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Heat Transfer with Pulsatiie Flow in a Tbbe
b~
Terry Moschandreou
Department of Appiied Mathematics
Submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Faculty of Graduate S tudies
The University of Western Ontario
London. Ontario
July 1996
@ Te- Moschandreou 1996
Acquisitions and Acquisitions et BiMiogmphic Senhœs services bibliographiques
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The partial differentid equations governing heat transfer with pulsatile flow in a
tube. which serves as a mode1 of a simple heat exchange device. are solved in this
t hesis.The governing equations axe the Navier Stokes equat ions and the convection-
diffusion equation. Two boundary value problems are solved.
tn the fint case a Xeumarin boundary condition is specified vvhich represents constant
heat flux at the waiI of the tube. with fluid entering a t h e m d region a t a constant
temperature. -4 regular perturbation e.xpansion is used to obtain higher order har-
monics downstream for the temperature field. The main assumption is that the
t ernperat ure field becomes fuli y developed downstream as the veloci t y field becomes
fully developed downstream. The perturbation parameter is the ratio of pressure
gradient amplitudes of unsteady 00w to that of steady flow. Using a Green's function
the first order solution is obtained. A s a measure of heat transfer enhancement. a
bulk temperature is formulated for the convective process involved and a change in
unsteady Yusselt number to that of steady Nusselt number is evduated anal-ytically
In the second part of the thesis the more difficult problem of heat transfer in pulsatile
flovv with constant wail temperature is considered. Although a complete solution is
not possible as in the first part, it is possible to use a combination of the Gener-
alized Integral Transforrn Techniques and Laplace transforms to solve this problem
upstream and downstream in the thermal region of the tube. The approximate solu-
tion indicates that a plane wave propogates down the tube and the phase of the wave
defines critical values in frequency and time dong the tube. Use of Dirac's distribu-
tion rnakes it possible to define a bulk temperature and a change in unsteady bulk
temperat ure from t hat of steady bulk temperat ure is presented. B y means of classical
analysis an inequality involving the two quantities is presented. -4s a result a measure .-. 111
of heat transfer in pulsatile flow compared to that of steady flow is presented.
To my father and mother
I wouid Like to express my thanks and appreciation to my supervisor Dr.bI.Zamir
for his guidance and direction over the past few years. I also appreciate helpful
discussions with Dr. Stan Deaiiin. I: also would Iike to thank National Sciences and
Engineering Research Council of Canada for their Financial support.
TABLE OF CONTENTS
CERTIFICATE OF EXAMINATION ii
ABSTRACT
DEDICATION v
ACKNOWLEDGEMENTS vi
TABLE OF CONTENTS vii
LIST OF TABLES x
LIST OF FIGURES 1
Chapter 1 Introduction 3
Li Motivation :3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Present Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
C hapter 2 Governing Equations 7 C, 2.1 Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 1 - - c. 2.3 Governing System of Equations . . . . . . . . . . . . . . . . . I
2.3 Preliminary Steady State problems . . . . . . . . . . . . . . . 11
2.3.1 Heat Thxirfer With SIug Flow in a Thbe . . . . . . . 11
2 - 3 2 Method of Solution . . . . . . . . . . . . . . . . . . . . . 12
2.3.3 The Graetz Problem . . . . . . . . . . . . . . . . . . . . 16
. . . . . 3.3.4 Asymptotic Method to solve Graetz problem 16
C hapter 3 Heat Transfer with Pdsatile FIow and Constant H a t Flux 21
.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 39 3.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Regular Perturbation Method . . . . . . . . . . . . . . . . . . 35
9.4 Steady Temperature . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 GreenYs Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.6 Oscillatory Temperature . . . . . . . . . . . . . . . . . . . . . . 33
3.7 Uniqueness of Solution . . . . . . . . . . . . . . . . . . . . . . . 39
. . . . . . . . . . . . . . . . . . . . . . . . . . i3.8 Riemann Surfaces 39
3.9 Verification of Boundary Conditions . . . . . . . . . . . . . . 43
3-10 Zero Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.11 Higer Order Perturbation terms - Convergence Criteria . 46
3 . 1 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 49
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Conclusions 53
Chapter 4 Heat Transfer with Pulsatile Flow and Constant Temperature 53
1 The Generdiaed Integral Transform Technique . . . . . . . 53
4 Heat Transfer in Pulsatile Flow with Constant Wdl Tem-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . perature 60
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Distributions 65
. . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Method of Solution 66
. . . . . . . . . . . . . . . . . . . . . . . 4.5 Results and Discussion Cl
C hapter 5 Concluding Remarks 81
Appendix A Maple Code for the Graetz Problem 83
Appendix B Maple Code for Constant Heat Flux Problem
Appendi* C Axial Gradient of Temperature Downstream
Appendix D Picard-Lindelof Theorem
Appendix E Properties of the Dirac Delta Function
REFEmNCES
VITA
LIST OF TABLES
2.1 Parameters associated rvith the Graetz problem . . . . . . . . . . . .
3.1 Zeros of the function !I! for zeroth order perturbation . . . . . . .
- 4.1 Zeros r.= of wave function and pulsatile velocity u = 1 + cost: t= n/4
- 4.2 Zeros r.= of wave function and pdsatile velocity u = 1 +cost:t =3i;/-L II
4.3 Zeros r. = of wave function and pulsatile velocity u = L + cost :t=3 i;/2
4.4 Zeros 7.5 of wave function and pulsatile velocity u = 1 + cost: t = 27r -
4.3 zeros ( r ). = of wave function and pulsatile velocity ti = 1 + cost + sint
A=,/;) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -
4.6 zeros (7). = of wave function and pulsatile velocity u = 1 + cosf + sint:
- 4 zeros ( r ) . = of wave function and pulsatile velocity u = 1 + cost + sint
LIST OF FIGPRES
2.1 lnterpoiated eigenfunction & of Graetz problern ...................... L9
2.2 interpolated eigenfunction RL of Graetz probkm ...................... 20
2.3 uiterpolated eigenfunction R2 of Graetz problem ...................... 21
*') 9 -1.4 [nterpolated eigenfunction R3 of Graetz problem ...................... -,
2.5 Temperature difference vs radial iength of Graetz problem ............. 23
2.6 Temperature difference vs axial Iength of Graetz problem .............. 24
2.7 Local Yusseit number vs axial length of Graetz problem ............... 25
3.1 Constant heat f l u model ................................... ,. .. .. .... 2 9
3 . 2 Bromwich contour associated with <p(x . r) ............................ 110
3.3 Riemann surface for log z ............................................. 15 .- 1-4 l.vu \ .sd : c = 0.1 ................................................... 3 1
........................... 3.3 A.\., VSJ* : ,C = 0.3 ..................... ,. 58
6 L V U vs C. : : = 0 . .j ................................................... 59
1.7 LI', V S ~ : : = L.0 ................................................... 60
.............................................................. 3.8 1 9 ~ v s u 61
....................................... ......... 3.9 Aivu vs LJ : P, = O . a .. 62
............................................... .3.10 AN, ~ S J . P,=.L.S 63
................................................ 1 &Vu v s u . P,=.j.O 64
4.0 Constant temperature mode1 ......................................... 72
4.1 Phase plane surfaces- Moderate Frequencies : g(r.t.r.sr)= const ......... 93
4.2 Phase plane surfaces-Higher Frequencies : gp.t.::*. t = consr . . . . . . . . . . . . 94
xii
Chapter L
Introduction
1.1 Motivation
T here are several practical situations where heat is being transfered under conditions
of pulsatile Bow. Examples occur in industria applications and in blood flow. Also
the problern may be important in the design of control systems for heat exchanger
equipment. One of the objectives in industrial applications has been to determine
whether the performance of a heat-exchange device could be improved with pulsatile
flow.
When pulsations are irnposed on Bow in a tube. it is believed that heat transfer would
be changed because the pulsations would alter the thichess of the bound- layer
and therefore the thermal resistance. This vieiv was put fonvard by Richardson [II who &ad shown chat the velocity profile near the <val1 is steeper in pulsatile Bow than
in Poiseuille Bow. It then follows from a consideration of Reynold's anaiogy that the
heat traosfer should increase under such conditions.
