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Stream function in polar coordinate

Stream function in polar coordinate. With. y = -v dx + u dy. + C = +. With. = - dr. + C= - dr + C. =. Source. Basic Flowfields 2- Source and Sink 2-1 Source. For the following flowfield. =. =. + C = - dr + C = - dr + C

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Stream function in polar coordinate

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  1. Stream function in polar coordinate With y = -v dx + u dy + C = + With = - dr + C= - dr + C

  2. = Source Basic Flowfields2- Source and Sink2-1 Source For the following flowfield = = + C = - dr + C = - dr + C = + C Let =0 at = 0 0 = 0 + C C = 0 which means that = = = - Sink 2-2 Sink In this case the stream function will be the inverse of source function = - =- = =

  3. Some Useful Combined Flowfields 2- Source and Sink of Equal Strength F= For sink = - For combined flow or = = ( For any point P , note that for the angle ( between r1 and r2 is =- For constant at any stream line, the 𝜶 is also a constant . That means all the stream line is circles through the source and sink and =

  4. Some Useful Combined Flowfields 3- Source and Sink of Equal Strength in Rectilinear Flow F = For Rectilinear Flow= U y So, For combined flow or = + U y = + U At point sL (-L/2 , 0) the velocity u = 0 = + U Or ( prove it ) at this point sL(-L/2 , 0) = + U (0) = () =

  5. Some Useful Combined Flowfields For the equation of the body = + U y Or 0 = + U y - The distance b could be find from the point (0, b/2) as follows From = + U y - = + U - = + U - = + U - = + U - =+ U - Solve by trail and error to find b/2 Note that

  6. Some Useful Combined Flowfields Example 3 : Plot the body contour formed by a source at the origin of 40 π m3/(s.m)in a uniform horizontal stream (from left to right ) of velocity 10 m/s. 1- calculate the velocity at the body contour for the value of θ = π , (5/6) π, (4/6) π , π/2 and (2/6) π . 2- calculate the velocity at the radial distance r = 2, 3 and 4 for θ = π.

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