1. Steady inviscid incompressible shear flow past a circular cylinder of radius a is given by...

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1. Steady inviscid incompressible shear flow past a circular cylinder of radius a is given by the stream function 4 = = Ursing (1-2)+ [1-(1-2)] 20 where (r, 0) are the radial and azimuthal components of a plane polar coordinate system, and U, w and a are positive constants. 1 (a) Show that r = a is an impermeable boundary. 2 (b) Determine the radial and azimuthal components of velocity ur and up. 4 (c) Stagnation points are points where the velocity u = 0. Let denote the angular position of a stagnation point on the surface of the cylinder r = a. Show that sin 0F(S) G(S). where S is the nondimensional parameter S = U Swa 0 and F(S) and G(S) are functions that you should determine. Determine the values of S for which there are (i) two, (ii) three and (iii) four stagnation points on the surface of the cylinder. Hint: Try plotting F(S) G(S). 2 (d) Plot the streamlines and surface stagnation points for S = 0.25 and 0.5. Make sure the streamline on the cylinder surface is visible in each case. 1. Steady inviscid incompressible shear flow past a circular cylinder of radius a is given by the stream function 4 = = Ursing (1-2)+ [1-(1-2)] 20 where (r, 0) are the radial and azimuthal components of a plane polar coordinate system, and U, w and a are positive constants. 1 (a) Show that r = a is an impermeable boundary. 2 (b) Determine the radial and azimuthal components of velocity ur and up. 4 (c) Stagnation points are points where the velocity u = 0. Let denote the angular position of a stagnation point on the surface of the cylinder r = a. Show that sin 0F(S) G(S). where S is the nondimensional parameter S = U Swa 0 and F(S) and G(S) are functions that you should determine. Determine the values of S for which there are (i) two, (ii) three and (iii) four stagnation points on the surface of the cylinder. Hint: Try plotting F(S) G(S). 2 (d) Plot the streamlines and surface stagnation points for S = 0.25 and 0.5. Make sure the streamline on the cylinder surface is visible in each case.