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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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Presentation Transcript
pressure force
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
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.
euler 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
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
Strain Rate Tensor
  • Rate of strain measures the amount of deformation in response to a stress.
    • Forms symmetric tensor
    • Based on the velocity gradient
stress and strain
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
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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