Stream Function - Boundary Layer

Stream function: with Through Bounday layer ( study) * Cusl ( grad) # @it on gfit + * E. / = o= Px it Dy's + O2k =7x70 V.O= 20x or + tay aby + oz a Dz. i k = 20' of by of a dd by ax by of = i Ap (of th) - 2g a tu (2(g) - of 2(2) + k (34th) - (2()) = i oyaz 22 (c) - -j(a8 t the bg ( only 20 - . Care (god) = O

any to God 1 Irvotational flow: flow in which each element of the moving fluid undergose no net rotation with respect to a chosen coordinate axes from one instant to other eg: Fewis wheel ( Garst wheel) Rotation of a fluid particle can be caused only by a torque applied by shear forces on the side of the particle. w . w.i +wg; think 4th, Curro 1/2 VXV = 1/2 (Cuslv) i j R = 2 a a a on 54 58 u V w a + du = 1/2 dw - 2 it az - aw dr j + k da ay og 2 If 7H = Pi+oj+ RE 1 j k Cus F = = 8XF a a ad/ on or 02 P Q R dif div I.F = ap + OQ + OR ax by az for X Cusl (divF) = (7.F) R i = 2 2 of sg 35for yeal i leave 2 22 on v DP OQ OR 02 on Dy 2 dr 0Q Carl (dw7). i( + j ond2 - and OR zp - ayer 2ydz 2Q 2. +i ( JP - on dy July ) =

Instational flow: " Clery to God" 03/11/2017 How which satisfes the equation Cul ( xxv) As irrotational flow can be described by using Velocity potential function ()) trust Q. consider a= the 35 show the by flow is instantial flow field 4 = at-ay2, where Solution: 2 Laplace If the flow is inotatord 4== equation to= O. I 2 Aut -0 3. V.Y = O. 310 2 2 W2 1/2 (av on av ) 54 + 04 + d 2n2 of 032 da - -2a O The flow is instational since 4.10=0 function for a certain components flow of is velocity gives. by 4 = The ax- stream ay 2 flow. , determine the for the give 04 V= of on U= oy U zay a (ax2 - ayj) = - 2ay V= -20xc 04 = U= by 04 = ag - or 2 (ox-ay) = - -2ax Determine - ox the velocity.vector V= J= 20x J

Q The stream function of a gass flow is gMs my P= ax'. ay ?. bidroist Determine Id velocity potential of the flow. not woll First peone that the flow is motatosal, if the flow is rotectional they there is no velocity potential flow is imstational: 2 V4 = O. 24 24 2 + + 04 =0. Jx2 dy' 232 2a - 2a =0 { the flow is instational The relationship between velocity component and velocity potestial U= - as ve = - as , ox First we have to ford U and V from the gas by streas fwaster U=-2ay V=-2age U= as V=-dq an g 4= do V= 20 dx Now of U= - do on payst 00 ox day dx = 20 Integrating dazy = $ 0= aazy. + fcy,

cristing V= east 20 an by #2ax - - as W2" 1/2 by a 20 so identity 00 - 2ax ay by' Interest o= = eazy + f(x) 0 = 4= ant-ay' - I 04 = U=- 2ay as = V= (2ax) by 2 - on Velocity potential 3 V= ad = 2000 o U. 20 = - 2ay ay da I d$=-1 = 2ay dr 2axy + f(y) was 1 Partially differenciating o = - w.rt. y and company 26 = - 29/2 + fly = - 20th sy figs f is a constant & - 2azy +C