.1)

governs the motion of an object that moves in one+x direction by velocity c, and the waveequation

∂2U

∂t2= c2

∂2U

∂x2(2.2)

represents the motion in both the+x and−x directions [6]. Because the human mental stateproceeds only in a forward direction in time and should not depend on any particular spacecoordinates, we impose the advection-like equation

∂Un

∂t= −

n∑

k=1

∂Un

∂τk, (n = 1, 2, · · · ) (2.3)

to be satisfied byUn which denotes then-th particular situation pertaining to an individualperson. In (2.3),τk’s (k = 1, 2, · · · , n) are virtual time variables that will be elaborated.

Previously, Shin and Cha proved by induction that the recursive Heaviside step functionUn(t, Tn) is a special solution of the advection-like equation [7], under the condition that thereal timet is not larger than any one of the virtual time variablesτks. The recursive Heavisidestep functionUn(t, Tn) with depthn is defined in terms of the usual Heaviside step functionH(−t+ t0) by [8]

Un(t, Tn) = H [−t+ τnUn−1(t, Tn−1)] (2.4)

MATHEMATICAL UNDERSTANDING OF CONSCIOUSNESS AND UNCONCIOUSNESS 77

FIGURE 1. Heaviside step functionH(−t+ τ).

where

U0(t, T0) = 1, H(−t+ t0) =

1 when t ≤ t00 when t > t0

whereTn in (2.4) is a set ofn virtual time variablesTn = (τn, τn−1, · · · , τ1) andT0 denotesthe null.

Fig. 1. shows the shape of the Heaviside step functionH(−t + τ). The shape of therecursive Heaviside step functionUn(t, Tn) is also same as the Heaviside step function shownabove. The vertical axis means the value of theUn which represents unconsciousness and thehorizontal axis represent the time,τn represent the present whileτn−1, τn−2, · · · denote thepast, andτn+1, τn+2, · · · denote the future.

3. UNCONSCIOUS-TO-CONSCIOUSPROCESS

We postulate that the recursive Heaviside step functionUn(t, Tn) represents individual un-consciousness about a situation from which one aspect of a person’s mental state can be formed.This postulate is conceived becauseUn(t, Tn) has an interesting property in which it gives aunique shape that is equal to that of the usual step functionH(−t+τ0), whereτ0 is the smallestamong the virtual time variablesτk s, regardless of either the depthn or the values of the otherτk s.

We next demand that the partial derivative of the recursive Heaviside step functionUn(t, Tn)with respect to the real timet should be interpreted as consciousness of a person who experi-ences the situation represented byUn(t, Tn). Therefore, it can be said that the human brain inaction is operating under a constant process of differentiation.

Figure 2 shows the typical example of recursive Heaviside step functions which representsthe unconsciousness function of three persons. Jane, Paul and Bill are walking down the streettogether when a lost dog appeared in front of them. Jane showsno interest as she has had

78 N. LEE, E.Y. KIM, AND C. SHIN

FIGURE 2. Unconsciousness function for Jane (U1), Paul (U2) and Bill (U3) withrespect to same situation and consciousness for Jane, Paul and Bill represented by∂U1/∂t, ∂U2/∂t, and∂U3/∂t, respectively.

no meaningful experience with a dog in the past. Paul, on the other hand, goes up to the dogand pats it on the head as he has spent a lot of time with dogs since childhood. However, Billwants to stay away from the dog as far as he can, because he was once bitten by a dog whenhe was a child. Three people’s conscious reaction to a lost dog may be different according totheir unconsciousness structure cumulated from past experiences. Their unconsciousness ofthe situation is comparable to functionsUn; U1 for Jane,U2 for Paul, andU3 for Bill. Threedifferent reactions (=consciousness) can be expressed as time derivatives (∂Un(t, Tn)/∂t) ofhuman unconsciousness using the recursive Heaviside step function.

4. CONCLUSIONS

This article proves that the process of transformation of unconsciousness into consciousnesscan be expressed using the infinite recursive Heaviside stepfunction and differentiation of re-cursive Heaviside step function with respect to current time. Differentiation of unconsciousnessfunctionUn(t, Tn) with respect to time is the process of being conscious. The human reactionsdepend on past times. For every moment we live and experience, the unconscious constituentsare accumulated. Then the human mind has heavy and innate overlapping of unconsciousnessat timesτk. For instance, if humans become unconscious ten trillion different times, there areten trillion squared different unconscious states including instances where various states co-exist. Consciousness derives from comparison of unconsciousness from the two consecutivemoments.

MATHEMATICAL UNDERSTANDING OF CONSCIOUSNESS AND UNCONCIOUSNESS 79

Lives are sum of the integral and differential of every moments and so is the consciousness,which we can never summon into our mind again, even though we have realistically experi-enced. Single seconds and millimeters we experience can be divided forever, which the inven-tion of scanning tunneling microscope shows. We need to redefine unconsciousness under theumbrella of modern mathematical theorem. Efforts to postulate the existence of unconscious-ness through mathematical formulas in the form of recursiveHeaviside step function may helpAI to imitate, assist, and potentiate human mental activities. We ardently wait further researchin the converging field of mathematics, physics, and psychiatry. Development of AI requiresalgorithmic regeneration of human mental activities on machines. Without numerical analysisof the unconscious potential energy, we can never build up any bridge between the limitlesspsychological and mechanical worlds.

ACKNOWLEDGMENTS

The authors would like to thank Professor Cha for his valuable suggestions and comments.

REFERENCES

[1] P. Davies,God and the New Physics, Simon and Schuster, New York, 1984.[2] P. Medawar,The Limits of Science, Harper and Row, New York, 1984.[3] S.A. Dadundshvili,A Coordinate System for Representation of the Information Phenomena, Gerogian Engi-

neering News (2005), 36–47.[4] C.G. Jung and T. Pauli,The Interpretation of Nature and the Psyche, Youam Seoga, Seoul, 2015.[5] M.C. Kefatos,Fundamental Mathematics of Consciousness Cosmos and History, The Journal of Natural and

Social Philosophy,11 (2015), 175–188.[6] G.F. Duff and D. Naylor,Differential Equations of Applied Mathematics, John Wiley and Sons, Hoboben,

1966.[7] C. Shin and D. Cha,An Attempt towards Mathematical Description of the Human Mental State, Advancements

and Developments in Applied Mathematics,5 (2015), 1–8.[8] E.J. Berg,Heaviside’s Operational Calculus, 2nd ed. Mc Graw-Hill, New York, 1936.



MINKYU KWAK 1 AND BATAA LKHAGVASUREN 2†

1DEPARTMENT OFMATHEMATICS, CHONNAM UNIVERSITY, KOREA


2DEPARTMENT OFMATHEMATICS, CHOSUN UNIVERSITY, KOREA


ABSTRACT. In this note, we show that the cone property is satisfied for aclass of dissipativeequations of the formut = ∆u+ f(x, u,∇u) in a domainΩ ⊂ R

2 under the so called exact-ness condition for the nonlinear term. From this, we see thatthe global attractor is representedas a Lipshitz graph over a finite dimensional eigenspace.

