1. Chapter 4 Fluid Mechanics Frank White
DERIVATION & SOLUTION METHODS FOR THE STEADY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS
Chapter 4
Fluid Mechanics
Frank White

2. INTRODUCTION [2]
In fluid mechanics, the Navier-Stokes equations are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases. For example: they govern the movement of air in the atmosphere, ocean currents, water flow in a pipe, as well as many other fluid flow phenomena.
The equations are derived by considering the mass, momentum and energy balances for an infinitesimal control volume. The variables to be solved for are the velocity components and pressure. The flow is assumed to be differentiable and continuous. The equations can be converted to equations for the secondary variables vorticity and stream function. Solution depends on the fluid properties viscosity and density and on the boundary conditions of the domain of study.

3. DERIVATION OF THE NAVIER STOKES EQUATIONS [1]
CONSERVATION OF MOMENTUM
Newton’s second law:
where F= applied force on a particle.
m= mass of the particle.
a= acceleration due to force.
Divide eq.(1) by the volume of the particle,
where f = applied force per unit volume on the fluid particle.
= body force
= surface force
= velocity of the fluid particle

4. DERIVATION OF THE NAVIER STOKES EQUATIONS [1] (continued)

5. DERIVATION OF NAVIER STOKES EQUATIONS [1] (continued)
The stress tensor can be represented as follows fig.(2.1):
where = stress in the j direction on a face normal to the i axis
The total force in each direction due to stress is:

6. DERIVATION OF NAVIER STOKES EQUATIONS [1] (continued)
For an equilibrium element,
Net force on the element in the x-direction,
or on a unit volume basis, dividing by , since
Thus the total vector surface force is

7. DERIVATION OF NAVIER STOKES EQUATIONS [1] (continued)
Conservation of momentum equation, now becomes
=density of the particle
=vector acceleration of gravity
For the fluid at rest,
where = hydrostatic pressure

8. DERIVATION OF NAVIER STOKES EQUATIONS [1] (continued)
Deformation Law:
Stokes three postulates are:
The stress tensor is at most a linear function of the strain rates
The fluid is isotropic;
When the strain rates are zero, , where =
Let be the principal axes, the deformation law could involve at most 3 linear coefficients,
The term is added to satisfy the hydrostatic (postulate 3 above).

9. DERIVATION OF NAVIER STOKES EQUATIONS [1] (continued)
From isotropic postulate 2 the cross flow effect of and be identical, i.e., that Therefore eq.(11) reduces to:
where for convenience. Note also that
equals
Transforming eq.(12) to some arbitrary axes where shear stresses are not zero and thereby find an expression for the general deformation law.
With respect to the principal axes let the axis have direction
cosines let the axis have the direction cosines and let axis have
for any set of direction cosines, then the transformation rule between a normal stress or strain rate in the new system and the principal stresses or strain rates is:

10. DERIVATION OF NAVIER STOKES EQUATIONS [1] (continued)
Similarly, the shear stresses (strain rates) are related to the principal stresses (strain rates) by the following transformation law:
Eliminating etc., from eq(13) by using the principal axis deformation law, eq(12), and the fact that
The result is:
where K= 2
is called the Lame’s constant and is given by the symbol
with exactly similar expressions for and Similarly, we can eliminate etc. from eqs.(14) to give
and exactly analogous expressions for and

11. DERIVATION OF NAVIER STOKES EQUATIONS [1] (continued)
Eqs.(15), (16) can be combined, using the initial notation, and rewritten into a single general deformation law for a Newtonian (linear) viscous fluid which is given in the next slide.
Deformation law for a Newtonian (linear) viscous fluid:

12. DERIVATION OF NAVIER STOKES EQUATIONS [1] (continued)
Substituting eqs.(17) into eq.(9) we get the desired Navier-stokes equations in the index notation.
INCOMPRESSIBLE FLOW
If the fluid is assumed to be of constant density, the Navier-Stokes equation reduces to:
Thus the Navier-Stokes equation is derived.

13. BOUNDARY CONDITIONS IN FLUID-FLOW:

14. SOLUTIONS OF THE NAVIER-STOKES INCOMPRESSIBLE FLOW [1]
Basically, there are two types of exact solution of Navier-stokes equations.
Linear solutions, where the convective acceleration V• vanishes.
Nonlinear solutions, where V• does not vanish.
It is also possible to classify solutions by the type of geometry of flow involved:
Couette (wall driven) steady flows.
Poiseuille (pressure driven) steady duct flows.
Unsteady duct flows
Unsteady flows with moving boundaries.
Duct flows with suction and injection.
Wind-driven (Ekman) flows.
Similarity solutions (rotating disk, stagnation flow, etc)
In this lecture, Couette (wall driven) steady flow between a fixed and a moving plate is discussed.

15. SOLUTIONS OF THE NAVIER-STOKES INCOMPRESSIBLE FLOW [1] (continued)
Almost all the known particular solutions are for the case of incompressible Newtonian flow with constant transport properties for which the equations reduce to:
Momentum:

16. SOLUTIONS OF THE NAVIER-STOKES INCOMPRESSIBLE FLOW [1] (continued)
COUETTE FLOWS: These flows are named in honor of M. Couette, who performed experiments on the flow between a fixed and moving concentric cylinder.

17. SOLUTIONS OF THE NAVIER-STOKES INCOMPRESSIBLE FLOW [1] (continued)
Two infinite plates are 2h apart, and the upper plate moves at a speed U relative to the lower. The pressure p is assumed constant.
The upper plate is held at temperature T1 and the lower plate at T0. These boundary conditions are independent of x or z (“ infinite plates”)
Equation (20) reduce to
Momentum:

18. SOLUTIONS OF THE NAVIER-STOKES INCOMPRESSIBLE FLOW [1] (continued)
Eq. (21)can be integrated twice to obtain
The boundary conditions are no slip, and ,
whence and Then the velocity distribution is
The shear stress at any point in the flow follows from the viscosity law:
Thus for this simple flow the shear stress is constant throughout the fluid, as is the strain rate.

19. EXAMPLE ON NAVIER-STOKES EQUATIONS
Problem 2:

20. CONCLUSIONS:
The Navier-Stokes equations are derived by considering the mass, momentum and energy balances for an infinitesimal control volume.
Boundary conditions in fluid flow need to be understood to obtain the solutions of the equations.
The solution of steady flow incompressible Navier-Stokes equation is discussed. It has been concluded that for the simple flow the shear stress is constant throughout the fluid, as is the strain rate.

21. EXAMPLE PROBLEMS ON NAVIER-STOKES EQUATIONS

22. REFERENCES:
[1] White, Frank M., Viscous Fluid Flow, McGraw-Hill Series in Mechanical Engineering, 2nd edition 2000.
[2] White, F.M., Fluid Mechanics, 5th edition, McGraw-Hill 2003.