It is the purpose of this thesis to determine whether ~ulsatile flow can give rise to an
increase in heat transfer in a tube. which is a basic elernent in many heat exchange
devices.
1.2 Previous Work
The mechanics of pulsatile Kow in a tube has received considerable attention starting
with the classical work of Uchida PI. Womersley [:3] and Atabek and Chung [;LI which
has led to an exact solution and much information about the flow problem. Studies of
the corresponding heat tramfer problem. hotvever. have been far less nurnerous and
existing results do not provide cornplete resolution of a l l the questions involved.
Siegel and Perlmutter (51 demonstrated the explicit dependence of overail heat trans-
fer on pulsatile frequenc- They found that for a constant d l temperature boundary
condition. t be resdting Nusselt number shotved penodk axial change which could en-
hance heat transfer.
Barnett and Vachon [6] analyzed fully developed pipe flow in the presence of lon-
gitudinal periodic pulsations by assuming that the radial part of the temperature
profile is additively independent from the axial part. Although they did not justify
this assumption they were able to conclude chat heat transfer effects with constant
flux at the mal1 are amplified by pulsations of fluids of Prandtl oumbers well below
unity for the limiting case of low frequency.
More recentiy CreE and Andce [ï] solved the Xavier Stokes and energy equations
using a finite difference met hod and an asymp totic development for the d ~ a m i c and
thermal quantities. Their mode1 shows the existence of an annular effect in the entry
region for the pulsatile part of velocity and temperature which is known as Richard-
son's effect. The work was based on the full Navier Stokes and energy equôtions which
made it difficult to obtain an explicit expression for the Nusselt number over one cycle.
Cho and Hyun [8] solved the time-dependent laminar boundaq layer equations nu-
merically over a large range of frequencies and amplitudes of pulsation. They found
that for certain frequencies the 'iusselt number increased over the steady-flow value.
-
They also found that the Susselt number trend is amplified as the amplitude of pul-
sation increases and the Prandtl number is much Iess than unity.
Haneke. Laschefski. Gro be-Gorgemann and Mit ra [9] solved the Xavier-S tokes and
eoergy equations numericaily for laminar pulsating flow in a chaanel. At the en-
trance region of the channel a sinusoidal pulse was applied. Their results indicated
a positive and negative overshoot (Richardson's effect) in the avid velocity profile
and flow separation near the w d . They concluded that appreciable heat t r a d e r
enhancement occurs in the channel.
Kim. Kang and Hyun (101 conducted a study of the heat transfer characteristics of
lully-developed pdsatile flow in a channel and found numericdy that changes in the
Nusselt number were pronounced in the entrance region. and only minor for down-
stream. -ils0 the effects of pulsatile frequency on heat transfer \vas found to be
noticeable when the frequency is s m d or moderate but not when it is high. They
concluded that oscillations may produce both heat transfer enhancement as well as
reduction at different a~ial locations in the channel.
Base. Campbell S; Hobbs [II] studied the heat transfer characteristics of pulsatile
Aow in a pipe experimentally and found there to be an optimal frequency at which
there is an increase over the steady value in heat trânsfer for fluids OF Prandtl number
near unity.
Some other experimental work in heat transfer with pulsatile Bow has b e n con- - -
ducted by Fallen [Io] showing that superposition of a pulsating flow in a pipe with
zero mean velocity on steady flow increased significantly the overall Nusselt number.
Experimental work has also been done by Genin, Koval. Manchkha and Sviridov
1131. Hapke [l4]. Andre. Creff and Crabol [~j] and Peattie and Budwig [16] , dl of
whom have reported increases in heat transfer rates with pulsatile flow.
1.3 Present Work
In this thesis the aim is to demonstrate the mechanics of heat transfer in pulsatile
ftow in a tube analytically. That is. ive must solve the energy equation coupled with
the Navier Stokes equat ions wit h a sinusoidal pressure gradient driving the flow.
In the first part of the thesis we use a regular perturbation expansion [1T] t o solve the
convection problem subject to a constant w d heat flux. CVe obtain. using Laplace
t ransforms and Green's functions. qualitative and quantitative insight into how the
above phenornenon depends on the nondimensional parameters involved such as pui-
sat ion frequenc- and amplitude. Prandtl and Reynolds numbers.
In the second part of the thesis we use an integral transform method known as the
geeoeeralized integral transform technique p3]. (-41. [25] to solve the problem of heat
transfer with pdsatile Bow in a tube as in the first part but with constant w d
temperature instead of constant heat flux. .An explicit fomulae is obtained for tem-
perature at any axial distance along the thermal region of the tube both upstream
and dosvnstream and the existence of a wave-like form in the solution is shotvn. The
bulk temperature for t his unsteady solution is iormulated and compared to the cor-
responding steady one.
Chapter 2
Governing Equations
2.1 Basic Assumptions
W e consider fully dewloped laminar pulsating flow of an incompressible. Sewtonia.
Buid. It is convenient to take cylindrical coordinates with x axis coinciding with the
center Line of the tube. The main assump tions are t hat ( 1) the velocity field is in the
r direction only and is assumed fu& developed: (2) the flow is axially syrnmetric: (3)
there is unifonn pulsating flow and hence the pressure gradient becomes a function
of time only: (4 ) viscous dissipation effects are negiigible compaxed to the convective
rate of heat transfer: ( 5 ) the fiuid is entering a thermal region with a constant uniforxn
t emp erat ure.
2.2 Governing System of Equations
The Xavier Stokes equations under the above assumptions and in cylindricai coordi-
nates are
a au àv I â p apv a 2 ~ 1 av - + u - + u - = --- at d~ br par
+Il(-+-+----) d x 2 dr* r d r r*
with the equation of continuity
where x.r are axial and radial coordinates and u.u are velocity components in x and
r directions respectively. t is time and p is density of the fluid.
If the Bow direction is assumed CO be parallel to the r-axis. as stated in our as-
sumptions. velocity has x component only and c is zero. The equation of continuity
gives
mhich indicates the velocity to be constant in the direction p a r d e l to the center line
of the tube. The pressure gradient is assumed to be a harmonic of some Fourier series
in t ime.
The temperature T' within the tube is governed by the simplified convection dif-
fusion equation
dT' ~IT' k (B*T- MT=) - + uœ(rl t t*)- = - - dtœ + ,- a ~ - pC, a ~ - * r dr*
where C,, k are specific heat. and thermd conductivity of the fluid, and the starred
quantities indicate dimensional fom of these variables. The second derivative of T'
with respect to x that appears in the laplacian is neglected on the assumption that
it is small compared with the derivatives of Tm with respect to r in the downstrearn
region of the fl ow being considered.
In the k t part of the thesis the solution is sought for a region of the tube down-
Stream. where the flow field is f d y developed and the heat BU dong the b o u n d q
of the tube is constant and fluid is entering the region at a uniform temperature Tt.
Introducing the nondimensional temperature dxerence
where q, is the constant heat B u at the wall. a is the radius of the tube. and k
is the thermal conductivity of the Buid. the problem can be put in the nondimen-
sionai form
with boundary conditions for B(x. r. t )
In the second part of the rhesis a constant mdi temperature is prescribed on the
i v d of the tube rvith Buid entering a thermai region at a different constant temper-
ature. The velocity field is assumed to be that of fu- developed pulsatile Bow as in
the first part of the thesis.The goveming equation is the same as in 2.6 but with
and where TtU. To are the waH temperature of the tube and the entry temperature of
the fluid respect ively.
The boundary conditions in this case are
-
3 Preliminacy Steady State pmblems
In what foiIows ive present resdts of steady state problems which are reguired for
solution of the heat transfer problem with pulsatile Bow.
