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p1132 - 11.32 CHAPTER 11 PROBLEM 32 41 11.32 Chapter 11...

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11.32. CHAPTER 11, PROBLEM 32 41 11.32 Chapter 11, Problem 32 Problem: A two-dimensional potential flow has streamfunction ψ ( r, θ ) = Jr 4 sin 4 θ where J is a constant of dimensions 1 / ( L 2 T ) . (a) Determine the velocity components u r ( r, θ ) and u θ ( r, θ ) . (b) Determine the corresponding velocity potential, φ ( r, θ ) . (c) Locate any stagnation points. Convert to Cartesian coordinates for the sake of clarity. (d) The body shape is given by ψ = 0 . If we confine our interest to the first quadrant excluding the y axis, i.e., 0 θ < π / 2 , what is the body shape? (e) Sketch a few streamlines. Include any stagnation points, body contours and flow direction along the streamlines. HINT: Find the flow direction on the body surface. Solution: (a) For the given streamfunction, ψ = Jr 4 sin 4 θ , the velocity components are u r = 1 r ∂ψ ∂θ = 4 Jr 3 cos 4 θ u θ = ∂ψ r = 4 Jr 3 sin 4 θ (b) Now, by definition of the velocity potential, we know that the velocity components are given by u r = ∂φ / r and u θ = (1 /r ) ∂φ / ∂θ . Thus, ∂φ r = u r
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