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# sol5 - Professor Philip S Marcus ME106 Solutions Problem...

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10/09/09 Professor Philip S. Marcus ME106 Solutions - Problem Set 5 Due in class Friday October 16, 2009 Solutions are intermingled with the lecture/problems and appear in bold font or mathematics font. Note that there was a typo in the part of problem O1 where you were asked to find the pressure at the origin! There is a piece of the statement of the problem (written in italics in the paragraph just before equation (2)) that got dropped in the final posted version. Without the piece in italics, you do not have enough information to solve for the pressure for all time. Some of you must have spend a lot of time worrying about this, so as a consolation –THERE WILL BE NO “O” PROBLEMS IN THE COMING HOMEWORK ASSIGNMENT . O1) Use Matlab to draw the streamlines at time t = 2 (by setting dy/dx = v y /v x ) of the unsteady velocity v ≡ ∇ ψ , where ψ = cos (3 t ) { ln [( x 1) 2 + y 2 ] ln [( x + 1) 2 + y 2 ] } . (1) All quantities have units of the MKS system. 1

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See plot at end of this set. Note that the entire velocity field oscillates in time as cos (3 t ). What is the vorticity ω = ∇ × v ? (Before you begin differentiating, remember that the curl of a gradient is zero.) Because the velocity is a gradient of a scalar, the curl of this velocity is identically zero. If the pressure at t = 0, x = 1 and y = 2 is 12, and the density of the fluid is 3 (for all locations and time), and if the pressure at infinity, that is, at | x | → ∞ and | y | → ∞ is constant in time , find a closed-form expression for the pressure at the origin as a function of time. Remember that Euler’s equation can be written in terms of the vorticity as v /∂t = v × ω − ∇ ( v 2 / 2) ( P ) g ˆ z (2) In the case where v ≡ ∇ ψ and where ρ and g are constant, we can move everything to the left-hand side of the equation and inside the gradient to write Euler’s equation as: ( ∂ψ/∂t + v 2 / 2 + P/ρ + gz ) = 0 . (3) What happened to the v × ω term? How does equation (3) relate to Bernoulli’s theorem? Is the quantity ( ∂ψ/∂t + v 2 / 2 + P/ρ + gz ) analogous to the Bernoulli function that we have used for more general flows? How does this new quantity vary along streamlines? How does it vary across streamlines? 2
Note that, in this case, the v × ω term vanishes from equation (2) because ω 0. Equation (3) tells us that ( ∂ψ/∂t + v 2 / 2+ P/ρ + gz ) does not vary in location, but can be a function of time. That is, for each instant in time t , ( ∂ψ/∂t + v 2 / 2 + P/ρ + gz ) = C ( t ), where C ( t ) is a function of time only. Note that ( ∂ψ/∂t + v 2 / 2 + P/ρ + gz ) is the same along streamlines, across streamlines, and everywhere. Because we are to compute the pressure, let’s find C ( t ). We start by evaluating C ( t ) at t = 0 by using the known value of the pressure at the point x = 1, y = 2 and z = 0. Because the flow is 2D, and z never appears in the description of the velocity, we can arbitrarily set z = 0. Note that at x = 1, y = 2 and z = 0 that v ( 1 , 2 , 0 ,t ) = ( 1 / 2) cos (3 t ) (ˆ x + ˆ y ) and [ v ( 1 , 2 , 0 ,t )] 2 / 2 = ( cos 3 t ) 2 / 4.

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sol5 - Professor Philip S Marcus ME106 Solutions Problem...

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