1. INTRODUCTION

An inertial manifold is a positively invariant finite dimensional Lipshitz manifold whichattracts all solutions with an exponential rate. Thus, it contains a global attractor and theexistence of an inertial manifold can explain the long time behavior of the solutions of evolu-tionary equations. Moreover, it allows for the reduction ofthe dynamics to a finite dimensionalordinary differential equation, which is called an inertial form. Inertial manifolds have beenconstructed for a wide class of partial differential equations. We refer to [2] and [5] for a moredetailed exposition of this theory. However, the theory stands incomplete since there are im-portant equations, including the Navier-Stokes equation,for which the inertial manifolds arenot known to exist. The main reason for this is the failure of the spectral gap condition for theeigenvalues of the leading partial differential operator.

In this note, we give a new observation that leads to the existence of an inertial manifold fora class of equations of the form

ut = ∆u+ f(x, u,∇u), x ∈ Ω ⊂ R2 (1.1)

with the Dirichlet boundary condition

u|∂Ω = 0.

Here,u = u(t, x) is a scalar function andΩ is a rectangular domain, for which the spectralgap condition is satisfied. Until now, it is known that the gapcondition holds only for special

Received by the editors March 29 2017; Accepted May 29 2017; Published online June 18 2017.2000Mathematics Subject Classification.35B40, 35B42, 35K55.Key words and phrases.Inertial manifolds, Global attractor, Cone property, Parabolic equations.† Corresponding author.

81

82 FIRST AND SECOND

bounded domains inR2 (see [5]). Furthermore, we impose a so called exactness condition onthe nonlinear term:

f(x1, x2, u, p1, p2)p1x2= f(x1, x2, u, p1, p2)p2x1

. (1.2)

The condition (1.2) is restrictive, but it allowsf to be a combination of functionsg(x, u),|∇u|2 and∇φ·∇u for some differentiable functionsg andφ in each variable. In the subsequentsections, we give the basic notations and the main results.

2. NOTATION AND THE MAIN RESULT

Let the spaceH be a Hilbert space with an inner product< ·, · > and a norm‖ · ‖. Letλjbe the eigenvalues of the operatorA = −∆+ boundary condition such that

λ1 < λ2 ≤ . . . ≤ λN < λN+1 . . . ,

andφj be the corresponding eigenvectors . Denote byP the orthogonal projection operatorfrom H to a finite dimensional space spanned byφ1, φ2, . . . , φN. If Q = I − P , thenH isorthogonally decomposed asH = PH ⊕QH. LetA be a global attractor for the solutions ofthe equation (1.1). Then we are interested in the question whether the projection operatorPrestricted toA is injective, i.e.,

P : A → PH is injective? (2.1)

Equivalently, if u1(t) = p1(t) + q1(t) and u2(t) = p2(t) + q2(t) are two solutions withp1(t), p2(t) ∈ PH andq1(t), q2(t) ∈ QH, then the question (2.1) can be rephrased as

p1(0) = p2(0) ⇒ u1(t) = u2(t), for all t? (2.2)

To state the main result, we recall the cone property ([4]). Let

u1(t) = p1(t) + q1(t), u2(t) = p2(t) + q2(t),

ρ(t) = p1(t)− p2(t), σ(t) = q1(t)− q2(t).

The coneCk is defined as a subset ofH by

Ck = (ρ, σ) ∈ H : ‖σ‖ ≤ k‖ρ‖ (2.3)

for somek > 0. Then the cone property is stated as

(i) If u2(0) ∈ u1(0) + Ck, thenu2(t) ∈ u1(t) +Ck for all t > 0.(ii) If u2(0) /∈ u1(0)+Ck, then eitheru2(t0) ∈ u1(t0)+Ck for somet0 and remains there

for all t > t0 or u2(t) /∈ u1(t)+Ck and‖u1(t)−u2(t)‖ → 0 exponentially ast → ∞.

It is well-known that the cone property is satisfied for the case of global Lipschitz nonlinearityunder the gap condition (2.6) below. More precisely, we consider the equation

du

dt= −Au+ F (u), (2.4)

and assume the nonlinear term is global Lipschitz continuous with the constantK:

‖F (u)− F (v)‖ ≤ K‖u− v‖, for all u, v ∈ H. (2.5)

THE CONE PROPERTY FOR A CLASS OF PARABOLIC EQUATIONS 83

First, we prove the cone property in an equivalent but concise form.

Proposition 1. Let u1(t) andu2(t) be any two orbits in the global attractor of the equation(2.4). Under the assumption of the spectral gap condition

λN+1 − λN >(1 + k)2

kK, (2.6)

we have‖σ‖ ≤ k‖ρ‖ for all t ∈ R and, therefore, the projectionP is injective.

Remark1. The proof is standard, however we provide a new simpler proofhere.

Proof. Let u1(t) = p1(t) + q1(t) andu2(t) = p2(t) + q2(t). Then, forρ(t) = p1(t) − p2(t)andσ(t) = q1(t)− q2(t), we have

ρt = −Aρ+ PF (u1)− PF (u2),

σt = −Aσ +QF (u1)−QF (u2).(2.7)

The standard estimates forρ andσ give

1

2

d

dt‖ρ‖2 =< ρt, ρ >= −‖A1/2ρ‖2+ < PF (u1)− PF (u2), ρ >

≥ −λN‖ρ‖2 −K(‖ρ‖+ ‖σ‖)‖ρ‖,(2.8)

and1

2

d

dt‖σ‖2 =< σt, σ >= −‖A1/2σ‖2+ < QF (u1)−QF (u2), σ >

≤ −λN+1‖σ‖2 +K(‖ρ‖+ ‖σ‖)‖σ‖.

(2.9)

From (2.8) and (2.9), it follows that

1

2

d

dt(‖σ‖2 − k2‖ρ‖2) ≤ −λN+1‖σ‖

2 + λNk2‖ρ‖2 +K(‖ρ‖+ ‖σ‖)‖σ‖

+k2K(‖ρ‖+ ‖σ‖)‖ρ‖.(2.10)

On the boundary of the cone‖σ‖ = k‖ρ‖, we find that

1

2

d

dt(‖σ‖2 − k2‖ρ‖2) ≤ −(λN+1 − λN )k2‖ρ‖2 + k(1 + k)2K‖ρ‖2 < 0, (2.11)

under the condition

λN+1 − λN >(1 + k)2

kK. (2.12)

This implies, onceu1(t) − u2(t) is in Ck, it will never leave it through the boundary‖σ‖ =k‖ρ‖ and stays in the cone for all time.

Furthermore, if two orbits sit outside of the cone for some timet0, then they must stay therefor all t ≤ t0. That is, if(‖σ‖2 − k2‖ρ‖2)(t0) > 0 for somet0 ∈ R, then

(‖σ‖2 − k2‖ρ‖2)(t) > 0

84 FIRST AND SECOND

for all t ≤ t0. From (2.9),1

2

d

dt‖σ‖2 ≤ −λN+1‖σ‖

2 + (1 +1

k)K‖σ‖2 = −aN‖σ‖2, (2.13)

with

aN = λN+1 −k + 1

kK > 0.

