Navier-Stokes

1
Navier-Stokes
2
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
3
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 Cauchys equation.

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Eulers 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 Eulers equation of motion for a fluid.

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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
6
Strain Rate Tensor

• Rate of strain measures the amount of deformation
  in response to a stress.
• Forms symmetric tensor
• Based on the velocity gradient

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Stress and Strain

• There is a general relation between stress and
  strain
• Constants a, b include viscosity
• An incompressible fluid has no velocity
  divergence.

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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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