2.3.1 Heat Transfer With Slug Flow in a Tube
in this section we consider the case of IaJninar flow in a circuiar tube with heat trans-
fer. Let the temperature of the fluid be To at the thermal entrance of the tube and the
tvall temperature be Tw. Also we assume in this section that a t the thermal entrance
to the tube the ve loc i - is uniform over the cross-section. t hat is
The energy equation for a fluid element is:
ivhere p is the Auid densit. C, is the specific heat at constant pressure. k is the
thermal conductivity of the fluid. cr = k/pC, is the thermal diffusivity and
in terms of nondimensional temperature 0 = (T' - Tw)/(To - Tw) and nondimen-
sional quantities x = x'/aP,- Re and r = r'/a the equation becomes:
86' a26' 1 de um- =a(-+--) ax dr* r d r
wi th b o u n d q conditions:
2 - 3 2 Method of Solution
%y separation of mriables
one obtains
The second equat ion has a solution
JO(&) = O Vi.
The constants -Ji are determineci by the condition that
Multiplying both sides of Eq. 2.16 by rJo(mr)
and integrating from O to L gives:
The following two results wiii be made use of in the solution.
If X and p are two different constants then.
Letting n = O and X = 3, in equation 2.19 above and given the identity rJn(r) =
-
nJ,,(r) - rJ,+Lir) rvhich implies that.
JO'(.) = J:(P)
we get
where
The unifonn temperature assumption f ( r ) = 1 gives:
and
The complete solut ion for slug flow becomes
3 The Graetz Problem
Heat transfer problems of forced convection in tubes have been studied for maoy y a x s
beginning with Graetz [lY] and later with Nusselt [191 and Leveque [-O].
In this section we assume poiseuille flow at the entrance of the thermal region and
to the left of this region we assume the Buid is isothermal. In the t hemal region a
prescribed temperature or heat flux is specified at the wall of the tube. This problern
is refened to as the Graetz problem.
'2.3.4 Asymptotic Method to solve Graetz problem
From Eq. 2.6 assuming a steady state form with u = 1 - r2. the Graetz problem
becomes
tvi t h boundary conditions :
rnhere 0 = (TR - T,) /(To - Tw) . Tu! To are rvall and entrance temperatures re-
spect iveI -
,\ssume O is in the form :
where A, are eigenvalues that satisfy the following differential equation:
satisfying the boundary equations:
The Sturm-Liouville theory gives us onhogonality with respect to the weight function
r( 1 - r2)? that is
for rn # n
It can be shown eaçily that
and the folloming identities are known to hold [lS]
1 a m a , l1 r ( l - r2)%dr = (-)(-- lx, ar ax, .=,
where the boundaiy conditions above have been used.
Lïsing these ttvo results we get at once that
-
Assume a solution to the above Sturm LiouviUe problem Eq. 2.29 in the fom:
R =
Substitution of R into the ordinary differential equation gives :
h" + hf2 + (L/+)hf + X2(1 - r") = O
An as-ymptotic solution is sought in the form:
h = Aho + hl + (A)-'h2 + ( ~ ) - ~ h ~ + . . .
where X is the eigenvalue of Eq. 2.35.
After straightfonvard substitutions a solution involving just ho and h is obtained in
the form
with A. B complex numbers.
Sellars. Tribus and Klein [-II have shown that the eigenvalues ase given as
and the eigenfunctions are as follows:
Year the center of the tube. r O
& ( r ) = Jo(kLr)
for intermediate r =z 0.5
and near the wall, r = 1
These 3 approximations can be patched together to produce continuous results. W e
used a Maple routine [Appendix Al in which eigenfunctions and temperature promes
of the Graetz problem can be obtained.
Refer to Fig. 2.1- Fig. 2.7 for eigenfunctions and temperature profiles.
The constants C, are given by
( - L )Y + 6 2 / 3 r ( - p ) x - ~ 3 C, = n - n = 0 . 1 . L . .
I l
Table 2.1 lists several eigenvaiues and important constants for the Graetz problem.
The Xusselt number for constant wall temperature is
Table 2.1: Parameters associated with the Graetz problern
Refer to Fig. 2.7 For the Nusselt number variation dong the tube.
Chapter 3
Heat Transfer with Pulsatile Flow and Constant Heat Flux
3 1 Overview
In this chapter an analysis of fuliy developed Bow in a tube in the presence of longi-
tudinal periodic pulsations with heat transfer is presented.
CTsing a regular perturbation method [ l T ] an anal-ytical solution is sought for the tem-
perature field domnstream. This expression is written as a steady terrn with higher
order hannonic terms superimposed on it. These higher order t e m s are due to veloc-
ity fluctuations in pulsatile Bow. It can be shown that these velocity pulsations cause
harmonic osciilat ions in temperature to occur t here b- spli t t ing the temperat ure field
into a steady part plus an oscillatory part.
Our aim in this thesis is to deterrnïne if the Bow field for pulsatile flow eobances
the overall heat transfer rate as rneasured by the Nusselt number.
The bulk temperature over one cycle and the change in unsteady Nusselt number
to that of steady Nusselt number is shown to depend on the dimensionless frequency
4 and a number rvhich ive obtain through dimensional andysis which involves a ratio
of unsteady pulsation amplitude to that of steady amplitude. this number which we
derive below is denoted by c.
3 Governing Equations
The Temperature T' within the tube is governed by the foilowing coupled partial
differentiai equations
DT' k cl-=-
ot' pCP of T=
where the materiai derivative operator is defined as
D d --- - + ~ ' - y & Dt* bt
The function u' satisfies the Xavier Stokes equations for a sinusoidd pressure gradi-
ent driving the flow with a no slip boundaq condition at the wall of the tube and a
fully developed velocity profie along the tube. The temperature of the fluid at the
thermal entrance region is constant with constant heat flux at the wall of the tube.
See Fig 3.1 on page 23.
The following boundary value problem results.
-
\ïscous dissipation effects as given by the dissipation function are assurned ne&
cible cornpared to the convective rare of heat transfer so that no nonlinear effects are z
present in the energy equation.
\Ve noow introduce the foiloiving nondimensional quantities
mhere q, is the constant heat flux at the boundary of the tube. a is the radius
of the tube and k is the thermal conductivity of the Buid.
The Prandtl number. Reynolds nurnber and the t hemal diffusivity are defined by
-
Substitution of Eq.3.4. i3.5 and 3.6 into 3.3 the governing equation finail' becomes.
in nondimensionai Form
with b o u n d q conditions for B(x, r. t )
Fig. 3.1
-
X3 Regular Perturbation Method
The flow field wit hin the thermal region of the tube is assumed to consist of a steady
part represented by f d y developed Poiseuille flow plus an oscillatory part represented
by the classical solution for pulsatile ff onr [3], PI. Thus with one h m o n i c of some
Fourier series elcpansion added to steady flow we mite
where A0 is the pressure gradient driving the steady part of the flow and AL is the
ampli tude of the oscillatory pressure gradient driving the oscillatory part of the flow.
The pressure gradient with respect to the x variable is written as a steady amplitude
plus one hannonic of some Fourier series in time.
Introducing a nondimensional frequency parameter.
the velocity downstream can be put in the nondimensional form
tr(r. t ) = uo(r) + fu t ( r . t )
iv here
Variables have been nondimensionalized as before. wi t h the normalizing velocity tio
now being taken as the maximum velocity in Poiseuille flow. namely
Due to the nonlinearity of the product of the velocity and the gradient of temperature
difference in the convection equation and since the veiocity for pulsatile flow depends
on the ratio of LI/.^^. the following regular perturbation e-xpansion is used to solve
for temperat ure downst ream.
ivhere 6 is the perturbation parameter defined previously and will be assumed small
(i.e. less than unity).
-
Substitut ing the velocity and temperature expressions in the governing systern yields
infinitel- ma- boundary kalue pmblem for Bo and 01 02. etc.
The first t hree are
a00 a2eo 1 ao, uo- = - + -- da: dr3 r dr
In view of the foim assumed for the temperature. these bound- d u e problems
govern the steady ( &(x. r) ) and oscillatory ( &(r, t ) , &(P. t) .etc.) parts of the tem-
perature field. respectively. and their solutions are considered separately in the next
t wo sections.
