Conv_hw1 - o The resulting transient velocity distribution...

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EML 6155 Homework 1 1) Consider a fluid element in a cartesian coordinate system with some property α where α = α (x,y,z,t), prove that D Dt α is equivalent to d dt in a frame of reference moving with the fluid element. 2) Derive the following equation, known as the kinetic energy equation for compressible flow, from the momentum equation in two-dimensional flow: 3) Derive the continuity equation for spherical coordinates by applying the conservation of mass to a differential control volume. Note the velocity vector is: r r V v e v e v e θ φ φ = + + r r r r 4) An infinitely large plate is suddenly moved parallel to its surface with a velocity U
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Unformatted text preview: o . The resulting transient velocity distribution of the surrounding incompressible fluid is given by 2 2 1 u Uo e d η π- =- ∫ where ( , ) 2 y y t t ν = Here t is time, y is the vertical coordinate, and ν is the kinematic viscosity. Streamlines are parallel to the plate. Determine the two components of total acceleration in the respective x and y directions , Du Dv Dt Dt x U o y yx xy yy xx 2 2 D 1 1 p p u v u v { ( + )}= -(u +v )+ ( + )+ ( + )+ X + Y u v Dt 2 x y x y x y τ ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂...
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