Fluid Flows due to Pure Mechanical Forces…

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Construction of Navier-Stokes Equations

Momentum Conservation Equations for Fluid Flows

tV

ij

tVtV

dVdVgdVvvt

v

.

ijgvvt

v

.

Localized Action & Reaction in a fluid Flow

Stress tensor is the more fundamental quantity, characterizing the fluid response

to an imposed deformation.

Relationship between Stress Tensor and Deformation Tensor

• The surface forces resulting from the stress tensor causes a deformation of fluid particles.

• An attempt to find a functional relationship between the stress tensor and the velocity gradient is an essential hypothesis to be invented !!!

vfij

TT vvvvv

2

1

2

1

ijijDv

ijijv

ijijij f ,

jiijjijiijjiij vveevveef ,,,, 2

1,

2

1

Deformation Law for a Newtonian Fluid Flow

• By analogy with hookean elasticity, the simplest assumption for the variation of viscous stress with strain rate is a linear law.

• These considerations were first made by Stokes (1845).

• The deformation law is satisfied by all gases and most common fluids.

• Stokes' three postulates are:

• 1. The fluid is continuous, and its stress tensor ij is at most a linear function of the strain rates.

• 2. The fluid is isotropic; i.e., its properties are independent of direction, and therefore the deformation law is independent of the coordinate axes in which it is expressed.

• 3. When the strain rates are zero, the deformation law must reduce to the hydrostatic pressure condition, ij = -pij.

Discussion of Stokes 2nd Postulates

• The fluid is isotropic; i.e., its properties are independent of direction, and therefore the deformation law is independent of the coordinate axes in which it is expressed.

• The isotropic condition requires that the principal stress axes be identical with the principal strain-rates ().

333231

232221

131211

ij

333231

232221

131211

ij

The Gradient of Velocity Vector

Tvv

2

1

j

i

i

jij x

v

x

v

2

1

These velocity gradients are used to construct strain-rates ().

z

w

z

v

z

u

y

w

y

v

y

ux

w

x

v

x

u

V

Intensive Nature of Stress : Invariants of Strain Tensor

3322111 I

231

223

2121133332222112 I

333231

232221

131211

3

I

Based on the transformation laws of symmetric tensors, there are three invariants which are independent of direction or choice of axes:

Combined Analysis of Stokes Postulate & Tensor Analysis

• As a rule the principal stress axes be identical with the principal strain-rate axes.

• This makes the principal planes a convenient place to begin the deformation-law derivation.

• Let x1, x2, and x3, be the principal axes, where the shear stresses and shear strain rates vanish.

• With these axes, the deformation law could involve at most three linear coefficients, C1, C2, C3.

Principal Stresses

333222111 CCCpii

•The term -p is added to satisfy the hydrostatic condition (Postulate 3).•But the isotropic condition 2 requires that the crossflow effect of 22 and 33 must be identical.

•Implies that C2 = C3.Therefore there are really only two independent linear coefficients in an isotropic Newtonian fluid.Above equation can be simplified as:

332211211 CKpii where K = C1 - C2

vCKpii

.211

33222111 CCpii

General Deformation Law

• Now let us transform Principle axes equation to some arbitrary axes, where shear stresses are not zero.

• Let these general axes be x,y,z.

• Thereby find an expression for the general deformation law.

• The transformation requires direction cosines with respect to each principle axes to general axes.

• Then the transformation rule between a normal stress or strain rate in the new system and the principal stresses or strain rates is given by,

2133

2122

2111 nmlxx

2133

2122

2111 nmlxx

0.121

21

21 nml

Shear Stresses along General Axes

• Similarly, the shear stresses (strain rates) are related to the principal stresses (strain rates) by the following transformation law:

213321222111 nnmmllxy

213321222111 nnmmllxy

These stress and strain components must obey stokes law, and hence

vCKp xxxx

.2

xyxy K

Note that the all direction cosines will politely vanished.