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Unformatted text preview: Fluid Dynamics 3 2011/12 Sheet 2 Homework to be handed in Friday 28th October: 1,3,5. 1. The Euler equation is ∂ u ∂t + ( u · ∇ ) u = 1 ρ ∇ p, where u ( x ,t ) is the velocity and p ( x ,t ) the pressure. Consider the coordinate transforma tion x = x V t , where V is a constant vector. What does this correspond to physically? What are the velocity and pressure fields u ( x ,t ) and p ( x ,t ) in the new coordinate system? Transform the Euler equation to the new variables u and p . 2. A steady twodimensional flow is given by u = αx , v = αy with α a constant. (i) Find the equation for a general streamline of the flow, and sketch a few streamlines. (ii) At t = 0, the fluid on the curve x 2 + y 2 = a 2 is marked with dye. Find the equation which governs the shape of the dyed fluid for t > 0. (iii) Does the area enclosed within the dyed region change over time? Is there a simple reason for this ?...
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 Fall '11
 Eggers
 Linear Algebra, Geographic coordinate system, Coordinate system, Spherical coordinate system, Polar coordinate system, Coordinate systems

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