Construction of Navier-Stokes Equations

1. Construction of Navier-Stokes Equations P M V Subbarao Professor Mechanical Engineering Department I I T Delhi A True Model for Analysis of Real Fluid devices……

2. Differential Form of Momentum Conservation Equations for Fluid Flows Newton’s Second law to a CV:

3. Fluid Mechanics as A Science • The knowledge of fluid flow cannot be called a discipline of science until Isaac Newton published his famous work Philosophiae Naturalis Principia Mathematica in 1687. • Newton stated in his principia that, for straight, parallel and uniform flow, the shear stress , between layers is proportional to the velocity gradient in the direction perpendicular to the layers. Newton’s Law of viscosity is the most Economic solution to highly complex truth.

4. Fluid Mecahnics as A Mathematics • With the development of calculus, many problems were solved in the frame of ideal fluid or inviscid fluid. • In 1738, Bernoulli proved that, the gradient of pressure is proportional to the acceleration. • Later, the famous differential equations, known as Euler’s equations, were derived by Euler.

5. Euler’s Equation • In a famous memoir of 1755, Euler obtained the equations of motion. • By equating the product of mass and acceleration for a cubic element of the fluid to the resultant of the pressures and external forces acting on and in this element. • In our notation this gives Unfortunately, this splendid theory led to absurd results when applied to realproblems of fluid resistance and flow retardation. Many researchers tried to add a friction term into Euler’s differential equations.

6. The First Five Births of the Navier-Stokes Equation • The Navier-Stokes equations are very complex and the behavior of its solutions are unpredictable. • The early life of this equation was as fleeting as the foam on a wave crest. • Navier’s original proof of 1822 was not influential, and the equation was rediscovered or re-derived at least four times, by Cauchy in 1823, by Poisson in 1829, by Saint-Venant in 1837, and by Stokes in 1845. • Each new discoverer either ignored or denigrated his redecessors’ contribution. • Each had his own way to justify the equation. • Each judged differently the kind of motion and the nature of the system to which it applied.

7. The Greatest Dispute !!!! • Navier started with molecular forces, but quickly jumped to the macroscopic level by considering virtual works. • Navier’s short-cuts from the molecular to the macroscopic levels seemed arbitrary or even contradictory. • Saint-Venant insisted that a clear definition of the concept of stress could only be molecular. • Venant justified the linearity of the stresses with respect to deformations by reasoning on hard-sphere molecules. • Cauchy and Poisson simply ignored Navier’s contribution to fluid dynamics. • Saint-Venant and Stokes both gave credit to Navier for the equation, but believed an alternative derivation to be necessary.

8. The Greatest Agreement • With very few exceptions, the Navier-Stokes equations provide an excellent model for both laminar and turbulent flows. • The anticipated paradigm shift in fluid mechanics centers around the ability today as well as tomorrow of computers to numerically integrate those equations. • We therefore need to recall (Realize) the birth of equations of fluid motion in their entirety.

9. 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  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.

10. 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 ().

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

12. 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:

13. 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.

14. 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 22and 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: where K = C1 - C2

15. 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,

16. 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.

17. Philosophy of Science • The goal which physical science has set itself is the simplest and most economical abstract expression of facts. • The human mind, with its limited powers, attempts to mirror in itself the rich life of the world, of which it itself is only a small part……. • In reality, the law always contains less than the fact itself. • A Law does not reproduce the fact as a whole but only in that aspect of it which is important for us, the rest being intentionally or from necessity omitted.

18. Newtonian (linear) Viscous Fluid: • Compare stokes equations with Newton’s Law of viscosity. The linear coefficient K is equal to twice the ordinary coefficient of viscosity, K = 2.

19. A virtual Viscosity • The coefficient C2, is new and independent of and may be called the second coefficient of viscosity. • In linear elasticity, C2, is called Lame's constant and is given the symbol , which is adopted here also. • Since  is associated only with volume expansion, it is customary to call it as the coefficient of bulk viscosity .

20. General deformation law for a Newtonian (linear)viscous fluid: This deformation law was first given by Stokes (1845).