Question

1. A source of strength +Λ is placed/t (-a,0), a silof strength-A/2 is placed at the origin, and a sínk of strength -A2 is placed at (+a,0), all in a uniform stream whose velocity is V-Voi. Find the location of the stagnation points as a function of Λ, a, and Uoo

Answer 1

(cp) complex potential due to uniform stream } w-1 = -U_infinity z along +ve x - axis is cp due to source (^) at (- a, 0) w_2 = -^ln(z + a) cp due to sink (-^/2) at (0, 0) w_3 = -^/2 ln(z) cp due to sink (-^/2) at (a, 0) w_4 = -^/2 ln(z - a) Net complex potential w(z) = w_1 + w_2 + w_3 + w_4 = -U_infinity z -^ln(z + a) - n/2 ln z -^/2 ln(z - a) The stagnation point (s) can be found from dw/dz = 0 implies - U_infinity -^/z + a -^/2z -^/2(z - a) = 0 implies U_infinity +^/2[2z(z - a) + (z + a)(z - a) + z(z + a)/(z + a)(z - a)z] = 0 implies U_infinity +^/2[2z^2 - 2za + z^2 - a^2 + z^2 + az/z(z^2 - a^2)] = 0 implies U_infinity +^/2[4z^2 - az - a^2/z(z^2 - a^2)] = 0 implies 4z^2