Question

Q1. (a) The velocity components of a certain two-dimensional flow field are claimed to be given by u =-Cy and v = Cx , where is a constant. (i) Does this velocity distribution satisfy continuity? If yes, what is the stream function y? (8 marks) (ii) Analyse whether the flow is rotational or irrotational. What is the velocity potential o ? (4 marks) (iii) Sketch several y and © lines (if exist) of the flow field. Briefly explain how you sketch it. (5 marks) (b) The velocity potential for a flow is given by 6=9 (x²-y?) where a is a constant. Determine the corresponding stream function. (5 marks) (c) The discharge of 30m²/s pollutants from a chemical plant into 10m deep river flowing at 0.3 m/s, can be modelled as a 2-D source across the river depth. It is found that the fishes in a certain zone die out, whereas those outside the zone are unaffected. Find out the extent of this critical zone, if the point of discharge is in the mid plane of a wide river. Justify your answer by giving a brief explanation with your own words. Sketch your diagram. (10 marks) (d) Describe a natural phenomenon that involves fluid flows where the streamlines are concentric circles. (8 marks)

Answer 1

Q:-) Us-cy & v=cn is for continuity equation to be satisfied: du + dv=o. In jy O to=o. Satisfied U=-dt & V=Jt Jy dre + cy=tty y = cy²+k Y=k ЈУ =0.= cn. dr (c=0 stream live function (1)=k. (i) We = Į (duay We = 1/3 (C-(-6)= 9 to Hence, it is rotational. velocity potential exists only for ideal flow & renotational flow.

(b) +- (n-ys). U=-de & v=-do Jor dy. V-1-2y) U= -9(20) lu=-an V= ay U=-de dy & v= de dre tan= tot dy to any+c] dt Ja ay stream line function (4)= any te, (c) d= 30m/ for Unit width. depth (h)=lom velocity (v) = 0.3 m/s '&=AV L-loom 30=2x0.3