The inequality (2.13) gives

d

dt(e2aN t‖σ‖2) ≤ 0 (2.14)

or

e2aN t0‖σ(t0)‖2 ≤ e2aN t‖σ(t)‖2, for all t ≤ t0. (2.15)

Since any orbit in the global attractor is bounded for all time, taking t → −∞, we get‖σ(t0)‖

2 = 0, which is a contradiction. Thus, we conclude that

‖σ‖2 ≤ k2‖ρ‖2, for all t ∈ R. (2.16)

Now, we state the main results of this note.

Theorem 1. Letu1(t) andu2(t) satisfy the cone property:

‖σ‖ ≤ k‖ρ‖, for all t ∈ R (2.17)

with v = u1 − u2 = ρ+ σ. Let us consider a nonlinear change of variable given by

V (x, t) = v(x, t)e−γ(x,t). (2.18)

If γ = γ(x, t) is any bounded smooth function forx ∈ Ω andt ∈ R, thenV (x, t) also satisfiesthe cone property with a different constant.

Remark2. The change of variable (2.18) was first used for one dimensional dissipative equa-tions from a different point of view in [3].

Proof. Denoteρ = PV andσ = QV . Recallingv = V eγ , we obtain

‖ρ‖‖ρ‖ ≥< ρ, ρ >=< ρ,PV >=< ρ,Pve−γ >=< ρ, (ρ+ σ)e−γ >

=

∫

Ω

ρ2e−γdx+

∫

Ω

ρσe−γdx ≥ m‖ρ‖2 −M‖ρ‖‖σ‖,(2.19)

wherem = min e−|γ(x,t)| andM = max e|γ(x,t)|.Thus

‖ρ‖ ≤1

m‖ρ‖+

M

m‖σ‖ (2.20)

and then from (2.17), we have

‖σ‖ ≤ k‖ρ‖ ≤k

m‖ρ‖+

kM

m‖σ‖. (2.21)


Without loss of generality, we may assumekMm

≤ 12

and then (2.21) becomes

1

2‖σ‖ ≤

k

m‖ρ‖. (2.22)

Similarly, we have

‖σ‖‖σ‖ ≥< σ, σ >=< QV eγ , σ >=< σ, (ρ+ σ)eγ >

≥ m‖σ‖2 −M‖ρ‖‖σ‖(2.23)

and

‖σ‖ ≥ m‖σ‖ −M‖ρ‖. (2.24)

Finally, combining (2.22) and (2.24), it yields

m‖σ‖ −M‖ρ‖ ≤ ‖σ‖ ≤2k

m‖ρ‖, (2.25)

thus, we get

‖σ‖ ≤mM + 2k

m2‖ρ‖, (2.26)

which completes the proof.

As an application of the Theorem 1, let us consider the equation (1.1) with the condition(1.2). Letv = u1 − u2. Then, from (1.1), we can write

vt = ∆v + α1(x, t)vx1+ α2(x, t)vx2

+ β(x, t)v, (2.27)

with

αi(x, t) =

∫

1

0

∂f

∂pi(x, u2 + τ(u1 − u2),∇u2 + τ(∇u1 −∇u2))dτ, (2.28)

for i = 1, 2, and

β(x, t) =

∫ 1

0

∂f

∂z(x, u2 + τ(u1 − u2),∇u2 + τ(∇u1 −∇u2))dτ. (2.29)

Now the nonlinear change of variableV (x, t) = v(x, t)e−γ(x,t) in (2.28) yields

Vt = ∆V +

2∑

i=1

(2γxi+ αi)Vxi

+ (∆γ + |∇γ|2 +

2∑

i=1

αiγxi+ β − γt)V. (2.30)

We see that the exactness condition

f(x1, x2, u, p1, p2)p1x2= f(x1, x2, u, p1, p2)p2x1

implies that the system

2γx1+ α1 = 0

2γx2+ α2 = 0

(2.31)

86 FIRST AND SECOND

can be solved forγ. Then the new equation becomes

Vt = ∆V + η(x, t)V, (2.32)

where

η(x, t) = ∆γ + |∇γ|2 − 2

2∑

i=1

γ2xi+ β − γt

andη(x, t) and its derivatives are bounded functions. Now we apply the Proposition 1 to theequation (2.32) and obtain the cone property under the spectral gap condition (2.6). Finally,the Theorem 1 gives the cone property for the equation (1.1).This implies the global attractoris indeed Lipschitz manifold with a finite dimension.

Since the cone condition (2.26) is satisfied, we can construct an N -dimensional inertialmanifold for (2.32) considering the negatively bounded solutions in [1].

Moreover, noting that

Vxi= vxi

e−γ − ve−γγxi= vxi

e−γ − γxiV (2.33)

(2.30) is rewritten as

Vt = ∆V +

2∑

i=1

(2γxi+ αi)vxi

e−γ + (∆γ + |∇γ|2 −

2∑

i=1

γ2xi+ β − γt)V. (2.34)

If we can solve the following linear first order partial differential equation forz = γ(x, y, t)on a rectangular domain:

2vx1γx1

+ 2vx2γx2

= −vx1α1 − vx2

α2, (2.35)

then we obtain a similar equation in the form (2.32).The coefficients in (2.35) are smooth in every variablesx1, x2 andt. Thus, we may write

(2.35) in the form

a(x, y, t)ux + b(x, y, t)uy = c(x, y, t). (2.36)

We look for a bounded smooth solutionz = u(x, y, t) at leastC2 in a space variable andC1 ina time variable. The standard method is solving the characteristic system (a gradient flow):

dx

ds= a(x(s), y(s), t(s)),

dy

ds= b(x(s), y(s), t(s)),

dt

ds= 0

(2.37)

and integrating along the characteristic:

du(s)

ds= c(x(s), y(s), t(s)). (2.38)


However, the geometry of the flow is unclear at this moment andit will be investigated in futureworks.

ACKNOWLEDGMENTS

The authors would like to extend their sincere gratitude forthe referees kind suggestions andcomments. This work was supported by Samsung, SSTF-BA1502-02.

REFERENCES

[1] Z.B. Fang and M. Kwak,Negatively bounded solutions for a parabolic partial differential equation, Bulletin ofthe Korean Mathematical Society,42 (2005), 829–836.

[2] C. Foias, G.R. Sell, and R. Temam,Inertial manifolds for nonlinear evolutionary equations, Journal of Differ-ential Equations,73 (1988), 309–353.

[3] I. Kukavica, Fourier parameterization of attractors for dissipative equations in one space dimension, Journalof Dynamics and Differential Equations,15 (2003), 473–484.

[4] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics,Applied Mathematical Sciences68 (1988) Springer-Verlag, New York.

[5] S. Zelik. Inertial manifolds and finite-dimensional reduction for dissipative PDEs, Proceedings of the RoyalSociety of Edinburgh: Section A Mathematics,144 (2014), 1245–1327.


