2. (8/31 points) In this problem we want to explore the dynamics of a pair of trailing vortices left behind by an aircraft while taking off. The trailing vortices are assumed to be close to the ground and therefore their dynamics will be affected by the ground topography. For simplicity, the take-off strip and surroundings are assumed to be perfectly flat and the ground is therefore represented by the x-axis, the real axis. Above the ground, two point vortices are located symmetrically with respect to the y-axis (the imaginary axis), on the right, at Ra+ib, and on the left, at z = a + ib, where a > 0, b > 0 and, in general, ab. The vortex to the right is a positive vortex of strength I, while the vortex to the left is a negative vortex of opposite strength -F. Recall that the complex potential of a positive (counter-clockwise) vortex located at z = zo is log(zzo). 2 (a) Construct the complex potential so that the no-penetration boundary condition on the x-axis is satisfied. (b) Verify that the real axis is a streamline. (c) Compute the velocity distribution along the real axis. Sketch the velocity distribution. Does the velocity distribution allow to detect the location of the vortices? (d) Compute the velocity of the right vortex located at ZR. Sketch the trajectories of the right and left vortices. Briefly, explain why the vortices move along the trajectories you sketched. How the velocity depends on the size of a and b and what is the impact of them? How the presence of the ground, i.e. when b < < a, affects the trajectories of the vortices? (e) Using Blasius' Theorem 2 i Fx - 31 dF dz dz, compute the force acting on the right vortex located at ZR. Briefly, discuss the physical meaning of your results, i.e., why the force is acting in the direction you found. How the force depends on the size of a and b? How the presence of the ground, i.e. when b < < a, affects the forces acting on the vortices?