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Navier-Stokes

Navier-Stokes. Viscosity. A static fluid cannot support a shear. A moving fluid with viscosity can have shear. Dynamic viscosity m Kinematic viscosity n. F. v x. y. Pipe. Steady flow in a pipe involves viscosity. The forces must be balanced. z. a. l.

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Navier-Stokes

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  1. Navier-Stokes

  2. 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

  3. Pipe • Steady flow in a pipe involves viscosity. • The forces must be balanced. z a l

  4. The flow in the pipe can be integrated. Zero velocity at the wall Another integration gives the total current. Poiseuille’s Law Poiseuille’s Law

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

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

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

  8. Make assumptions about flow to approximate fluid motion. Incompressible Inviscid Irrotational Force from gravity Apply to Navier-Stokes The result is Bernoulli’s equation. Bernoulli Rederived

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