NUMERICAL SOLUTIONS OF AN UNSTEADY 2-D INCOMPRESSIBLE FLOWWITH HEAT AND MASS TRANSFER AT LOW, MODERATE, AND HIGH

REYNOLDS NUMBERS

V. AMBETHKAR † AND D. KUSHAWAHA

DEPARTMENT OFMATHEMATICS, FACULTY OF MATHEMATICAL SCIENCES, UNIVERSITY OF DELHI , IN-DIA

E-mail address: [email protected] ([email protected]),[email protected]

ABSTRACT. In this paper, we have proposed a modified Marker-And-Cell (MAC) method toinvestigate the problem of an unsteady 2-D incompressible flow with heat and mass transferat low, moderate, and high Reynolds numbers with no-slip andslip boundary conditions. Wehave used this method to solve the governing equations alongwith the boundary conditionsand thereby to compute the flow variables, viz.u-velocity, v-velocity,P , T , andC. We haveused the staggered grid approach of this method to discretize the governing equations of theproblem. A modified MAC algorithm was proposed and used to compute the numerical solu-tions of the flow variables for Reynolds numbersRe = 10, 500, and50000 in consonance withlow, moderate, and high Reynolds numbers. We have also used appropriate Prandtl (Pr) andSchmidt (Sc) numbers in consistence with relevancy of the physical problem considered. Wehave executed this modified MAC algorithm with the aid of a computer program developed andrun in C compiler. We have also computed numerical solutionsof local Nusselt (Nu) and Sher-wood (Sh) numbers along the horizontal line through the geometric center at low, moderate,and high Reynolds numbers for fixedPr = 6.62 andSc = 340 for two grid systems at timet = 0.0001s. Our numerical solutions foru andv velocities along the vertical and horizontalline through the geometric center of the square cavity forRe = 100 has been compared withbenchmark solutions available in the literature and it has been found that they are in good agree-ment. The present numerical results indicate that, as we move along the horizontal line throughthe geometric center of the domain, we observed that, the heat and mass transfer decreases upto the geometric center. It, then, increases symmetrically.

1. INTRODUCTION

The problem of 2-D unsteady incompressible viscous fluid flowwith heat and mass trans-fer has been the subject of intensive numerical computations in recent years. This is due to

Received by the editors March 10 2017; Revised June 2 2017; Accepted in revised form June 12 2017; Publishedonline June 21 2017.

2000Mathematics Subject Classification.35Q307, 6D05, 76M20, 80A20.Key words and phrases.Heat transfer, Mass transfer, Velocity, Marker-And-Cell (MAC) method, Reynolds

number, Nusselt number, Sherwood number.† Corresponding author.

89

90 V. AMBETHKAR AND D. KUSHAWAHA

its significant applications in many scientific and engineering practices. Fluid flows play animportant role in various equipment and processes. Unsteady 2-D incompressible viscous flowcoupled with heat and mass transfer is a complex problem of great practical significance.

NOMENCLATURE

∆t time spacing n pertains to then iteration∆x grid spacing alongx-axis P dimensionless pressure∆y grid spacing alongy-axis p pressure (N/m2)∂

∂ndifferentiation along the Q volumetric flow rate (m3/s)

normal to the boundary Ra Rayleigh numberu, v pseudo-velocity components inx, T dimensionless temperature

y coordinate directions, respectively t non-dimensional timeu, v dimensionless velocity components inx, tn time level aftern iterations

y coordinate directions, respectively µ viscosity of the fluid (N ·m2/s)∇ · u divergence of pseudo-velocity vector ν kinematic viscosity (m2/s)∇ · ~u divergence of dimensionless velocity vector Nu average Nusselt numberun x component of the dimensionless Sh average Sherwood number

velocity aftern iterations ρ fluid density (kg/m3)vn y component of the dimensionless x, y coordinates

velocity aftern iterations ~u dimensionless velocity vectorNu local Nusselt number C dimensionless concentrationPr Prandtl number,ν/k D mass diffusivity (m2/s)Re Reynolds number, uL/µ Sc Schmidt number,ν/Di index used in tensor notation j index used in tensor notationSh local Sherwood number k thermal diffusivity (m2/s)

This problem has received considerable attention due to itsnumerous engineering prac-tices in various disciplines, such as storage of radioactive nuclear waste materials, transfergroundwater pollution, oil recovery processes, food processing, and the dispersion of chem-ical contaminants in various processes in the chemical industry. More often, fluid flow withheat and mass transfer are coupled in nature. Heat transfer is concerned with the physicalprocess underlying the transport of thermal energy due to a temperature difference or gradi-ent. All the process equipment used in engineering practicehas to pass through an unsteadystate in the beginning when the process is started, and, theyreach a steady state after sufficienttime has elapsed. Typical examples of unsteady heat transfer occur in heat exchangers, boilertubes, cooling of cylinder heads in I.C. engines, heat treatment of engineering components andquenching of ingots, heating of electric irons, heating andcooling of buildings, freezing offoods, etc. Mass transfer is an important topic with vast industrial applications in mechani-cal, chemical and aerospace engineering. Few of the applications involving mass transfer are

NUMERICAL SOLUTIONS OF AN UNSTEADY 2-D INCOMPRESSIBLE FLOW 91

absorption and desorption, solvent extraction, evaporation of petrol in internal combustion en-gines etc. Numerous everyday applications such as dissolving of sugar in tea, drying of woodor clothes, evaporation of water vapor into the dry air, diffusion of smoke from a chimney intothe atmosphere, etc. are also examples of mass diffusion. Inmany cases, it is interesting tonote that heat and mass transfer occur simultaneously.

Harlow and Welch [1] used the Marker-And-cell (MAC) method for numerical calculationof time-dependent viscous incompressible flow of fluid with the free surface. This methodemploys the primitive variables of pressure and velocity that has practical application to themodeling of fluid flows with free surfaces. Ghia et al. [3] haveused the vorticity-stream func-tion formulation for the two-dimensional incompressible Naiver-Stokes equations to study theeffectiveness of the coupled strongly implicit multi-grid(CSI-MG) method in the determina-tion of high-Re fine-mesh flow solutions. Issa et al. [5] have used a non-iterative implicitscheme of finite volume method to study compressible and incompressible recirculating flows.Elbashbeshy [6] has investigated the unsteady mass transfer from a wedge. Sattar [7] has stud-ied free convection and mass transfer flow through a porous medium past an infinite verticalporous plate with time-dependent temperature and concentration. Sattar and Alam [8] haveinvestigated the MHD free convective heat and mass transferflow with hall current and con-stant heat flux through a porous medium. Maksym Grzywinski, Andrzej Sluzalec [10] solvedthe stochastic convective heat transfer equations in finitedifferences method. A numericalprocedure based on the stochastic finite differences methodwas developed for the analysis ofgeneral problems in free/forced convection heat transfer.Lee [11] has studied fully developedlaminar natural convection heat and mass transfer in a partially heated vertical pipe. Chiriacand Ortega [12] have numerically studied the unsteady flow and heat transfer in a transitionalconfined slot jet impinging on an isothermal plate. De and Dalai [14] have numerically studiednatural convection around a tilted heated square cylinder kept in an enclosure has been studiedin the range of1000 ≤ Ra ≤ 1000000. Detailed flow and heat transfer features for two dif-ferent thermal boundary conditions are reported. Chiu et al. [15] proposed an effective explicitpressure gradient scheme implemented in the two-level non-staggered grids for incompressibleNavier-Stokes equations. Xu et al. [16] have investigated the transition to a periodic flow in-duced by a thin fin on the side wall of a differentially heated cavity. Lambert et al. [17] studiedthe heat transfer enhancement in oscillatory flows of Newtonian and viscoelastic fluids. Al-harbi et al. [18] presented the study of convective heat and mass transfer characteristics of anincompressible MHD visco-elastic fluid flow immersed in a porous medium over a stretchingsheet with chemical reaction and thermal stratification effects. Xu et al. [20] have investigatedthe unsteady flow with heat transfer adjacent to the finned sidewall of a differentially heatedcavity with conducting adiabatic fin. Fang et al. [21] have investigated the steady momentumand heat transfer of a viscous fluid flow over a stretching/shrinking sheet. Salman et al. [22]have investigated heat transfer enhancement of nano fluids flow in micro constant heat flux.Hasanuzzaman et al. [23] investigated the effects of Lewis number on heat and mass trans-fer in a triangular cavity. Schladow [24] has investigated oscillatory motion in a side-heatedcavity. Lei and Patterson [25] have investigated unsteady natural convection in a triangular en-closure induced by absorption of radiation. Ren and Wan [26]have proposed a new approach


to the analysis of heat and mass transfer characteristics for laminar air flow inside vertical platechannels with falling water film evaporation.

