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Fluid Flow 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 dz p dV dx dy p p dx dydz dV x x dFx p p p dF xˆ dV yˆ dV zˆ dV x y z dF pdV Equation of Motion A fluid element may be subject to an external force. • Write as a force density • Assume uniform over small element. dF fdV dv dm dF dt x dv F 2k l x l l x dV fdV pdV dt dv p f dt dv P f dt 2 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. 2 2 2 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. dv 1 f p dt 1 v f v v p t 1 v f v v P t Momentum Conservation dv dV pdV fdV dt d v dV f p dV dt v dV • Euler equation with force density • Mass is constant d v dV fdV pdV dt V V V d v dV fdV nˆpdS dt V V S The momentum is found for a small volume. Momentum is not generally constant. • Effect of pressure The total momentum change is found by integration. • Gauss’ law Energy Conservation The kinetic energy is related to the momentum. d dt • Right side is energy density 2 2 2 • Total time derivative • Volume change related to velocity divergence 1 2 v dV v f p dV 2 d pdV dp dV p ddV dt dt dt x p F 2k l x l l x v pdV p v dV t v pdV Some change in energy is related to pressure and volume. d dt 1 2 2 d pdV p p v dV dt t p v dV pdV v fdV p v dV t 2 Work Supplied The work supplied by expansion depends on pressure. dW ddV p p v dV dt dt • Potential energy associated with change in volume dW d d udm udV dt dt dt F 2k l 2 x 2 l x l x d p v dV udV dt This potential energy change goes into the energy conservation equation. d dt 1 2 2 2 p v dV pdV udV v fdV t 2 Bernoulli’s Equation f gzˆ Gravity is an external force. • Gradient of potential • No time dependence v fdV v dV d dV dt t d dV dV dt t The result is Bernoulli’s equation. • Steady flow no time change • Integrate to a constant d 1 2 p dV dV 2 v dV pdV dV udV dt t t 2 1 p d 1 2 p v p 2 v u gz u k dt 2 t t next