Navier-Stokes

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# Navier-Stokes - PowerPoint PPT Presentation

Navier-Stokes. Viscosity. A static fluid cannot support a shear. A moving fluid with viscosity can have shear. Dynamic viscosity m Kinematic viscosity n. F. v x. y. Pipe. Steady flow in a pipe involves viscosity. The forces must be balanced. z. a. l.

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### Navier-Stokes

Viscosity
• A static fluid cannot support a shear.
• A moving fluid with viscosity can have shear.
• Dynamic viscosity m
• Kinematic viscosity n

F

vx

y

Pipe
• Steady flow in a pipe involves viscosity.
• The forces must be balanced.

z

a

l

The flow in the pipe can be integrated.

Zero velocity at the wall

Another integration gives the total current.

Poiseuille’s Law

Poiseuille’s Law
Strain Rate Tensor
• Rate of strain measures the amount of deformation in response to a stress.
• Forms symmetric tensor
• Based on the velocity gradient
There is a general relation between stress and strain

Constants a, b include viscosity

An incompressible fluid has no velocity divergence.

Stress and Strain
Navier-Stokes Equation
• The stress and strain relations can be combined with the equation of motion.
• Reduces to Euler for no viscosity.
Make assumptions about flow to approximate fluid motion.

Incompressible

Inviscid

Irrotational

Force from gravity

Apply to Navier-Stokes

The result is Bernoulli’s equation.

Bernoulli Rederived