The Art of Comparing Force Strengths…

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Diagnosis of NS Equations

Mach View of NS Equations• Reference velocity : Velocity of sound in a selected fluid.

• Velocity sound, c:

For an ideal gas: RTc

For a real gas:

v

p

Tz

Tz

Tz

Tz

n nzRTc

Property Inertial

property Elastic

B

c

For an incompressible liquids:

Average bulk modulus for water is 2 X109 N/m2.

In Mach’s view it is not possible to differentiate the flows with Ma<0.2.

Reduces the resolution of flow solutions….

Honor of Osborne Reynolds

• Consider the Navier-Stokes equations with constant density it their dimensional form:

i

j

j

i

jii

j

ij

i

x

v

x

v

xx

pg

x

vv

t

v

0 v

Define dimensionless variables as:

U

vv

*

L

xx

*2

*

U

pp

•Here U, L are assumed to be a velocity and length characteristic of the problem being studied. •In the case of flow past a body, L might be a body diameter and U the flow speed at infinity.

L

Utt *

Non-dimensionalization of NS EquationIn low Mach number region with constant viscosity:

i

j

j

i

jii

j

ij

i

x

v

x

v

xx

pg

x

vv

t

v

1

*vUv

*xLx

*2 pUp

U

Ltt

*

Non-dimensional of NS Equation

i

j

j

i

jij

ij

i

x

v

x

v

xULx

p

U

Lgk

x

vv

t

v*

*

*

*

**

*

2*

**

*

*ˆ

i

j

j

i

jii

j

ij

i

x

v

x

v

xULx

p

U

Lg

x

vv

t

v*

*

*

*

**

*

2*

**

*

*

Importance of Reynolds Number

• The Reynolds number Re is the only dimensionless parameter which is always important in the equations of fluid motion.

• The Reynolds number is very small compared to unity, Re<< 1.

• Since Re = UL/, the smallness of Re can be achieved by considering

• extremely small length scales, or

• by dealing with a highly viscous liquid, or

• by treating flows of very small velocity, so-called creeping flows.

The Creeping Flows

• The choice Re << 1 is an very interesting and important assumption.

• It is relevant to many practical problems, especially in a world where fluid devices are shrinking in size.

• A particularly interesting application is to the swimming of micro-organisms.

• This assumption, unveils a special dynamical regime which is usually referred to as Stokes flow.

• To honor George Stokes, who initiated investigations into this class of fluid problems.

• We shall also refer to this general area of fluid dynamics as the Stokesian realm.

• This is of extreme contrast to the theories of ideal inviscid flow, which might be termed the Eulerian realm.

Ancient Fluid Dynamics

Stokesian Kingdom Eulerian Kingdom

An experimental investigation of the circumstances which determine whether the motion of water in

parallel channels shall be direct or sinuous and of the law of resistance in parallel channels

The Principle Characteristics of the Stokesian Realm

• Re is indicative of the ratio of inertial to viscous forces.

• The assumption of small Re means that viscous forces dominate the dynamics.

• That suggests that to drop entirely the term Dv/Dt from the Navier-Stokes equations.

• This renders the linear system.

• The linearity of the problem will be a major simplification.

i

j

j

i

jij

ij

i

x

v

x

v

xx

p

Frk

x

vv

t

v*

*

*

*

**

*

*

**

*

*

ReReˆRe

It is tempting to say that the smallness of Re means that theleft-hand side of the first equation can be neglected.

This leads to the reduced (linear) system

ij

i

x

p

Frk

x

v*

*

2*

*2

ReReˆ

i

j

j

i

jij

ij

i

x

v

x

v

xx

p

Frk

x

vv

t

v*

*

*

*

**

*

*

**

*

*

ReReˆRe

ii

j

j

i

j x

p

Frk

x

v

x

v

x *

*

*

*

*

*

*Re

Reˆ

Modification of Non-Dimensional Terms

• Re define the dimensionless pressure as pL/(μU) instead of p/(U2).

ij

i

x

Up

UL

Frk

x

v*

2

2*

*2 Reˆ

ij

i

x

UpUL

Frk

x

v*

2

2*

*2 Reˆ

ij

i

x

UpL

Frk

x

v*2*

*2 Reˆ

ij

i

x

p

Frk

x

v*

*

2*

*2 Reˆ

Stokes Flow

• The basic assumption of creeping flow was developed by Stokes (1851) in a seminal paper.

• This states that density (inertia) terms are negligible in the momentum equation.

• Under non-gravitational field.

ij

i

x

p

x

v*

*

2*

*2

ij

i

x

p

x

v

2

2

In dimensional form

pv 2 with 0 v