The above mentioned literature survey pertinent to the present problem under considerationrevealed that, to obtain high accurate numerical solutionsof the flow variables, we need todepend on high accurate and high resolution method like the modified Marker-And-Cell (MAC)method, being proposed in this work. Furthermore, a modifiedMAC algorithm is employedfor computing unknown variablesu, v, P , T , andC simultaneously.

What motivated us is the enormous scope of applications of unsteady incompressible flowwith heat and mass transfer as discussed earlier. Literature survey also revealed that, the prob-lem of 2-D unsteady incompressible flow with, heat and mass transfer in a rectangular domain,along with slip wall, temperature, and concentration boundary conditions has not been studiednumerically. Furthermore, it has also been observed that, there is no literature to concludeavailability of high accurate method that solves the governing equations of the present problemsubject to the initial and boundary conditions. Moreover, in order to investigate the importanceof the applications enumerated upon earlier, there is a needto determine numerical solutionsof the unknown flow variables. In order to fulfil this requirement, we present numerical aninvestigation of the problem of unsteady 2-D incompressible flow, with heat and mass transferin a rectangular domain, along with slip wall, temperature,and concentration boundary con-ditions, using the modified Marker-And-Cell (MAC) method. Though there is a well knownMAC method for solving the problem of 2-D fluid flow, we presenta suitably modified MACmethod as well as a modified MAC algorithm to compute the accurate numerical solutions ofthe flow variables to the problem considered in this work.

Our main target of this work is to propose and use the modified Marker-And-Cell (MAC)method of pressure correction approach to investigate the problem of unsteady 2-D incom-pressible flow with heat and mass transfer. We have proposed and used this method to solvethe governing equations along with no-slip and slip wall boundary conditions and thereby tocompute the flow variables. We have used a modified MAC algorithm for discretizing the gov-erning equations in order to compute the numerical solutions of the flow variables at differentReynolds numbers in consonance with low, moderate, and high. We have executed this modi-fied MAC algorithm with the aid of a computer program developed and run in C compiler. Wehave also computed numerical solutions of local Nusselt(Nu) and Sherwood(Sh) numbersalong the horizontal line through the geometric center at low, moderate, and highRe, for fixedPr = 6.62 andSc = 340 for two grid systems at timet = 0.0001s.

The summary of the layout of the current work is as follows: Section 2 describes mathemati-cal formulation that includes physical description of the problem, governing equations, and ini-tial and boundary conditions. Section 3 describes the modified Marker-And-Cell method, alongwith the discretization of the governing equations. Section 4 describes a modified Marker-And-Cell (MAC) algorithm, along with numerical computations. Section 5 discusses the numericalresults. Section 6 illustrates the conclusions of this study. Section 7 provides validity of ourcomputer code used to obtain numerical solutions with the benchmark solutions.


2. MATHEMATICAL FORMULATION

2.1. Physical Description. Fig. 1 illustrates the geometry of the problem considered inthisstudy along with no-slip and slip boundary conditions. ABCDis a rectangular domain aroundthe point(1.0, 0.5) in which an unsteady 2-D incompressible viscous flow with heat and masstransfer is considered. Flow is setup in a rectangular domain with three stationary walls and atop lid that moves to the right with constant speed(u = 1).

FIGURE 1. Rectangular cavity

We have assumed that, at all four corner points of the computational domain, velocity com-ponents(u, v) vanish. It may be noted here regarding specifying the boundary conditions forpressure, the convention followed is that either the pressure at boundary is given or velocitycomponents normal to the boundary is specified [2, pp.129].

2.2. Governing equations. The governing equations of 2-D unsteady incompressible flowwith heat and mass transfer in a rectangular domain are the continuity equation, the two com-ponents of momentum equation, the energy equation, and the equation of mass transfer. Theseequations (2.1) to (2.5) subject to boundary conditions (2.6) and (2.7) are discretized using themodified Marker-And-Cell (MAC) method. Taking usual the Boussinesq approximations intoaccount, the dimensionless governing equations are expressed as follows:

Continuity equation∂u

∂x+

∂v

∂y= 0, (2.1)

x-momentum∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= −

∂P

∂x+

(

1

Re

)(

∂2u

∂x2+

∂2u

∂y2

)

, (2.2)

y-momentum∂v

∂t+ u

∂v

∂x+ v

∂v

∂y= −

∂P

∂y+

(

1

Re

)(

∂2v

∂x2+

∂2v

∂y2

)

, (2.3)

Energy equation∂T

∂t+ u

∂T

∂x+ v

∂T

∂y=

(

1

Pr

)(

∂2T

∂x2+

∂2T

∂y2

)

, (2.4)


Mass transfer equation∂C

∂t+ u

∂C

∂x+ v

∂C

∂y=

(

1

Sc

)(

∂2C

∂x2+

∂2C

∂y2

)

. (2.5)

whereu, v, P , T , C, Re, Pr, and,Sc are the velocity components inx andy- directions, thepressure, the temperature, the concentration, the Reynolds number, the Prandtl number, and theSchmidt number respectively.

The initial, no-slip and slip wall boundary conditions are given by:

for t = 0, u(x, y, 0) = 0, v(x, y, 0) = 0, T (x, y, 0) = 10, C(x, y, 0) = 10. (2.6)

for t > 0, on boundary AB:u = 0,∂v

∂x= 0,

∂T

∂x= 0,

∂C

∂x= 0,

on boundary BC:u = 0, v = 0, T = T0 +∆T , C = C0 +∆C,

on boundary CD:u = 0,∂v

∂x= 0,

∂T

∂x= 0,

∂C

∂x= 0,

on boundary AD:u = 1, v = 0, T = T0 −∆T , C = C0 −∆C.

