1. Fluid Flow

2. Pressure Force  Each volume element in a fluid is subject to force due to pressure. Assume a rectangular boxAssume a rectangular box Pressure force density is the gradient of pressurePressure force density is the gradient of pressure VV p xx yy zz

3. Equation of Motion  A fluid element may be subject to an external force. Write as a force densityWrite as a force density Assume uniform over small element.Assume uniform over small element.  The equation of motion uses pressure and external force. Write form as force densityWrite form as force density Use stress tensor instead of pressure forceUse stress tensor instead of pressure force  This is Cauchy’s equation.

4. Euler’s Equation  Divide by the density. Motion in units of force density per unit mass.Motion in units of force density per unit mass.  The time derivative can be expanded to give a partial differential equation. Pressure or stress tensorPressure or stress tensor  This is Euler’s equation of motion for a fluid.

5. Momentum Conservation  The momentum is found for a small volume. Euler equation with force density Mass is constant  Momentum is not generally constant. Effect of pressure  The total momentum change is found by integration. Gauss’ law

6. Energy Conservation  The kinetic energy is related to the momentum. Right side is energy densityRight side is energy density  Some change in energy is related to pressure and volume. Total time derivativeTotal time derivative Volume change related to velocity divergenceVolume change related to velocity divergence

7. Work Supplied  The work supplied by expansion depends on pressure. Potential energy associated with change in volumePotential energy associated with change in volume  This potential energy change goes into the energy conservation equation.

8. Bernoulli’s Equation  Gravity is an external force. Gradient of potential No time dependence  The result is Bernoulli’s equation. Steady flow no time change Integrate to a constant next