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Navier-Stokes

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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  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 Cauchy’s equation.

  4. 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

  5. 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

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

  8. Navier-Stokes Equation • The stress and strain relations can be combined with the equation of motion. • Reduces to Euler for no viscosity. next

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