A solution to (:3.19) is sought downstream. where we assume that 2 = const . The
-
bound- condition is physicaUy reasonable since far downstream the effects of idet
temperature are negiïgible where primary heating of Buid is due to the heat flac at
the maii done-
In the next section ive claim that the derivative of Bo with respect to x downstream
becornes a constant. Since the series is e'cpanded in { which is variable then the
partial derivatives of 62, etc with respect to x. approach O downstream.
3.4 Steady Temperature
The zerot h order problem. 60. is solved \vit h the use of Laplace transforms and residue
theo. W e have described the solution in appendix C . where we obtain the following
result for the temperature
where some talues of 32 and W(-32) are given in the following table
Table 3.1:
and
Zeros of the function Q for zeroth order perturbation
W e observe that for tb is zeroth order problem far downstream
tVe make use of this result in the solution of the k t order problem.
:Lj Greenk Eùnctions
The Green's function merhod is one of the most important approaches for solv-
ing b o u n d q d u e problems wi t h noohomogeneous ordinaq different ial equat ions.
Sturm Liouville pmblems w hich involve self adjoint different id operators are writ ten
as
The idea is to obtain a solution of the boundary value problem
Lh = g
in the form
where Ii turns out to be an integral operator. This kernel is cded the Green's
function for the boundary value problem.
Ordinarly q.g are real d u e d but the method can be extended to cornplex values
for q. g. The theory of global analytic functions allows one to study the complex
solution of ordinary different ial equations. [22]
This extension is necessary in Our problem for the solution for oscillatory temper-
at ure. Construct iveiy. a standard technique for solving Sturm-Liouville problems is
t O use tariat ion of parameters. Suppose O: V are two linearly independent solutions
of the corresponding homogeneous equation L h = 0.
A solution is sought in the form h = oU + GV upon which substitution gives
If g is continuous and the CVronskian CV = p ( L V - VU') is not zero then h = OU+ c V
is a solut ion for the differential equation. more e-xp ki t -
is the Green's function associated wit h the problem.
In the next section the t-ype of Sturm-Liouville problem associated with oscillatory
temperature involves a differential equation with complex valued arguments as CO-
efficients with the nonhomogeneous part being complex d u e d as well. -4s already
mentioned a solution can be obtained in the complex phne-
3.6 Oscillatory Temperature
The result of the previous section ( Eq. :323 ) maks it possible to seek a solution far
downstream for BI in the form
Substituting this into boundaq value problem Eq 3.19 Leads to the following bound-
ary talue problern
CPh I d h - + -- - iwh = g ( r ) dr* r d r
h f ( l ) = 0 . h l (0 ) = O
In addition ive assume that at zero frequency. the solution for h is bounded and
finite.
The differential equation is of regular cornplex valued Sturm-Liouville type
where G is the Green's fuoction of the above Sturm-Liouville problem and K is a
specid integral operator '
The most general solut ion is
The Wronskian is defined as follows
K is a compact cornplex Hermitian operator acting on L2(0. l ) , 1; : L2(0. 1) + L'[O. 1)
Xow the boundaq conditions. of s y m e t r y and flux are
The first condition gives
and since the derivative of the Xeumann function is unbounded at the centreline
of the tube.
Differentiation of the function h ( r ) ivith respect to r gives
The second boundary condition gives us at once that
c1U'(l) + V ' ( I ) G V - ~ LL U ( r ) g ( r ) d r = O
from which cl is determined as
Hence the solution for the function h becomes
The functions U. V are Iinearly independent solutions of the corresponding homo-
geneous different id equat ion
-
Lh = O
The function H is Heaviside's function. defined on [O.Il.
Substitution of the linearly independent sohtions L-. C;' the function g, the Wron-
skian W and the Green's function G leads to the following solution:
where.
Jo(r\l-iwl Pr) 1' 1W
3.7 Uniqueness of Solution
The solution obtained in the previous section involves Bessel functions 1; of a com-
plex argument =. and since these in tum depend on log2 each part of the solution is
multi\dued and thus not unique. In this section we discuss the notion of a Riemann
surface upon which each part of the solution h l , hz, h3 becomes single-dued and thus
rvell-defined.
W e introduce the idea of a branch in order to study rnuiti vaiued functions properly.
Two function elements (fi. R I ) and ( f2, R2) determine the same branch at a point in
the complex plane z0 E ( R I n RÎ) whenever fl = f2 in a neighbourhood of the point
3, which holds if the functions are analytic continuations of each other. The branch
at the point 20 given by the function element (f. R) is denoted as (f: 20) . The set of
branches ( f. t ) is defined as the Riemann Surface.
-4 Riemann surface is simply a generalization of the complex plane where a multiple
valued function is defined on a surface of many sheets where the function becomes
single d u e d on this surface. To describe a surface of many sheets we mil1 consider
the Riemann surface for the function logr which is comected to our Green's function
solut ion obtained previously.
A Surface for Log 2
For z in polar form
- = re - id
log= = Logr + i9
where Log r is the principal value of log =.
The function log z is obviously many d u e d at each point in the complex plane
since the angle 19 is increased by multiples of 2 1 and -Zn respectively.
To turn this Function into a single valued function. the rplane is replaced by its
Riemann surface on wbich new points are represented denever the argument of the
point z is either increased or decreased by 2n radians.
By considering the =-plane as a sheet &, cut along the positive x axis. let the angle
O range frorn O to 2;;. -4 second sheet Ri is cut ideotically and placed in front of the
sheet 5. The lower edge of slit in & is joined to the upper edge of the dit in RI.
On RI the angle 0 ranges from 21 to 4;1. The same is done for sheets R2, R3, etc.
A sheet R-I placed behind Ra. on which B varies from O to -21. is cut with the lower
edge of it's dit connected to the upper edge of the slit in Ra. and this is coatinued
for R 4 , R+ etc-
'iow the origin is cornmon to ail sheets. In the above construction as the point com-
pletes a cycle around the origin on &, the angle ranges frorn O to 27i. -1s it rnoves
across 6 = 2;; the point passes to the sheet Ri ol the surface. As the point completes
a cycle in RI. the angle 6 ranges from 27 to -Lx and so on. Refer to Fig 3.3 for the
Riemann surface for log:.
-
A U parts of the solution h. chat is hlth2.h3 are ma- valued functions in the
complex plaae and therefore require a Riemann surface consisting of infinirely many
sheets for the Iogarithmic function involved. As mentioned previously the functions
becorne single d u e d on the Riemann surface and hence are well defined functions.
The solution obtained however for h via the Green's Function is unique. Despite
the multivaluedness of each function h 1, h2, hJ, their sum h adds up to be single
valued. .-i qui& calculation shows that the logarithmic parts cancel out as they are
summed up to obtain h.
To show this note that since logr = Logr + i0. the multi-valued part of hl is
L L - ~ i ~ j ~ > & i 3 1 2 . j o i ~ i 3 ~ 2 r ) ~ ~ 0 ( & i ~ / ~ 7 ) ~ [l - Joc~-, -iu/ P r ) d r (3.12)
the multi-valued part of h2 is
and the multi-valued part of h3 is
Adding the three parts together gives us zero identically and thus the function h
becomes single-kalued in the complex plane. Hence we m i v e at a unique solution.
1.9 Verification of Boundary Conditions
One of the properties of the solution for h obtained previously is that in addition to
satis@ing the complex differential equation. it satisfies both flux and symmetry Line
boundary conditions.
Different iation and use of the first version of the fuidamental t heorern of calculus
gives us the followiog
then it follows immediately t hat
&(rd-iw/ P r ) -- ah, - &< fii312r)(a/z)~i3/2 dr 1
Adding up the three tems above gives
Next the flux boundary condition is checked. Differentiating and using the first
version of the fundamental theorem of cdcuius we get
Adding the above parts gives.
Hence bot h boundary condit ions are satisfied by the Green's funct ion solution.
3.10 Zero Frequency
At zero frequency the term. - V ( l ) / U ' ( I ) l in hl ( a.39 ) is unbounded because its
imagin- pa.rt is unbounded. The real part provides a d i d solution however and
therefore. O& the real part of -C"(1)/Uf('(1) is taken for calculation purposes.