(2.7)

3. NUMERICAL METHOD AND DISCRETIZATION

3.1. Modified Marker-And-Cell (MAC) Method. Our main purpose in this work is to pro-pose and use an accurate numerical method that solves the governing equations of the presentproblem subject to the initial and boundary conditions. In order to solve the equations (2.1)-(2.5) which are semi-linear coupled partial differential equations, we propose and use the mod-ified Marker-And-Cell (MAC) method based on the MAC method ofHarlow and Welch [1].The present modified MAC method (algorithm) is a generalization of the original MAC method(algorithm) in the sense that the original one enable us to discretized equations (2.1)-(2.3) andhence to compute the flow variablesu, v, andP . Where as, the modified MAC method (algo-rithm) enable us to discretized equations (2.1)-(2.5) and hence to compute the flow variablesu, v, P, T, andC. Consider a modified MAC staggered grid foru, v, and a scalar node (Pnode), where pressure, temperature, and concentration variables are stored as shown in Fig. 2.Thex-momentum equation is written atu nodes, and they-momentum equation is written atv nodes.The energy and the mass transfer equations are written at a scalar node. Accordingly,the various derivatives in thex-momentum equation (2.2) are calculated as follows:

(

∂u

∂t

)n+1

i+1/2,j

=(

un+1

i+1/2,j − uni+1/2,j

)/

∆t, (3.1)

(

∂2u

∂x2

)n+1

i+1/2,j

=(

un+1

i+3/2,j − 2un+1

i+1/2,j + un+1

i−1/2,j

)/

∆x2, (3.2)

(

∂2u

∂y2

)n+1

i+1/2,j

=(

un+1

i+1/2,j+1− 2un+1

i+1/2,j + un+1

i+1/2,j−1

)/

∆y2, (3.3)


(

∂P

∂x

)n+1

i+1/2,j

=(

Pn+1

i+1,j − Pn+1

i,j

)/

∆x. (3.4)

Using the modified MAC method, the other terms in thex-momentum equation (2.2) arewritten as

(

u∂u

∂x

)n

i+1/2,j

= uni+1/2,j

(

uni+3/2,j − uni+1/2,j

)/

∆x, (3.5)

(

v∂u

∂y

)n

i+1/2,j

= vni+1/2,j

(

uni+1/2,j+1− uni+1/2,j

)/

∆y. (3.6)

Herevni+1/2,j is given by

vni+1/2,j =(

vni,j+1/2 + vni,j−1/2 + vni+1,j+1/2 + vi+1,j−1/2

)/

4. (3.7)

FIGURE 2. Modified MAC staggered grid system

For simplicity, we will implement a fully explicit version of the time-splitting (fractionaltime-step) method, for both the viscous and the diffusion terms. When this method is appliedto thex-momentum equation on the staggered grid from time leveltn to t yields the followingequation at the intermediate step:

ui+1/2,j − uni+1/2,j

∆t= uni+1/2,j

uni+1/2,j − uni+3/2,j

∆x+ vni+1/2,j

(

uni+1/2,j − uni+1/2,j+1

)

∆y

+uni+3/2,j − 2uni+1/2,j + uni−1/2,j

Re∆x2+

uni+1/2,j+1− 2uni+1/2,j + uni+1/2,j−1

Re∆y2.

(3.8)


Similarly for they-momentum equation we obtain the following equation:

vi,j+1/2 − vni,j+1/2

∆t= uni,j+1/2

vni−1,j+1/2 − vni,j+1/2

∆x+ vni,j+1/2

vni,j+1/2 − vni,j+3/2

∆y

+vni+1,j+1/2 − 2vni,j+1/2 + vni−1,j+1/2

Re∆x2+

vni,j+3/2 − 2vni,j+1/2 + vni,j−1/2

Re∆y2.

(3.9)

Practical stability requirement obtained from the Von Neumann analysis for the Euler explicitsolvers are given by Peyret and Taylor [4, 148], given in equations (3.10) and (3.11). It maybe noted here that these relations establish stability requirement of the discretized equationsassociated with the original MAC method. Where as the present physical problem containsadditional flow variables associated with heat and mass transfer equations ((2.4) and (2.5)).For which the discretized equations are in (3.19) and (3.20). However there is a need to definestability requirement which are proposed by us in equations(3.12) and (3.13). In fact, thesetwo relations containPr andSc, associated in heat and mass transfer equations.

(

|u|+ |v|)2∆tRe ≤ 4, (3.10)

∆t

Re

[

1

∆x2+

1

∆y2

]

≤ 5, (3.11)

∆tmaxi,j

[(

ui,j

∆x+

vi,j

∆y

)

1

Re+

2

Pr

(

1

∆x2+

1

∆y2

)]

≤ 1, (3.12)

∆tmaxi,j

[(

ui,j

∆x+

vi,j

∆y

)

1

Re+

2

Sc

(

1

∆x2+

1

∆y2

)]

≤ 1. (3.13)

Now advancing fromtn to t and thent to tn+1 one obtains the elliptical pressure equation:

∇ · u

∆t= ∇2Pn+1. (3.14)

The corresponding homogeneous Neumann boundary conditionfor pressure is given by:

∂pn+1

∂n= 0. (3.15)

Now using central difference scheme:

Pn+1

i+1,j − 2Pn+1

i,j + Pn+1

i−1,j

∆x2+

Pn+1

i,j+1− 2Pn+1

i,j + Pn+1

i,j−1

∆y2

=1

∆t

[

ui+1/2,j − ui−1/2,j

∆x+

vi,j+1/2 − vi,j−1/2

∆y

]

.

(3.16)

We obtain the velocity field at the advanced time level(n + 1), for each velocity component,this equation gives

un+1

i+1/2,j = ui+1/2,j −∆t

∆x

(

Pn+1

i+1,j − Pn+1

i,j

)

, (3.17)


vn+1

i,j+1/2 = vi,j+1/2 −∆t

∆y

(

Pn+1

i,j+1− Pn+1

i,j

)

. (3.18)

The modified MAC staggered grid quotients of different derivatives which appeared in theenergy equation is written as follows:(

∂T

∂t

)n+1

i,j

=(

T n+1

i,j − T ni,j

)

/∆t,

(

∂T

∂x

)n

i+1,j

=(

T ni+1,j − T n

i,j

)

/∆x,

(

∂T

∂y

)n

i,j+1

=(

T ni,j+1 − T n

i,j

)

/∆y,

(

∂2T

∂x2

)n

i+1,j

=(

T ni+1,j − 2T n

i,j + T ni−1,j

)

/∆x2,

(

∂2T

∂y2

)n

i,j+1

=(

T ni,j+1 − 2T n

i,j + T ni,j−1

)

/∆y2.

The discretized form of the energy equation (2.4) is given by:

T n+1

i,j − T ni,j

∆t= uni,j

T ni,j − T n

i+1,j

∆x+ vni,j

T ni,j − T n

i,j+1

∆y

+T ni+1,j − 2T n

i,j + T ni−1,j

Pr∆x2+

T ni,j+1 − 2T n

i,j + T ni,j−1

Pr∆y2.

(3.19)

The modified MAC staggered grid quotients of different derivatives which appeared in the masstransfer equation is written as follows:(

∂C

∂t

)n+1

i,j

=(

Cn+1

i,j − Cni,j

)

/∆t,

(

∂C

∂x

)n

i+1,j

=(

Cni+1,j − Cn

i,j

)

/∆x,

(

∂C

∂y

)n

i,j+1

=(

Cni,j+1 − Cn

i,j

)

/∆y,

(

∂2C

∂x2

)n

i+1,j

=(

Cni+1,j − 2Cn

i,j + Cni−1,j

)

/∆x2,

(

∂2C

∂y2

)n

i,j+1

=(

Cni,j+1 − 2Cn

i,j + Cni,j−1

)

/∆y2.

