Navier-Stokes

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# Navier-Stokes - PowerPoint PPT Presentation

Navier-Stokes. Pressure Force. Each volume element in a fluid is subject to force due to pressure. Assume a rectangular box Pressure force density is the gradient of pressure. d V. d z. d y. p. d x. Equation of Motion. A fluid element may be subject to an external force.

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## Navier-Stokes

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Presentation Transcript

### Navier-Stokes

Pressure Force
• Each volume element in a fluid is subject to force due to pressure.
• Assume a rectangular box
• Pressure force density is the gradient of pressure

dV

dz

dy

p

dx

Equation of Motion
• A fluid element may be subject to an external force.
• Write as a force density
• Assume uniform over small element.
• The equation of motion uses pressure and external force.
• Write form as force density
• Use stress tensor instead of pressure force
• This is Cauchy’s equation.
Divide by the density.

Motion in units of force density per unit mass.

The time derivative can be expanded to give a partial differential equation.

Pressure or stress tensor

This is Euler’s equation of motion for a fluid.

Euler’s Equation
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

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.

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