Question

Given the velocity potential for a 2-D incompressible flow, (x, y) = xy + x2 - y2 (a) Does the potential satisfy the Laplace Equation (i.e. V20 = 0)? What is the physical intepretation of this? (b) Find u(x,y) and v(x,y) (the corresponding velocity field of the flow). (c) Does the stream function y (x,y) exist? If so: (a) Find the stream function. (b) Find the implicit equation of streamline that passes through (x,y) = (1, 2).

Answer 1

0 (8,4)= dy +02²-y² (velocity potentiel) equation According Laplace's to continuity equation is a) 11 pe = y + 22 bet = 0.-27 ; e 2 ; -2 - too do t = 2-2-0 dsca . Jy2 - * so, yes This Velocity potential is satisfying The Laplace equation > This Flow is satisfying Laplace egy which means - it is a incompressible & irrotational flow field U = -2¢ (velocity field in V = -2d (velocity field in Tu=-(y +2x)) x direction) y direction) V=29-x) Ans(b)

. ) There will be an existance of stream lines. Function egr of Streams function is I dy = ŭ , U= -y-2x V=2y 2x tu I 1-21 Yab 2+14) ->0

To solve this differential egy . let's Assume 1/2=t) - yaxt 1 du = tradito Put ② 4 ② in egr 0. tt x dt = 1-2 (). 2tt ad y = 1-2t ut enes l-2t -t(2++) 2tt redte 1-27-27-2 x t +2 oc (dt) = 1-4t-t² 타 (t+2) dt. THE-zz a dol OL dal c. -2(t+2) dt = (-2) (1-4t-t?) I (1-4t-t?) eles 1(-2) (1-4t-t2) =

en(1-4+-+2) = (-2). 41 () + luce) . where ane is a integration constant] en (1-44-2²) = ln (9²) + luc en (1-4t-f²) = ln (22) 1-42-2² x?c. (t="o) put it back into this eqh) 1-41%82) - (you? 2-2 (1-41 Yl) –(Wa)) = constant (c) be (Mos+ 41 You)-1)= constant = c. x² ( 724 +44 2-1) =( | 4²+ 4 Y.,.-x²=2) x AMS (())) Implicite egh of Stream line. = constant

-Now, If point & find this stream line passing through (112) them pur x=1 &y=2 the value of constant ((). y²+ky,x-x²=c 2 + 4(2).(1)-co=C 4+8-1=0 (=11) final ean of Stream line is Ty²+uya-ol² = 11 Ans(C)(2)

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