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# Potential Flow Theory and Transportation

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I need some help with these three problems: (see attached file for better symbol representation)
Problem 1
a) Show that an elliptical cylinder is obtained when a circle of radius "a" in the "z" plane is transformed conformally into the "" plane using

Assuming irrotational flow, obtain a general expression for the velocity of flow crossing the vertical axis of symmetry. The circle is centered at (0, 0).
b) Show that the velocity on this line is everywhere greater than U and that a maximum value of 1,15 U occurs on the elliptical boundary at the point of intersection with vertical axis.
Find also the distance from this intersection to the point at which the velocity is q()=1,1 U .

Problem 2:
A solar car design calls for a body shape which is symmetrical airfoil of length 6m and maximum thickness 0,35m. The driver is to be positioned 1,5m back from leading edge.
The maximum thickness to chord ratio is given by: (see attachment)

a) Confirm the derivation that the real and imaginary components of the shape are given by respectively.
b) Show that maximum thickness occurs at
c) Determine the flow velocity on the airfoil surface at the site proposed for the driver when the vehicle is travelling at 100 km/h on a horizontal road in still air.

It is assumed that the airfoil performance is not influenced by its proximity to the road and that the edge effects associated with its width are negligible.

Problem 3:
Due to an error in construction, the airfoil shape in Question 5 is inclined at an angle of 6 with respect to the horizontal road surface. This nose up condition generates lift which is opposed by 18 bolts which connect the body to the chassis. Calculate the average loading which these bolts must carry. Data provided in Question 5 applies identically in Question 6 but the Kutta-Joukowski condition should be applied. Since the lift generated is the same in the  and z plane, the answer should be structured in the z plane only. Assume air = 1,2 kg/m3.