3.11 Higer Order Perturbation terms - Convergence Criteria
The perturbation e-vansion in section 3.3 led to infinitely many coupled partial dif-
ferentiai equations of which we have solved only the first two downstream.
A naturd question to ask is whether or not the perturbation series converges down-
streaxn for s m d perturbation parameters c.
In order to answer this we must first solve a,il of the remaining higher order terms
in the perturbation series starting with the second order and proceeding to the kth
order one and so on.
This of course means that infinitely many boundary value problems must be solved
from the second order system and so on.
Recall the second order probIem
ae, ae, adz a2e2 i de2 - + u ~ ( r . t ) - + u&)- = p at + -- dx t3x r
Downstrearn this simplifies to
In fact the generai k'th order boundq value problern downstrearn becomes
The solution of this problem is readily obtained as
where
and prime denotes differentiation with respect to r.
The eigenfunctions R(&, r ) are the solutions to the FoHowing eigenvalue problem:
where the 3, are the zeroes of J L .
Yow one sees that the k'th order solution is bounded over one cycle and across the
tube radially.
Since the k'th order problem in the perturbation e.xpa,nsion is bounded with 5 < 1
t hen the series converges downstream.
Hence the perturbation e-xpansion converges downstream.
3-12 Results and Discussion
To determine the effect of the oscillations on the rate of heat t ramier ive define the
local Xusselt number
and the corresponding Yusselt number in steady Boiv
where R denotes the real part of a cornples valued function. the subscript 6 refers to
bulk properties of the fluid. and the overbar indicates time average over one cycle.
that is
In terms of the nondimensional temperature difference 0 the expressions for the Nus-
selt numbers become
in these. we have that
and
w here
where only the Çst two terms are considered in the perturbation expansion since the
expression for veiocity is exact. (Eq. 3.12)
As a measure of the effect of oscillations on the heat transfer rate we now consider
the relative difference quant ity
Substituting for the Nusselt numbers from equation 3.52 and 3.53 and for the unsteady
bulk temperature from equat ion 3.57 t his becomes finally
In Fig 3.4-Fig :3.ll ive gaphed our resuits for &ous frequencies and Prandtl num-
ben as iveil as for different values of C.
In particular. frequencies in the range of 0-50 are considered wit h Prandtl numbers
P, at 0.5. 2.5 and 5.0. The perturbation parameter is taken as 0.1. 0.3. 0.5 and 1.0.
3.13 Conclusions
Csing a regdar perturbation method. the coavection equation coupled to the clas-
sical pdsatile solution was solved using Green's huictions in the cornplex plane and
Laplace transfonns upon which a bulk temperature and relative Nusselt nmber were
obtained. Since heat transfer fiom the tube is constant because of the boundaxy
condition. the effect of osciiiations can only occur in tems of change of buik ternper-
ature tvithin the tube- From the definition of Nusselt number we see that the Xusselt
number and the relative 'iusselt number depend on the change in bulk temperature
due to the pulsations. It is believed that Iower viscosity and hence srnalier Prandtl
numbers are associated with s m d momentum and thermal b o u n d q Iayers hence
by Reynold's andogy an increase in heat t ransfer shodd occur. The results obtained
indicate that pdsatile floiv contributes to an enhancement in the overalI heat transfer
rate. An important observation is that there exists a critical value in frequency where
a maximum occurs in the relative Nusselt n u b e r . The Prandtl number dependency
is such that k a t transfer increases by pulsations of fluids with Prandtl number less
t han unity and decreases for Prandtl numbers greater than unit.
Since the Green's function requires us to evaiuate a number of tedious integrais in-
rolving Bessel functioos it was appropriate to wi te a Maple program to compute the
bulk temperature and relative 'iusselt number. [AppendixB]
t i r
Chapter 4
Heat Transfer with Pulsatile Flow and Constant Temperature
4.1 The Generalized Integral 'Itansform Technique
In the previous chapter a constant flw b o u n d q condition [vas specified and we
were able to obtain an anaiflical solution. The corresponding problem wit h constant
temperature specified at the 1vaU is more difficult to sohe and hence a compIete as-
alyticai solution is not possible as in the previous section. However because of the
discontinuity due to the boundary conditions at the entrance of the thermal region
of the tube the use of the Laplace transform seerns appmpriate for this ~robIem as
well as the following integral transform rnethod.
In the foilowing the General Integral Transform technique ['Z3]. ['XI. ['25] [30]-[X]
is discussed which \ d l be used to solve the convection equation with pulsatile Bow.
This technique is a hybrid rnethod [34]. [43].[44] applied to the convection diffu-
sion equation.
Consider the following general transient Iinear convection diffusion problem in the
-
where cc. I<. d. P have continuous partial derivat ives in some closed bounded volume
C'. rvith the initiai condition
and b o u n d q conditions
The technique of the generalized integral transform method is to start by choos-
ing the foilowing au'ciliary problem.
tvi t h boundary conditions
a d it is assumed that the solution of the above auxiliary problem is known.
tif
-
The appropriate integral transfonn pair obtained from this e igendue problem is
Inverse:
Operating on the general transient convection-diffusion equation with the operator.
we obtain
- where the function g i ( t ) can be shown to be
The integrai appearing in the transformed convection-diffusion equation is cdculated
by making use of the inversion formula to give the foiiowing system of coupled ordi-
nùry differentid equat ions
where the matrix -4; as a function of t is
and the normaiization integral. Xi, is given by
-
The initiai condit ion equation is also transformed using the operator
which gives us the following
These equations. are an infinite system of coupled ordinary differential equations
for the transformed funct ions. ie the 'S.
Since the system is infinite it is not always possible to obtain an exact solution.
If this system can be solved however. the inversion formula above can be used to give
LIS the solution T(.E t ) of the convection-diffusion equation. In practice the system to
be solved can be tmncated at the N'th row and column. with N siifiiciently large for
the required accurac. and then the fuute system can be solved by standard numerical
techniques. [3S]
In matrix form. the tmncated version of the system becornes
-4 special case occurs when the velocity u in Eq. 4.11 is independent of t . In this
case the matrix of coefficients. -4. becomes constant. and the solution of the system
of differential equations above can be expIicitly written as the following
where the eaponential marrix can be computed once the eigenvalues and eigenvectors
of -4 have been obtained through solving the dgebraic problem
Scientific subroutines are available to accurately accornpiïsh this ta&.
In applications obtaining an approximate explicit solution c m be very important
and of special interest in getting qualitative insight into a problem. -4 Iowest order
-
solut ion can b e O btained provided the non-diagonal elements of the coefiicients matrix
-4 are neglible as compared to those of the main diagonal tvhich wouid then approx-
imately correspond to a decoupled system. Therefore by keeping o d y the diagonal
elements of the matrix an appro-cimate solution is obtained from the fouowing initial
d u e problem
This type of problem arises for the case of heat transfer ivith pdsatile flow and
constant wall temperature which is considered in the next section.
-
4.2 Heat %ansfer in Pulsatile Flow with Constant Wall Temperature
In t his section the convection equation with pulsatile Bow and constant temperature
prescribed dong the wall of the tube is solved using the integral transform technique
discussed in the previous section. The governing equation derived from Eq. 4.1 be-
cornes
where 0 = ( T - TW)/(To - Tw). T, is the constant wall temperature of the tube
and To is the constant temperature of the fluid at the thermal entrance region. Refer
to Fig. 4.0 for a sketch of the model.
Fig. 4.0
-4s in Chapter 2 the velocity u satisfies the Navier Stokes equations for full44 developed
axiai flow with the pressure gradient set equd to a harmonic of some Fourier series
in time, It foilows as before that
Taking the Laplace transform of Eq. 4-18 with respect to x. that is operating on
this partial differential equation with Jo e-"'( -)dx mhere s is cornplex and Rs >
for some 7 positive . we get
where we have integrated by parts and used the boundary condition at x = O .
