Navier stokes
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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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Navier-Stokes

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Navier stokes

Navier-Stokes


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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