1. Study of Navier-Stokes Equations
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Understand the Mathematical Behaviour of Real Flows……

2. General deformation law for a Newtonian (linear) viscous fluid:
This deformation law was first given by Stokes (1845).
Initial form of N-S equations:

3. Thermodynamic Pressure Vs Mechanical Pressure
Stokes (1845) pointed out an interesting consequence of this general Equation.
By analogy with the strain relation, the sum of the three normal stresses xx , yy and zz is a tensor invariant.
Define the mechanical pressure as the negative one-third of this sum.
Mechanical pressure is the average compression stress on the element.

4. Stokes Hypothesis
The mean pressure in a deforming viscous fluid is not equal to the thermodynamic property called pressure.
This distinction is rarely important, since v is usually very small in typical flow problems.
But the exact meaning of mechanical pressure has been a controversial subject for more than a century.
Stokes himself simplified and resolved the issue by an assumption:
Above equation leads to
This relation, frequently called the Stokes’ relation,.
This is truly valid for monoatomic gases

5. The Controversy Stokes hypothesis simply assumes away the problem.
This is essentially what we do in this course.
The available experimental evidence from the measurement of sound wave attenuation, indicates that  for most liquids is actually positive.
 is not equal to -2/3, and often is much larger than .
The experiments themselves are a matter of some controversy.

6. Incompressible Flows Again this merely assumes away the problem.
The bulk viscosity cannot affect a truly incompressible fluid.
In fact it does affect certain phenomena occurring in nearly incompressible fluids, e.g., sound absorption in liquids.
Meanwhile, if .v0, that is, compressible flow, we may still be able to avoid the problem if viscous normal stresses are negligible.
This is the case in boundary-layer flows of compressible fluids, for which only the first coefficient of viscosity  is important.
However, the normal shock wave is a case where the coefficient  cannot be neglected.
The second case is the above-mentioned problem of sound-wave absorption and attenuation.

7. Bulk Viscosity Coefficient
The second viscosity coefficient is still a controversial quantity.
Truly saying,  may not even be a thermodynamic property, since it is found to be frequency-dependent.
The disputed term, divv, is almost always so very small that it is entirely proper simply to ignore the effect of  altogether.
Collect more discussions on Births of N-S Equations & Bulk Viscosity prepare a report : Date of submission: 29tthSeptember 2016.

8. The Navier-Stokes Equations
The desired momentum equation for a general linear (newtonian) viscous fluid is now obtained by substituting the stress relations, into Newton's law.
The result is the famous equation of motion which bears the names of Navier (1823) and Stokes (1845).
In scalar form, we obtain

9. Tensor Notation for Fluid Flow Analysis
The tensor Notation is a powerful tool that enables the reader to study and to understand more effectively the fundamentals of fluid mechanics.
Enables to recall all conservation laws of fluid mechanics without memorizing any single equation.
All the quantities encountered in fluid dynamics are redefined as tensors.

10. Einstein Notation : 1916
Range convention: Whenever a subscript appears only once in a term, the subscript takes all possible values. E.g. in 3D space:
Summation convention: Whenever a subscript appears twice in the same term the repeated index is summed over the index parameter space. E.g. in 3D space:

11. Scalar Product : Work & Energy
Scalar or dot product of two vectors results in a scalar quantity .
Apply the Einstein's summation convention to work or energy scalars.
Rearrange the unit vectors and the components separately:

12. Kronecker delta
In Cartesian coordinate system, the scalar product of two unit vectors is called Kronecker delta, which is:
Using the Kronecker delta,

13. Vector or Cross Product : Creation of Torque
The vector product of two vectors is a vector that is perpendicular to the plane described by those two vectors.
Apply the index notation
With ijk as the permutation symbol with the following definition

14. Using the above definition, the vector product is given by:

15. Scalar Triple product
For every three vectors A, B and C, combination of dot and cross products gives

16. Non repeated subscripts
Non repeated subscripts remain fixed during the summation. E.g. in 3D space
one for each i = 1, 2, 3 and j is the dummy index.