Problem 4. Vortex in a Corner A two-dimensional, point vortex of strength is located in a 90 corner with vertex at the origin and walls along the positive x and y axes. The vortex is placed at 20 = roexp (10) where is the distance from the origin and is the angle with respect to the x-axis, see Fig. 3. (a) If the vortex is fixed in location or "bound", what is the force on it? (b) If the vortex is free to move, and is at position z(t) = r(t)exp (t()) at > 0 show that dr dt 4wr2 [Hint: use the method of images to construct df/dz = u-iv, where u = dx/dt, v = dy/dt, x = r cose, y = rsine. (c) Hence show that the vortex moves along the trajectory sin (20) sin (20) (d) Sketch this trajectory de dt Zicot (26) Zo To 8. . 90 Fig. 3 Problem 4. Vortex in a Corner A two-dimensional, point vortex of strength is located in a 90 corner with vertex at the origin and walls along the positive x and y axes. The vortex is placed at 20 = roexp (10) where is the distance from the origin and is the angle with respect to the x-axis, see Fig. 3. (a) If the vortex is fixed in location or "bound", what is the force on it? (b) If the vortex is free to move, and is at position z(t) = r(t)exp (t()) at > 0 show that dr dt 4wr2 [Hint: use the method of images to construct df/dz = u-iv, where u = dx/dt, v = dy/dt, x = r cose, y = rsine. (c) Hence show that the vortex moves along the trajectory sin (20) sin (20) (d) Sketch this trajectory de dt Zicot (26) Zo To 8. . 90 Fig. 3