2. Fluid Flows due to Pure Mechanical Forces… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations

3. Momentum Conservation Equations for Fluid Flows

4. Localized Action & Reaction in a fluid Flow Stress tensor is the more fundamental quantity, characterizing the fluid response to an imposed deformation.

5. 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 !!!

6. 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 = -p  ij.

7. Discussion of Stokes 2 nd 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 (  ).

8. The Gradient of Velocity Vector These velocity gradients are used to construct strain-rates (  ).

9. Intensive Nature of Stress : Invariants of Strain Tensor Based on the transformation laws of symmetric tensors, there are three invariants which are independent of direction or choice of axes:

10. 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 x 1, x 2, and x 3, 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, C 1, C 2, C 3.

11. Principal Stresses 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 C 2 = C 3. Therefore there are really only two independent linear coefficients in an isotropic Newtonian fluid. Above equation can be simplified as: where K = C 1 - C 2

12. 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,

13. Shear Stresses along General Axes Similarly, the shear stresses (strain rates) are related to the principal stresses (strain rates) by the following transformation law: These stress and strain components must obey stokes law, and hence Note that the all direction cosines will politely vanished.