Question

5) The velocity field for a 2D U=(x-2y)t Ň - (2x+y)t flow Ý is: a) Is this How incompressible? irrotational? 6) Find the acceleration of a Fluid element in this c) Find 0 and 4 for this flow . flow

Answer 1

uz (x-2y)& â - (anty) t ý fo, incomprensible and posof atjanal flow, ay + ore 20 de ani эх я U = (x-2y) t = -(24 ty) + e a (xt-2yt) + 2 (ant "yt) 1 = x + (-+) zo hence, flow Å incompressible and protational. Acceleration components are: = (x^2y)+ (24-2yt) +- (2x+y)+. A (-2xt-yt) + af at (R-2y) + = (1-2y) ++-(22+4) + (-+) + (x+24), = (-2y) x + (paty) + + (x-2y) -(3x-7)+'+(x-2y) = (x294. a (-xat-y!)–(anty)+. . (-347-yt) + (-227-yt) = (x-2y)+ (-2+)-(auty)t (-+) - (24+y] = (2-24 X-24²) + (2x+y)+² - (2x+y). = 5y + - (2x+y) az anilt að = [134y) x2 + (729Ji + [5y42-681+y))

Let it is potential function, 1. u= xt - 2y+ , r=-2ut -yt - -42 Lyx-x+ - the 2-10 = lat ty+ - Integrating ① we get 0 = 2nyt - 4* + C - differentiating in wat t - han det + But, from (in) 24 24t tyt 34+ + y Besk tyt - cs geht to, Sjuce, G+ f2.). this taking C120, we get c= yet Putting in Cin , we get Let it be the stream function, then, auf der = -aut-yt - Ay Integrating ① we get : | - -x^4 - + + c (1)

differentiating my wont y But from a lot 24+ -** Thus, et + t 22yt-pt 1 integrating, cz gt + Ci sunce, G + f(x,y), this. Gozo So, putting in Sin we get [p = pt -nyt tyet 0 W )