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newassign2 - FLUID MECHANICS II Assignment 2 Question 1 a...

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FLUID MECHANICS II Assignment 2 Question 1 a) By following the motion of a fluid parcel in a two-dimensional, planar flow (parcel moves from (x,y) to (x+Δ x, y+Δ y) in time Δ t), show that Lagrangian and Eulerian acceleration in 2 D ( ~a = a x ˆ i + a y ˆ j ) are related by a x = Du ( t ) Dt = ∂u ∂t + u ∂u ∂x + v ∂u ∂y , a y = Dv ( t ) Dt = ∂v ∂t + u ∂v ∂x + v ∂v ∂y , where u, v on RHS are the Eulerian velocities, i.e. ~u ( x, y, t ) = u ( x, y, t ) ˆ i + v ( x, y, t ) ˆ j. b) Work out the corresponding connection between the parcel rate of change of temperature DT ( t ) Dt and the Eulerian temperature field T(x,y,t). Question 2 By expanding out the vector equality ~a = D~u ( t ) Dt = ∂~u ∂t + ( ~u. ~ ) ~u, where ~ = ∂x ˆ i + ∂y ˆ j + ∂z ˆ k work out the three-dimensional equivalents of 1(a); i.e. find a x , a y and a z in terms of Eulerian quantities.
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