Question

= (c = constant) is c2(xj-yi) (i) Test whether the motion specified by V x2 +y2 of the potential kind. (ii) If so determine the velocity potential. (iii) Determine the equation of streamlines. (iv) Show that Vº: VŲ = 0 (v) Sketch the equipotentials and streamlines.

Answer 1

са + с . Na 2 (я- ур 1 - - с* U= х?у? 1+y- 8 9. - хуу- х2 + - curl = 0XT=0 flow is potential kind. (irrotational) au (2 = 0 ) 09 - су дях? +u-) 39 °+y then an | | а -Аз ура[ 1) — [1, 2 - 1 (2) = c [ 8-1" =y- - С2 + (з'tu') -- *9*( 2 ) (12+) 1 (x+y)- + 07. means it is is rotational / Potential ЦЕ - Оф а) (Б) д дф - g, Оф да 29 +су - идф я* ч д су xxy- • Integrates partial diff. as with respect to x. iy tar" (4) + (9) ca гу x2+y- до 2-с-х ду ху. - 0 ( Ja*a to: а) to Я 19. — дф Now consider

x Integrate it with respect to yo фа - с tah (2) = f (x) ф, -с? tar" (2) + f(x) from do Фа tar" (5) (2) c² tan & 2 tent Constant (ii) equation of stream line » hop )) da u 하면 da с? Я ху с nda 2 Х 2. с t c' ) x+y- -ydy - , (x2 + y2) = 26' = c" (x² + y² = "7 Eq" of streamline - (ау + 2 (2 is) Үф : 9р - (дф у 1 1 ): ( фе he дф . дф т ОФ ОФ оу ду ди ду fripomos ussef :. siemen eq" » дҳ = + до 4 дф бч бу - - до he 2 che che ке the te dy Бg xe о год од