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Unformatted text preview: Excerpt for Chapter II, Handout 4 from Dr. L. Sankar Modified by Dr. S.M. Ruffin Bernoulli Eqution : The u-, v- and w- momentum equations are nonlinear PDEs and are hard to solve. Fortunately, for inviscid flows without body forces, these equations yield a simple algebraic equation linking pressure and velocity. We derive this simple equation below. We start with the u- momentum equation. It has many forms. But the form that we use is: 1 = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ x p z u w y u v x u u ρ (3) This form is valid for steady, inviscid flows with no body forces. It is valid for compressible and incompressible flows. Multiplying this equation by dx we get: 1 = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ dx x p dx z u w dx y u v dx x u u ρ Recall that for a streamline, , = − = − = − = = × wdy vdz wdx udz vdx udy Or dz dy dx w v u k j i s d V r r r r r (1) Along a streamline, according to equation (1), udy=vdx, udz=wdx. The above momentum equation becomes: 1 = ∂ ∂ + ∂ ∂...
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- Summer '07