Question

11.32 (C) Consider the rotary viscous coupling. Derive an equation that indicates how far the side wall effect extends into the coupling.

incompressible flow panton chapter 11

Answer 1

Side wall effect in rotary viscous damping:

For showing the side wall effect, solution is given below step by step

-h- ov r ( R Anower: This is to find the flow of fluid which can enists in thing gap b/w the end Wall of rotating with R. vo (2,2-0)= 0 Vo (9,2=h) = 2 { w up B. c} » Volr-R, 2) = 0 from o momentum een Si [í ag (100)] + drvo when gap esthin h/ko small gap Vo charge due to 0 [ da2 and momentin eg can be written as 222 rozrz with arvo & by solving etris egn ra) this is velocity profile of notary coupling It is valid for small E=h/R and for or not B.C close to R. i also does not satisfies VolR,2)=0

we will find the solution which is valid for all er. from Non dimensional variables. r 2 - R Vt - vo rer R gap. radius or width ratio .& is a small numbe. radius h E=h so Bic V* ( 2* , 2) =0 juta ( 14 , 2 * - 1) = 1 * No ( 2 = 1, 2+=0=0 assume vor is function of vot (r+ , 2") from non dimensional momentomeon this 0=E ad 1 2 (revo rar* (v*ve*)] + + dot dot 022 e to o outer if than only second tous will be then re r & z4 Turn is called outer solution at side walls es gies Vor (r*-1, 2*) - 1 instead of no slip condition voert - 1, 2*) 20 solution is singular The flow near the sidew all is called inner problem New Radial cordinate Y=R-r

y*= J h 8* = 1-Ey* drt -E on oy* from momentum equatic 2 -eye ayel + * ) ] ((-Ey*) Vot 2² Ug* 0 242 ㅗ Si llega of E for small valus + ayin On Opvot zuot Jy* dz? by using same symbol for Vot (1, 25 and vot ( 4*, I) Bc will become Vote (y *, s* )-o, volya, 2% -1)-1-ley* vot ly* -0,2*)=0 As replace equation meds closed domain BC so wis matching supplies Vogutar ( 2*, &*) arts to match the inner answer. voinner ly*200 ; 2) - Vogueal &* _1, 2*) we know outer auswer as vo ret 2+

o common Part - 24 we can complete Bo Vor inner ly*300, z 20, Z)> z* n n=1 neighbourhood of Sidewall. conditias oy compa he exp from ea @ by sepuation of variable No inner (y * 2*) = 2 + 2 { Ell" exp(-nay*) sin(no It is only valid only in det satisfies sidewall Bunday can write the uniform valid answer Voumposite: Vot Vonna (y + 2*) + U. douter (4 * , 2 ) - ot comon part with yt 1-rt postition near the side walls, last two terms cances and first term gines the correct selut on. vot we ( z*) E for postition