Question

6.102 The flow field for a plane source at a distance h above an infinite wall aligned along the x axis is given by 20[x2 + y - 12|xi +(y-hil 9 2;[xi + (y + hil * 29x2 + (y + h) 21 where q is the strength of the source. The flow is irrotational and incompressible. Derive the stream function and velocity potential. By choosing suitable values for 4 and h, plot the streamlines and lines of constant velocity potential. (Hint: Use the Excel work book of Example 6.10.)

Answer 1

ANSWER Given that [xit (y-2)]] W = q at (x²7cy-h) tq . 20 (1214th 27 [xil + cyth)]l7 The velocity component in * - direction. - U= 1 . t qu . 24 (2419+632) 207 (x24cyth)] Distance from the plane source is b o strength of source is q. The velocity component in y-direction v=q14-h) qlyth) 27 (8²714-hj? 27 (x²4 lyth/2] The distance from the plane source ish - strength of the source is q. Determine the stoream function in te direction 4=d60 oy The stream function is a Determine the staream function in y-direction

225 2x Here, 'The stream Function is 4 Determine the stream function by the intergration the velocity in x-direction - 4 = Juckyddy Substitute e qx +qx for u only) 217 (224c9-632] 21 (x?7 (4th 327 4 = 18 22 (271(x4 (4-6) 2T (x²+(ya hf 1 quay laxdy Jotat cyt) 2 4 Bu? cyths" ] “m tan ( =))+ ta" (++f() Determine the stream function by Integrating The velocity component in y-direction. y = - svorcy] d > Substitute q ly-h) + 214th) for variy) 977 (x2+cy-ho) 2TT (x+14+h)] 4= -1 acych) + q6y+b) | 217(x2+ cy-hj%] 2T (84cy+h)]] I acy-hida .q (Yth) dx - 271 (x2+64-ho? 277 (x² + cythy 2

tun! ( 436 ) +2. tan! ( hay )+ 914) ::The Simple stream function is given. dx su Determine the velocity function in x-direction 45- Herre, The velocity function is a Delcormine the velocity Punetion in y-disection V=-30 Here, the velocity function is a Determine the velocity function by the Integ freding The velocity x- divjection 0 = - Jucx,y) dx. substitute I qr + 9x - for ulary) 217 6x?+C4-hy?] 27 (2469+h32] - 92 qudy 0 - - {[opertency-h3] * 2001+04+hi))* 5- Suome st) - Single Cent) = 922 [-112 log (153 2hy tx ?y? ] + 2 [12 loy chishy *x2_427+(45 2 TT

There looy Ch-y)? 22) + log (hty 32722])+(43) Determine the velocity function by Integreeting velocity component in X-divection. 0 =- Sucury) dy gly-h) 9 (4th) for uchi (x4(y-h) ] 277(xhly+h)? qa 11 211 qxdy 271 qxda 0 Serie y tie ti (59 ]ose -- I love those 5-S 469452 - 29 419 ths] =-q [- log Ch? zhy + x² + y²)) + q [-1 log - Cha2hy + x² 4?]+ g(x) =-[ logCh-yi?+2+log (htyj*+x2)] +9(x) ... the simple velocity furetion is given equation mq (tog (hy)}+x+log chtyj+2237] ZT IT

2uoc plot the graph of storam function e velocity potential function excel. =) open the Excel workbook of 6th chapters of Example 6-10 5) Type the stream function in the gst text box named as #NAME =) Type The velocity poten tid function in and text bou again # NAME =) To generate the plotes, press the bircle which is used to goraph the function plot the graph of stream function for q as 100 & I four ho by use of excel wart book re

7 stream function plot the graph of velocity function for a as 100 del for Velocity potential