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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

• 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.

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

• 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

Constants a, b include viscosity

An incompressible fluid has no velocity divergence.

Stress and Strain

• The stress and strain relations can be combined with the equation of motion.

• Reduces to Euler for no viscosity.

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