DERIVATION

1
DERIVATION SOLUTION METHODS FOR THE STEADY
INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

• Vishalakshi Kuppa
• MEEN 5330 CONTINUUM MECHANICS

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
• Newtons 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 Lames 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

• Problem 1

22
REFERENCES

• 1 White, Frank M., Viscous Fluid Flow,
  McGraw-Hill Series in Mechanical Engineering,
  2nd edition 2000.
• 2 Doering, Charles R., Applied Analysis of
  the Navier-Stokes Equations, Cambridge Texts In
  Applied Mathematics, 2nd edition 2002.
• 3 White, F.M., Fluid Mechanics, 5th
  edition, McGraw-Hill 2003.