The velocity potential for the flow of a fluid of constant density p in the absence of gravity is \phi=\frac{1}{2}\left(x^{2}-y^{2}\right) (i) Find the velocity field for this flow. (ii) Show

that the flow is incompressible. (iii) Find the stream function for this flow. (iv) Show that the streamlines are everywhere orthogonal to the equipotential lines for constant o. In the xy-plane sketch carefully the streamlines and the equipotential lines. Clearly indicate the direction of flow. (v) A particle is released into the flow at the point (xo, Yo). Write down two differential equations describing the subsequent motion of the particle. (vi) Solve for the particle path. How long does it take the particle to reach the line y = 0 if yo # 0? (vii) Calculate the hydrodynamic force F due to pressure, where \boldsymbol{F}=-\int_{C} p \boldsymbol{n} d s and C is the unit circle centred at the origin, with s arc length along C and n the unit normal vector to C that points away from the origin.