Yow using the Generalized Integral Transform technique [4l].[Q]. Ive introduce the
-
following alLui1iary problem ivhich is independent of the parameter s and whose solu-
tion is the zero order Bessel hnctioa as a function of 3 where the 3 s are the zeros
of JO.
1 d dd($, - - [r rdr dr
"1 + d211r(13. r ) = O
The following integral transform pair is dehed as:
Eq. 4.19 is now operated on wieh the operator
to obtain the foIloiving:
Yow. using the auxiliary problem the Laplacian on the right hand side can be written
as
From this we obtain the following initial value problern with s cornplex.
-L L 9; (O) = -
S
-
where the matris A , is caiculated from the following
and the function .\;, is simply the n o m so that system of eigenfuoctions becomes
an ort honormal system.
Recall in the discussion of the generaiized integral transform technique that if the
rnatrix -4 is diagoody dominant then a lower order solution is possible. The main
difficulty is that the matrix -4 is infinite and hence an infinite system of coupled dif-
ferential equations in s arises for which the solution is impossible to obtain. In what
follows. to make the problem tractable. we assume that the off diagonal terms of the
matris =Li j can be neglected. This is clearIy an approximation to the solution of Eq.
4. LS. Based on t his approximation. we arrive at the following system of O D E k
-L 1 6; ( O ) = -
S
where u is left general for now. .ln existence and uniqueness theorem for this type of
problem is found in Appendix D.
4-3 Distributions
The solution to Eq. 4.26. a s wiU be shown Iater involves a generalized solution b o w n
as Dirac's distribution-
P hysicai quant it ies are usuaiIy t hought of as funct ions. But in an experimental set ting
it is quite difficult if not impossible to observe instantaneous values of the function at
every instant of space or time. A measuring instrument would record the effect chat
the function produces on it over some interval of space or time with nonzero length.
An alternate description for some physical quantity is to specify it as a functional. a
rule which assigns a number to each function in a set of testing functions. Continuous
linear functionals d e h e d on the space of testing functions are cdled distributions.
[-61 The distribution theory is a very powerfd tool for studying linear partial differen-
tial equations mith smooth kariable coefficients. It t m s out that using distributions
allows for anal!-zing types of physical phenornena more n a t u r d y than using the func-
tion concept. The Dirac delta function is an example of a distribution rvhich aises
naturaily in physical settings in particular in the following section.
The delta function ( S ) is defined hy
where ~ ( t ) is a test function. an infinitely smooth function that vanishes outside a
finite interval.
The delta function 6(t - r). selects the value of a test function at the point t = T.
Refer to the Appendix E for some important identities involving the delta function.
4.4 Method of Solution
From Eq.4.26. the infinite system of cornplex o r d i n q differential equations in s with
initiai condition in s is
The solution of this system is
Let
Refer to Appendix E for the inverse Laplace transform of the exponential function in
s, Kence we obtain
where CS is Dirac's generalized function.
As in the case of a Green's function the intenal is extended onto the real axis by
introducing the Heaviside function as a kernel.
From t his we obtain
ivhere a formula for the delta funct ion as a funct ion of a general argument has been
used from Appendix E and the phase g is
If Ive Let the phase equal a constant then a set of critical points is defined in the
- - phase plane (r. T ) . Since the velocity also depends on frequency a set of critical fre-
quencies eBst in the phase plane.
The sequence of terms ~ i ( t . t ) are the zeros of the function g(r), assuming they eGst
for a given function - A i i ( r ) .
Tables of zeros. Z and velocity are included for different axial d u e s as well as for
different values of time in a cycle. Various velocities in puisatire flow are considered.
Phase plane surface plots for frequency variation are included.
In the foilowing section the results for pulsatile flow are discussed. Recail from the
integral transfonn pair
The series in the solution for 8 is unifonnly convergent since r j < t and thus the
inverse Laplace transform of gL ( r. t. s) exists.
From the generalized integral transform method. the solution for forced convection
heat tramfer ivith a general velocity term u(t ) becomes
After t e m s rlii(r) concel out we are left with the following expression for temperature
difference which depends explicitly on the zeros of g.
rvhere rve are surnming up over a l the zeros of g.
-
4.5 Results and Discussion
Ln this section a bulk temperature is defined as
-4 solution for the corresponding steady problem was given in chapter 2: the bulk
temperature in that case is given by
where
As in the constant flux solution. a measure of the effect of pulsations on the overall
heat transfer rate is required. The following ratio of steady buik temperature to un-
steady buik temperature will be considered.
o b s i=- @h
From the steady state solution of Chapter 2 and the generalized integrai transform
technique of t his chapter rve have
It can be shown chat there exists a zero ri = r of g ( ~ ) for each z. t for the pul-
satile flow u ( t ) = I + c r c o ~ ( ~ f ) ) where g is specified in Eq. 4.35
In fact ive can generaiize this to the case of an arbitrary Fourier series in t . It
can be shown that there always exists a zero of g(r) at any axial distance dong the
tube and at any t ime t in a complete cycle.
If there is only one zero then
For some r positive and
Since the 7,'s are cornputable and positive then the following inequaiity results
This proves that i is strict- less than unity and thus we have heat transfer en-
hancement due to the pulsations in the flow field.
Yumericd integration techniques can be used to calculate i for different foms of
w-
It is also interesting to note that if we define the bulk temperature as in Eq. 3.56,
3.5; of Chapter 3 then we have heat transfer enhancernent as well.
- Table 4.1: Zeros 7.- of wave function and pulsatile velocity u = 1 + cost: t= ; ~ / 4
x
0.1
0.2
0.3
0.3
0.5
Zeros r
0.727.5
0.6709
0.6153
0.360'7
0.5069
- - = - 1
. -1 - -3
-4
. 5
- - - - -
Velocity u ( r )
1.0953
1, LS22
1.26198
1,3334
1.4033
a 1 Zeros r 1 S ( Velocity u ( r ) / ,
Table 4.2: Zeros r.S of wave function and pulsatile velocity u = 1 + cost:t = 3a/4
Table 4.3: Zeros r. Z of wave function and pulsatile velocity u = 1 + cost :t=3 r / 2
x Zeros r E 1 Velocity u ( r )
Table 1.4: Zeros r . of wave function and pdsatile velocity u = 1 + cost: t = Zn
- Table 4.5: zeros ( r ) . = of wave function and pulsatile veiocity u =
- Table 4.6: zeros ( T ) , = of wave function and pulsatile velocity u =
x Zeros T S Velocity u
0.1 2.2127 . I A013 - 0.2 12.1332 .- -4668
--
0.3 2.0598 -3 -5302 i I
- Table 4.7: zeros (r) ' ; of wâve function and pulsatile velocity u =
L + cost + sint : t = 3x12
Phase p h e sur faces irequencies
Chapter 5
Concluding Remarks
In the first part of the thesis. a method of anaiysis is presented for the problem of heat
transfer in pulsatile flow with constant heat transfer in a tube. -4 regdar perturbation
expansion is used to solve for the temperature field downstream. A cornplex-valued
Green's function is utilized to brmuiate a b u k temperature and a relative Nusselt
number. Since heat transfer from the tube is constant the effect of pulsation only
occurs by means of a change in the b u k temperature within the tube. The resuits
indicate that a pulsating Bow field enhances the overd heat transfer rate. In par-
ticular there exists a critical value in frequency where the relative Nusselt number
attains a maximum value. The trend in relative Nusselt number with Prandtl number
is that there is an increase with decreasing values of Prmdtl number less than unity
and a decrease mith increasing values greater than unit- The perturbation param-
eter plays an important role in the mechanism of heat transfer with pulsatile flow.
The results indicate that increases of pulsatile pressure gradient amplitude over that
of steady pressure gradient amplitude give rise to increases in relative Nusselt number.
In the second part of the thesis. ac approxïrnate solution is presented for the problem
of heat transfer in pulsatile Borv with constant wall temperature in a tube. Using
the generalized integral transform method and the Laplace transform a system of
ordinaxy cornplex differential equations arises tvhose solution is presented with the
aid of Dirac's distribution t h e o . A bulk temperature of this solution is formulated . - and an increase of unsteady buk temperature over that of steady bulk ternperature
is presented.