The discretized form of the mass transfer equation (2.5) is given by:

Cn+1

i,j − Cni,j

∆t= uni,j

Cni,j − Cn

i+1,j

∆x+ vni,j

Cni,j − Cn

i,j+1

∆y

+Cni+1,j − 2Cn

i,j + Cni−1,j

Sc∆x2+

Cni,j+1

− 2Cni,j + Cn

i,j−1

Sc∆y2

(3.20)

We note thatuni,j is not defined onu node andvni,j is not defined onv node. Therefore in orderto obtain these quantity, we use averaging as given below:

uni,j =1

2

(

uni+1/2,j + uni−1/2,j

)

and vni,j =1

2

(

vni,j+1/2 + vni,j−1/2

)

. (3.21)


4. NUMERICAL COMPUTATIONS

We have used a modified MAC algorithm to the discretized governing equations in orderto compute the numerical solutions of the flow variables at different Reynolds numbers inconsonance with low, moderate, and high. This modified MAC algorithm has an advantageover the original MAC algorithm in discretizing and computing the solutions of any number ofgoverning equations and the flow variables augmented to the basic physical problem of a 2-Dsimple fluid flow. We have executed this modified MAC algorithmwith the aid of a computerprogram developed and run in C compiler. The input data for the relevant parameters in thegoverning equations like Reynolds number (Re), Prandtl Number (Pr), and Schmidt number(Sc) has been properly chosen incompatible with the present problem considered.

4.1. Modified MAC Algorithm. We summarize the sequence of computational steps involvedin the modified MAC algorithm as follow:

FIGURE 3. The rectangular staggered computational grid

4.1.1. Prediction Step.

• Using (3.8) and (3.9), calculateu andv at their respective grid point locations.• Apply the initial and boundary conditions given in equations (2.6) and (2.7) respec-

tively.• These equations will be solved algebraically because time advancement is fully ex-

plicit.• Linear stability conditions (3.10), (3.11), (3.12), and (3.13) must be obeyed.• Divergence of the velocity field must be calculated at every time step, using the velocity

field at the advanced time level,(n+ 1),

∇ · ~u =un+1

i+1/2,j − un+1

i−1/2,j

∆x+

vn+1

i,j+1/2 − vn+1

i,j−1/2

∆y.


The sum of the divergence magnitude at all grid points shouldbe satisfied to machine zeroat each time step. If this quantity increases, the calculation should be terminated and restartedwith a smaller time step.

4.1.2. Pressure Calculation.

• Calculate pressure from Pressure-Poisson equation (3.14).• Boundary condition applied to the pressure equation at all boundaries is the homoge-

neous Neumann boundary condition given by (3.15). This equation is solved at thepressure(ij). Note that the Euler explicit time-advancement calculatesthe actual ther-modynamic pressure (scaled by the constant density) and nota pseudo-pressure.

4.1.3. Velocity Correction.

• Obtain the velocity field at the advanced time level(n + 1), for each velocity compo-nent, this equation gives

un+1

i+1/2,j = ui+1/2,j −∆t

∆x

(

Pn+1

i+1,j − Pn+1

i,j

)

,

vn+1

i,j+1/2 = vi,j+1/2 −∆t

∆y

(

Pn+1

i,j+1− Pn+1

i,j

)

.

4.1.4. Temperature Calculation.

• Calculate the numerical solutions for temperature profilesby using equations (3.19)and (3.21).

4.1.5. Concentration Calculation.

• Calculate the numerical solutions for concentration profiles by using equations (3.20)and (3.21).

5. RESULTS AND DISCUSSION

We used the modified Marker-And-Cell (MAC) method to carry out the numerical compu-tations of the unknown flow variablesu, v, P , T , andC for the present problem. We haveexecuted the modified MAC algorithm mentioned above with theaid of a computer programdeveloped and run in C compiler. To verify our computer code,the numerical results obtainedby the present method were compared with the benchmark results reported in [3]. It is seenthat the results obtained in the present work are in good agreement with those reported in [3]at low Reynolds numberRe = 100. This indicates the validity of the numerical code that wedeveloped.

Based on the numerical solutions foru-velocity, Fig. 4 illustrates the variation ofu-velocityalong the vertical line through the geometric center of the rectangular domain at low, moderate,and high Reynolds numbersRe = 10, 500, and50000. We can see that, for a givenRe = 10andRe = 500, u-velocity first decreases from the bottom boundary of the rectangular domain.


It, then, increases to the upper boundary. But, forRe = 50000, u-velocity increases from thebottom boundary to the upper boundary of the rectangular domain. We also observe that, theabsolute value ofu-velocity decreases with increase in Reynolds number.

FIGURE 4. u-velocity along the vertical line through the geometric center ofthe domain at low, moderate, and highRe for a fixedPr = 6.62 andSc = 340for grid 32× 32 at timet = 0.0001s.

Based on the numerical solutions forv-velocity, Fig. 5 illustrates the variation ofv-velocityalong the horizontal line through the geometric center of the rectangular domain. It is clear thatfor a givenRe, v-velocity decreases from the left boundary to the right boundary. Further, theabsolute value ofv-velocity decreases with increase in Reynolds number.

FIGURE 5. v-velocity along the horizontal line through the geometric centerof the domain at low, moderate, and highRe for a fixedPr = 6.62 andSc = 340 for grid 32× 32 at timet = 0.0001s.

Based on the numerical solutions for pressure, Fig. 6 illustrates the variation of pressure inthe rectangular domain. We observed that, forRe = 10, the pressure is oscillatory in nature.


However, forRe = 500 andRe = 50000, we observed the pressure decrease from the leftboundary to the right boundary. Further, the absolute valueof pressure increases with increasein Reynolds number.

FIGURE 6. Pressure variation at low, moderate, and highRe for a fixedPr =6.62 andSc = 340 for grid 32× 32 at timet = 0.0001s.

Based on the numerical solutions for temperature, Fig. 7 illustrates the variation of tempera-ture at different Reynolds numbers (Re = 10, 500, and50000), along the vertical line throughthe geometric center of the rectangular domain. It is clear that for a givenRe, temperatureincreases from the bottom boundary to the upper boundary.

FIGURE 7. Temperature variation at low, moderate, and highRe for a fixedPr = 6.62 andSc = 340 for grid 32× 32 at timet = 0.0001s.

Based on the numerical solutions of concentration, Fig. 8 illustrates the variation of con-centration at different Reynolds numbers (Re = 10, 500, and50000), along the vertical linethrough the geometric center of the rectangular domain. It is clear that for a givenRe, concen-tration increases from the bottom boundary to the upper boundary.


FIGURE 8. Concentration variation at low, moderate, and highRe for a fixedPr = 6.62 andSc = 340 for grid 32× 32 at timet = 0.0001s.

