A Generalized Frame work Viscous Fluid Flow… P M V Subbarao Professor Mechanical Engineering...
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![Page 1: A Generalized Frame work Viscous Fluid Flow… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations.](https://reader036.fdocuments.in/reader036/viewer/2022082816/56649d6f5503460f94a51ae5/html5/thumbnails/1.jpg)
A Generalized Frame work Viscous Fluid Flow…
P M V SubbaraoProfessor
Mechanical Engineering Department
I I T Delhi
Construction of Navier-Stokes Equations
![Page 2: A Generalized Frame work Viscous Fluid Flow… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations.](https://reader036.fdocuments.in/reader036/viewer/2022082816/56649d6f5503460f94a51ae5/html5/thumbnails/2.jpg)
Deformation Law for a Newtonian Fluid
• 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.
![Page 3: A Generalized Frame work Viscous Fluid Flow… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations.](https://reader036.fdocuments.in/reader036/viewer/2022082816/56649d6f5503460f94a51ae5/html5/thumbnails/3.jpg)
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
![Page 4: A Generalized Frame work Viscous Fluid Flow… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations.](https://reader036.fdocuments.in/reader036/viewer/2022082816/56649d6f5503460f94a51ae5/html5/thumbnails/4.jpg)
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 ().
![Page 5: A Generalized Frame work Viscous Fluid Flow… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations.](https://reader036.fdocuments.in/reader036/viewer/2022082816/56649d6f5503460f94a51ae5/html5/thumbnails/5.jpg)
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:
![Page 6: A Generalized Frame work Viscous Fluid Flow… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations.](https://reader036.fdocuments.in/reader036/viewer/2022082816/56649d6f5503460f94a51ae5/html5/thumbnails/6.jpg)
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.
![Page 7: A Generalized Frame work Viscous Fluid Flow… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations.](https://reader036.fdocuments.in/reader036/viewer/2022082816/56649d6f5503460f94a51ae5/html5/thumbnails/7.jpg)
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
![Page 8: A Generalized Frame work Viscous Fluid Flow… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations.](https://reader036.fdocuments.in/reader036/viewer/2022082816/56649d6f5503460f94a51ae5/html5/thumbnails/8.jpg)
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
![Page 9: A Generalized Frame work Viscous Fluid Flow… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations.](https://reader036.fdocuments.in/reader036/viewer/2022082816/56649d6f5503460f94a51ae5/html5/thumbnails/9.jpg)
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.