Problem 5.43 [Difficulty: 2] Given: X component of a 2-dimensional transient flow. Find: Y component of flow and total acceleration. Solution: Governing Equations: 0 z w y v x u (Continuity Equation for an Incompressible Fluid) t V z V w y V v x V u Dt V D a p (Material Derivative) Assumptions: Incompressible fluid No motion along the wall ( x = 0) limited to two dimensions ( w = 0). The given or available data is: T t Ax u 2 sin0 w Simplify the continuity equation to find v: T t A x u y v 2 sin Integrate: C T t Ay v 2 sin Use the boundary condition of no flow at the origin to solve for the constant of integration Give the velocity in vector form: j y i x T t A V ˆ ˆ 2 sin Use the material derivative to find the acceleration. Start with the convective terms. y V v x V u a conv p
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