Grid dependence tests have been conducted on two grid systems (16 × 16 and32 × 32).In order to evaluate the effect of the two grid systems on the natural convection flow in therectangular domain, three representative quantities are evaluated numerically: the volumetricflow rate(Q), average Nusselt number(Nu), and average Sherwood number(Sh) [26] acrossthe horizontal centerline of the rectangular domain, whichare defined as (also see [24, 25])

Q =1

2

∫

L2

−L2

|v|dx, (5.1)

Nu =

1

L

∫

L2

−L2

∣

∣

∣

∣

vT −∂T

∂y

∣

∣

∣

∣

dx

(

2∆T

H

) , (5.2)

Sh =

1

L

∫

L2

−L2

∣

∣

∣

∣

vC −∂C

∂y

∣

∣

∣

∣

dx

(

2∆C

H

) , (5.3)

where local Nusselt number(Nu), and local Sherwood number(Sh) is defined as follows:

Nu = vT −∂T

∂y, (5.4)

Sh = vC −∂C

∂y. (5.5)

In the present problem length,L of the rectangular domain varies from 0 to 2, height,H variesfrom 0 to 1,∆T = 0.5, and∆C = 0.5. Then the volumetric flow rate(Q), average Nusselt


number(Nu), and average Sherwood number(Sh) across the horizontal centerline of therectangular domain, take the form

Q =

∫

2

0

|v|dx, (5.6)

Nu =

∫

2

0

∣

∣

∣

∣

vT −∂T

∂y

∣

∣

∣

∣

dx, (5.7)

Sh =

∫

2

0

∣

∣

∣

∣

vC −∂C

∂y

∣

∣

∣

∣

dx. (5.8)

In order to investigate heat transfer from the bottom to the top wall of the rectangular do-main, we have computed the numerical solutions of local Nusselt number for two differentgrid systems along the horizontal line through the geometric center of the domain at differentReynolds numbers. Fig. 9 illustrates comparison of computed local Nusselt number (Nu) fromthe hot bottom wall to the cold top wall at timet = 0.0001s.

(a) (b)

FIGURE 9. Variation of local Nusselt number (Nu) profiles along the hori-zontal line through the geometric center at low, moderate, and highRe for afixedPr = 6.62 andSc = 340, (a) for grid16 × 16 and (b) for grid32 × 32,at timet = 0.0001s.

From this figure, we observe that, as we move along the horizontal line through the geomet-ric center of the domain, heat transfer decreases up to the geometric center. It, then, increasessymmetrically.

In order to investigate mass transfer from the bottom to the top wall of the rectangulardomain, we have computed the numerical solutions of local Sherwood number for two differentgrid systems along the horizontal line through the geometric center of the domain at differentReynolds numbers.

Fig. 10 illustrates comparison of computed local Sherwood number (Sh) from the hot bot-tom wall to the cold top wall at timet = 0.0001s. From this figure, we observe that, as we


(a) (b)

FIGURE 10. Variation of local Sherwood number (Sh) profiles along the hor-izontal line through the geometric center at low, moderate,and highRe for afixedPr = 6.62 andSc = 340, (a) for grid16 × 16 and (b) for grid32 × 32,at timet = 0.0001s.

move along the horizontal line through geometric center of the domain, mass transfer decreasesupto the geometric center. It, then, increases symmetrically.

6. CONCLUSIONS

In this study, we have proposed a modified Marker-And-Cell (MAC) method to investigatethe problem of an unsteady 2-D incompressible flow with heat and mass transfer at low, mod-erate, and high Reynolds numbers with no-slip and slip boundary conditions. We have usedthis method to solve the governing equations along with the boundary conditions and therebyto compute the flow variables, viz.u-velocity, v-velocity, P , T , andC. We have used thestaggered grid approach of this method to discretize the governing equations of the problem.A modified MAC algorithm was proposed and used to compute the numerical solutions ofthe flow variables for Reynolds numbersRe = 10, 500, and50000 in consonance with low,moderate, and high Reynolds numbers.

Numerical solutions foru-velocity illustrates the variation ofu-velocity along the verti-cal line through the geometric center of the rectangular domain at low, moderate, and highReynolds numbersRe = 10, 500, and50000. We have observed that, for a givenRe = 10andRe = 500, u-velocity first decreases from the bottom boundary of the rectangular domain.It, then, increases to the upper boundary. But, forRe = 50000, u-velocity increases from thebottom boundary to the upper boundary of the rectangular domain. We also observed that theabsolute value ofu-velocity decreases with increase in Reynolds number. The numerical solu-tions forv-velocity illustrates the variation ofv-velocity along the horizontal line through thegeometric center of the rectangular domain. We have observed that, for a givenRe, v-velocitydecreases from the left boundary to the right boundary. Further, we also observed that theabsolute value ofv-velocity decreases with increase in Reynolds number.


Numerical solutions for pressure illustrates the variation of pressure in the rectangular do-main. We have observed that, forRe = 10, the pressure is oscillatory in nature. However,for Re = 500 andRe = 50000, we observed the pressure decrease from the left boundaryto the right boundary. Further, we have also observed that,the absolute value of pressure in-creases with increase in Reynolds number. The numerical solutions for temperature illustratesthe variation of temperature at different Reynolds numbers(Re = 10, 500, and50000), alongthe vertical line through the geometric center of the rectangular domain. We have observedthat, for a givenRe, temperature increases from the bottom boundary to the upper boundary.The numerical solutions of concentration, illustrates thevariation of concentration at differentReynolds numbers (Re = 10, 500, and50000), along the vertical line through the geomet-ric center of the rectangular domain. We have observed that,for a givenRe, concentrationincreases from the bottom boundary to the upper boundary.

Based on the computed local Nusselt number (Nu) from the hot bottom wall to the cold topwall at timet = 0.0001s, we have observed that, as we move along the horizontal line throughthe geometric center of the domain, heat transfer decreasesup to the geometric center. It, then,increases symmetrically. Based on the computed local Sherwood number (Sh) from the hotbottom wall to the cold top wall at timet = 0.0001s. We have observed that, as we move alongthe horizontal line through geometric center of the domain,mass transfer decreases upto thegeometric center. It, then, increases symmetrically.

7. CODE VALIDATION

To check the validity of our present computer code used to obtain the numerical results ofu-velocity andv-velocity, we have compared our present results with those benchmark resultsare given by Ghia et al. [3] and it has been found that they are in good agreement.

FIGURE 11. Comparison of the numerical results ofu-velocity along the ver-tical line through the geometric center of the square cavityfor Re = 100.


FIGURE 12. Comparison of the numerical results ofv-velocity along the hor-izontal line through the geometric center of the square cavity for Re = 100.

ACKNOWLEDGMENTS

The first author acknowledge the support from research council, University of Delhi forproviding research and development grand 2015–16 vide letter No. RC/2015-9677 to carry outthis work.

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VOLUME 21 NUMBER 2 / June 2017


ContentsVOL. 21 No. 2 June 2017



DULTUYA TERBISH, ENKHBOLOR ADIYA, MYUNGJOO KANG ············ 63


UNCONCIOUSNESS

NAMI LEE, EUN YOUNG KIM, CHANGSOO SHIN ······························ 75


MINKYU KWAK, BATAA LKHAGVASUREN ····································· 81




V. AMBETHKAR, D. KUSHAWAHA ················································· 89

ISSN 1226-9433(print)ISSN 1229-0645(electronic)

Vol. 21 No. 2 June 2017 The K

orean Society for Industrial and Applied M

athematics K

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