Courant and Hilbert [45] have defmed a prog-ressing plane wave as a solution of the -
fonn u = f(S) where J = x - d is the phase of the solution and f is any function.
The form of Eq. 1.35 indicates that rve have a plane wave propagating down the tube.
-
For future rvork any possible connections that can b e made between the convection
equation and the wave equation fiom this solution can be studied. The existence of
mave-like phenornena could prove to be usefd in describing physicdy the increase in
Yusselt number of pulsatile flow over that of steady Bow. .Us0 the frequency chacac-
teristics for the convection equation coupled to the full Navier-S tokes equations with
a general sinusoidal pressure gradient can be investigated. In addition solutions us-
ing the generalized integrai transform technique can be studied for the Xavier-Stokes
equations without neglecting off diagonal t e m s in the method.
Appendix A
Maple Code for the Graetz Problem
# Program to calculate Eigenfunctions for Graetz Problem
# and to calculate Temperature Distribution t heta
# R:=array(U-.3.l..fOl): # Subroutine to calculate point of intersection of graphs
# of funetions B(delta) and G(de1ta)
My'iewton := proc(rn.delto)
deltold := delto:
for i to rn do
deltnerv:= deltoid - evalf(H(de1told) )/evalf(K(deltold) ):
deltold:=deltnew:
od:
end:
# Subroutine to calculate the matching point eps of the # solution
# of the asymptotic formula and the sirnilarity transformation
# solution of the Graetz problem.
'iIyNewton2:= proc(s.eps10)
epslold := epslo:
for i to s do
epslnew := epslold - ed(f(epslold))/evaIf(fp(epslold));
epslold:=eva.lf(epslnew ) ;
od:
end:
# Dimensionalize array for successive approximations via Newton's # Method
deltt:=array(l..3):
delto:=0.11:
# Loop calculates the Eigenfunct ions for the Graetz problem
-
# for n=0..3 using Lagrangian Interpolation.
for n from O to 3 do
delta:='delta':
Digits:=I6:
lamda :=4'n+d/3:
G1:= delta - > sqrt(2/(PiXlamda"delta)); G2:= delta - > cos((lamda/Z)*(delta*sqrt(1- deltaz) + atcsin(delta))-(Pi/4))/(1 - delta2)(1/1);
G:= GI"G2:
E:= x - > exp(-lamda' * c); FI:= delta - > diff(G(delta)?delta);
B : = delta - > Bessel J(O.larnda'de1t a):
F-:= delta - > diff(B(delta).delta):
H:= unapply( (G-B) (delta).delta);
K:= unappl-( ( F 1-F'I)(delta).delta);
F3:= delta - > sqrt(2'(I-de1ta)/3)*(-l)"'Besse1J(1/3.1amda"sqrt(S)' ( 1-delta)(3/'2)/3)
F4:= delta - > diff(F'3(~ielta)~delta);
L:= unapply( ( G-F3 )(delta) .delta):
M:= unapply((F1-F.L)(delta).delta): SI:=plot(B(delta).C;(delta).F3(delta),delta=O..l.titIe=-R(O)'):
m:=10:
Boolean BIock assigns array values deItt[n] in Xewton's
Method according to the value of n.
if n = O then
intl:= 0.10:
int2:= 0.45:
int3:= 0.50:
int4:= 0.53:
int5:= 0.65:
int6:= 0,933:
elif n = I then
deltt [n]:=MyYewton(m.delto):
delto:=deltt [n]:
int4:= 0,.53:
int5:= 0.65:
int6:= 0.935;
else
hdel:=evalf(delto/'l) :
delt t [n]:=MyNewton(rn+hdel):
delto:=deltt[nj:
int4:= 0.53:
int.5:= 0.65:
int6:= 0.935;
fi:
Lagrangian Interpolating p o l ~ o m i d s for interpolation of
functions B(de1ta) G(de1ta). F3(delta) on [O. 11.
Quadratic fit
L 1 := delta - > ( (delt a-int2)'(deIta-intJ) )/( (int 1-int.l)'(int 1-int3) ) ;
L2:= delta - > ((delta-int l )'(delta-int3))/((in%int l)*(int2-int3)):
L3:= delta - >((delta-intl) '(delta-int2))/((int3-intlint2));
L4:= delta - >((delta-int5)'(delta-int6))/((int.L-int5)'(intPintG)); L5:= delta - >((del ta- int4) ' (del ta- int6)) / (( int5-int~int6)) ;
L6:= delta - >((delta-int4)x(delta-int5))/((int6-int4)x(int6-int5)): writeto('eig4.dath):
if n = O then
k:=.Ol:
for i From 1 to 101 do
delta:=(i-L)"k:
if delta >= O and delta <= intl then
R[n.i]:= B(delta);
ri nt (delta,evalf(R[n.i]) ) :
elif delta > int 1 and delta <= int3 then
R[n.i]:= B(int l)'Ll(delta)+G(int2)*L3(delta)+
C(int3)*L3(delta);
print (delta.evalf(R[n.i]) );
elif delta > int3 and delta <= int4 then
od:
Loop to print out 2 dim a a y R for n Tom O to -5 do
for i from 1 to 100 do print(ed(R[n.i]));
od;
od ;
Loop to print out temperature theta(x,r) vaild for n>=eps.
We can prove that eps is in the neighbourhood of 0.01
Define on array for theta(x?r)
p:=may(l..lOl);
q:=array(l..lOl):
theta:= arra+y(1..100.1..101);
writeto('3 D.datb):
k:=.01;
1:= 2;
j:=l:
for j from 1 to 50 do &:=l"j;
for i from 1 to 101 do i:=101:
delta:=(i-l)'k:
init:=O.O;
for n from O to 3 do
theta[i,i]:= init + ((2/Pi) * (-l)n * 6(2/3) * GAMMA(2/3) (4 * n + 8/3)( - 213) ) * R[n, il * exp(-(4 * n + 813)' * axi);
init:=thetab,i];
od; print (axi,delta,evaL€(thetab,i]));
od;
od: p[i]:=delta;
q[i]:=ed(thetaDj]):
od:
od:
writeto(terminal);
writeto( 'out lg-dat') :
for j fkom 1 to 101 do
print( plil 7 q U ' I 1; od:
writeto(terminal);
Routine to calculate local Nusselt modulus for t emperat ure
distribution given above
C := n- > (-l)n * 2 * 6(2/3) * GAMM.4(2/3) * (4 * o + 8/3)( - 2/3)/Pi:
r:=array(1..3000);
1:=0*001:
init:=O.O;
for j from L by 10 to 3000 do wi:=j*l;
for n fiom O to 100 do
Nus := init + (-2 * C(n) * (- l )(n + 1) * 2(2/3) * (4 * n + 5 / 3 ) ( 1 / 3 ) / ( G ~ ~ ~ r \ ( 4 / 3 )
9 ( 5 / 6 ) ) ) * e q ( - ( 4 * n + 8/3)* * a.); init:= Nus:
od:
init:=0.0;
rb] :=(axi,eMir(Nus));
od;
writeto('Nus2.dat');
for i from 1 by 10 to 3000 do
print(r[i]);
od; writeto(termina1) ;
Routine to coldate local Nusselt modulus for the temperature
distribution theta for constant heat flux q.
