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Navier-Stokes

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Navier-Stokes

- 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

- 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.

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

F

vx

y

- 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.

- The stress and strain relations can be combined with the equation of motion.
- Reduces to Euler for no viscosity.

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