Chapter6_Potential_Flow - 57:020 Mechanics of Fluids and...

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57:020 Mechanics of Fluids and Transport Processes Chapter 6 Professor Fred Stern Fall 2006 1 1 Chapter 6 Differential Analysis of Fluid Flow Inviscid flow: Euler’s equations of motion Flow fields in which the shearing stresses are zero are said to be inviscid, nonviscous, or frictionless. for fluids in which there are no shearing stresses the normal stress at a point is independent of direction: x xy yz z p σ σσ −= = = For an inviscid flow in which all the shearing stresses are zero, and the normal stresses are replaced by p, the Navier-Stokes Equations reduce to Euler’s equations () gp t ρρ −∇ = + ⋅∇ V VV In Cartesian coordinates: x p uuu u gu v w xt x y z ⎛⎞ ∂∂ + + + ⎜⎟ ⎝⎠ y p vvv v v w yt x y z + + + z p www w v w zt x y z + + + The Bernoulli equation derived from Euler’s equations The Bernoulli equation can also be derived, starting from Euler’s equations. For inviscid, incompressible fluids, we end up with the same equation 2 2 pV gz const ρ ++ =
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57:020 Mechanics of Fluids and Transport Processes Chapter 6 Professor Fred Stern Fall 2006 2 2 It is often convenient to write the Bernoulli equation between two points (1) and (2) along a streamline and to express the equation in the “head” form by dividing each term by g so that 22 11 2 2 12 pV p V zz gg γγ ++ =++ The Bernoulli equation is restricted to the following: inviscid flow steady flow incompressible flow flow along a streamline The Irrotational Flow and corresponding Bernoulli equation If we make one additional assumption—that the flow is irrotational 0 ∇× = V —the analysis of inviscid flow problems is further simplified. The Bernoulli equation has exactly the same form at that for inviscid flows: 2 2 p V
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Chapter6_Potential_Flow - 57:020 Mechanics of Fluids and...

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