!-:=am-( 1 .-1000):
0~l:=25.639;
agn2:=Y4.624:
gm3:=1'76.40;
H1:=0.008854:
H2:=O1002O62;
H3:=OO0009435:
1:=0.001:
for j from 1 by 10 to 1000 do &:=j'l:
?Iu:=1/((11/4S) + (1/2)*(exp(-gml"axi)/(-gm12 ' Hl)+ exp(-gm2 ' axi)/(-gd2 ' H2)+
exp(- gm3' axi)/(-gma2 ' H3)):
yij]:=(axi.evalf(Nu));
od:
print('Local Nusselt Valueso):
wri teto(cNuss.dat6);
lor j from L by 10 to 1000 do
print (Y Li] ) ; od:
writeto( terminai):
Appendix B
Maple Code for Constant Heat Flux Problem
writeto('output'):
hsum := proc(w.Pr,N):
Iambda := s ~ ~ ( w ) V ~ / ~ :
alpha := sqrt ( w / ~ r ) * i ~ / ~ :
ulc := convert(series(BesseIJ(0Jambda*r) .r=0~m):
u2c := convert (series(Besse1Y (0,lambda'r) , r = O , N o m ) :
ulprime := diff(u1c.r):
u2prime := diff(u2c.r):
b := evalc(subs(r=i.u'lprime)):
u l tc := convert (series( BesselJ(Otaiphaxt ),t =O.N),poiynom):
u3tc := convert (series(BesselJ(OJambdart),t=O~N)~pol~~om):
u4tc:= convert (series(Besse1 J(O.alphatr) ? r = O . N ) m ) :
u2tc := convert(series(BesselY(O~IambdaLt) . t=O~m):
c l := evd(sum (((-1). * (~qrt(w/Pr)/2)'~~)/(factorial(2 * n ) 2 ) , n = 0.25)) :
c2 := evaif(sum (((-1)" * (~qrt(w/Pr)/2)'~+~)/(factoria.l(2 * n + I ) ~ ) , n = 0.25)) :
c := c l +I*cS:
hl l :=Re(edc(-b/a)):
hl2 := evalc( Pi*~~rt(~)*0.5*1~/~)*~1~):
h l 3 := edc(int((u3tc/(I*w))'(l-ultc/c),t=0..l)):
h l := evalc(hll*hl2*hl3):
ha := ulc*Pi*sqrt(w)*0.5* I3l2: hb := int((u2tc/(I*w))*(l-ultc/c),t=r..l):
h2 := evalc(ha*hb):
hc := ~2c*Pi*s~rt(w)*0.5'1~/~:
hd := int((u3tc/(I'~v))*(l-ul tc/c),t=O..r):
h3 := evalc(hc'hd):
h l + h3 +h3 :
h := evdc("):
ha1 := Re(h): ha2 := Im(h):
g := ( l/(I'm))'( 1-u4tc/c):
u := edc(g):
u l := Re(u):
u2 := Im(u): part0 := ulWhal:
part':, := uPha2:
part I := evalf(ht((part0) * (r), r = 0.J. 5. NCrule) ) :
part3 := e d ( h t ( ( p a r t 2 ) * (r). r = O--1.5. NCruie)) :
#print ('the b d k temperature isœ):
Yx(part 1) + Yx(part3):
end:
for i from 1 to 10 do
P := i*0.5:
for j from 1 by 10 to 50 do
print ('The frequency wb:The Prandt 1 No-?*The b u k temperature is ' ) :
print(j IP.hsum(j.P.25)):
od:
od:
quit:
Appendk C
Axial Gradient of Temperature Downstream
In this appendix we show that the pa.rtial derivative of the zerogth order solution
Bo tvith respect to the x Mliable approaches a constant downstream. The result is
used in section 3.6 and makes it possible to obtain a solution of the first order problem.
To show this let the Laplace transform of t - to be :
If the wdl temperature is given by t,(z), using the principle of superposition the temperature difference t - to is expressed as the foliowing Stieltjes Integral
where 0 is a solution of the Graetz problern:
Define the foIlowing transforms
Applying the convolution theorem to Eq. C 2 ' and using the boundary conditions
above we get the foilowing
Let
A pplying the convolution t heorem and using the foilowing identities
we arrive at the foilowing
Letting
lIff(s) = C { F ( r ) } and g(s) = Ç{G(t)} then f(s)g(a) = C { H ( x ) } where H ( x ) = I: F(<)G(x-
W C
t hen:
t hen
Yext ive let,
and
(C. 15)
Using the convolution theorem it can be shown that
k L - ' L { / = ~ ( = ) ~ ( z - t, r)dc} = C 1 { r ( s 7 r)-1
O ro
which irnplies directly t hat:
efore. if we are given the heat flux q, then the temperatur
( C . 18)
s given by
(C.20)
To End @(x, r ) we apply the following inversion formula and use residue theo- in
order to perform the complex integration * - - - - - - - -
?o(x, r) = Res(esfO(s. r)} at the poles of @(S. r)
Cornplex htegration is taken dong the Bromwich contour:
Fig. 3.2
R e c d t hat .
where A, = 472 + Y/3
Refer to section 2.3.1 and Table 2.1 for the eigenfunctions &, eigendues A, and constants C,.
This leads to:
t hen t his imp lies t hat :
Xow Q(s) and i y ( s ) have poIes at s = O and s = -A:. Therefore 8/9 hos p o l e
o d y at s = O and the zeros of %(s). Let 9: be nich that 4(-dd:) = O
Then the following holds:
The zeros of q ( s ) can ody be found nurnericdy.
'iow we compute the residues.
First we set.
Computing the residue Res[G. O] we get:
L Res[G. O] = lim -sesz[s C,&(l)-]
s 4 0 n=O s +A:
Thus the residue of G at zero is:
-
W e wiU m d e use of the result to cornpute the residue of the entire expression
associated with o.
Since ik has a simple zero at -,Jm then the residue of G at 2 = -3; is
Consider the other part of the term
in Eq. C.27 that is
The residue of this part is
Hence rve obtain the following solution for c$(x. r )
where we can solve For 0: numericdy for ad values of m from the following
31f f. g are andytic at so and if f (so) # O and g has a simple zero at so, then Res [ f/g,so ] = f (so ) l9'h
where An = 4n + 813
It can be shown that @(O) = 114 [46]
Now returning to the formula denved above for temperature t - to in Eq. C.20.
for q = const.
Therefore.
t hen
Appendix D
Picard-Lindeiof Theorem
In this section an existence theorem for first order differential equations of the form
given in Eq. 1.36 is given.
Consider the Cauchy Problem
where (7. €) E D is a nonempty open subset of R x C
.-\ssurne that f is continous on D and satisfies the Lipschitz condition
I / ( W - f ( t , ~ ) ISCR l x - y 1
on each rectrangle R c D and CR is a constant depending on the size of the rectangle.
Then there exists a unique solution to the Cauchy problem above on I = (r-a. r+a),
w here a, 6 are chosen appropriately s m d so t hat for the rectangle centered at ( r, c ) ,
and
where
iM = max 1 f ( t , x ) 1
and the maximum is taken over ( t - x ) E ka.
Proof
LVe show that it is possible to define
- ( I r = 0.1.2 ....) for t E I = [r-a.r +a] and derive the estimates
for some c > O. This would prove that (&) is a Cauchy sequence in the normed
Linear space of continous complex valued functions C(7), with the uniform metric
nom. Nor since this space is known to be a complete metric space (ie. Banach
Space) t hen
dt -t d .for some . q5 E C(T),
Taking Limits of the functions & it foUows that
It is a simple matter to see that r$ provides the required solution.
W e prove the above rigorously
Define c i o on 7 using the initial condition already given.
The sequence of functions t$k defhed above continue to successively defhe O,. &, 4, . . . on T Trivial l~ ive have that
for ail t E 7. Hence. because of the definition of a . the c u v e ( t. ~ ( t ) ) . t E remains
in Ra,b and. in particular. the integral to be used in defining 01 exists. Induction is
used for all k. Thus assume that
Now 4k+l must be shown to satisfy the same inequality. But from the first step
of induction method? &+, is defined on Ï, and using the definition of the maximum
LM, leads to
The next step is to estabiish the estimate
( k = 1,2, . . .) , t E 7. where c is the Lipschitz constant associated with R = ki.
Now using the Lipschitz inequality, it follows t hat
Hence the above inequality is proven by induction. Reapplication of this inequd-
ity gives:
Hence the sequence of hinctionri oc is a Cauchy sequence and the sequence must
converge to a sohtion d .
The uniqueness of the solution foIIows readily using the Lipschitz condition for f
Appendix E
Properties of the Dirac Delta hnction
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