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On the Plane Couette Flow of a Viscous Compressible Fluid with Transpiration Cooling

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On the Plane Couette Flow of a Viscous Compressible Fluid with Transpiration Cooling ON THE PLANE COUETTE FLOW OF A VISCOUS COMPRESSIBLE FLUID WITH TRANSPIRATION COOLING J. L. BAI~AL AND N. C. JAIN [Oepartment of Mathematics, University of Jodhpur, Jodhpur {India)] Received Dr 13, 1973 (Communicated by Prof. J. N. Kapur, v..~.sc.) Ansr~cr The exact solution for the plane couette flow of a viseous compres- sible, heat conducting, perfect gas with the same gas injection at the stationary plate and its corresponding removal at the moving plate has been studied. It is found that the gas injection is very helpful in reducing the temperature recovery factor. Effects of injection on the shearing stress at the lower plate, longitudinal velocity pro¡ and the enthalpy arr shown graphically. 1. NOMENCLATURE E Eckert number, = (~ -- 1 ) M| ~ TI . T1 -- To b distance between the plates; h* defined by equation (30); H enthalpy, = cpT ; H* = H/Ha ; Ha* dimensionless adiabatie wall enthalpy; M,, Maeh number ; P pressure; p* dimensionless pressure, = p/plU ~; Pr Prandfl number ; Ha* -- 1 rŸ recovery factor, = (7- 1)M,, ~ ; A l~July, 74 2 J. L. BANSAL AND N. C. JAtN Re Renolds number, = piUb//zx ; T tcmpcrature; U vclocity componcnt along the x axis; U* dimensionless axial velocity, = u/U; U vclocity of the moving platc; q) vclocity componcnt perpendicular to the x-axis dimensionless trartsverse velocity, = v/U ; X axial distance from the entrance of the chanrtel; Y perpendicular distance from the x-axis; Greek Symbols : 7 isentropic exponent, = c~/cv ; dimensionless transverse distance, = y/b ; injection parameter, ---- povob/tz~ ; tz viscosity; t. dimensiouless viscosity, = /~/tzl ; P density ; p* dimensionless density, = p/pl ; T o dimensionless shearing stress de¡ by equation (16); Subscripts : 0, 1 values of the variables at the stationary and moving plate respectively. 2. INTRODUCTION IT is well known that mi exact solution of the Navier-Stokes equations exists for a viscous incompressible fluid in the case of plane Couette flow. This solution has been generalized by Nahme, ~ Hausenblas 8 and Groff 4 when the viscosity of the fiuid depends on tempemture. The further exten- sion to a compressible fluid was given by Illingworth5 and Morgan. ~ Aja Plane Couette Flow of a Compressible Fluid important problem in the development of missiles, satellites and spaceships is the excessive heating of the skin of these vehicles by friction in the high velocity air stream. A method which appears very effective is called trans- piration cooling. Although the nature of the problem is purely of boun- dary layer, effort has also been made by Eckert 7 to describe the reduction of heat transfer by Couette flow in the case of an incompressible fluid by injecting the fluid into the flow field from the stationary plate and corres- pondingly removal from the moving plate. Moreover, the Couette flow model proved to be very helpful, as has been reported by Rannie 8 and Mickley,9 to describe the conditions in the laminar sublayer of a turbulent boundary layer. Since at high speeds the compressibility of the gas should also be taken hato account and the viscosity of the gas depends on tempera- ture the corresponding Couette flow needs ah investigation. In the present paper an exact solution for tho Gouette flow of a viscous compressible fluid with the same fluid injection at the stationary plate and its removal at the moving plate has been studied. The assumption made in the present study are (i) the specific heat of the gas is constant (ii) for solid walls the Prandtl number Pr is constant but arbitrary (iii) for porous walls the value of Pr has been taken as 3/4, which is ah important special case since it is very near to the value of air and (ir) the viscosity of the gas is directly proportional to the temperature. 3. STATEMENT OF THE PROBLEM Let us consider the two-dimensional laminar flow of a viscous compressible fluid between two parallel plates, one in uniform motion and the other at rest with uniform injection of the same ¡ at the stafionary plate anda corresponding uniform suction at the moving plate. The axis of x is taken along the stationary plate and y is measured at right angle to it. In such a case all variables of the gas motion can be taken as timctions of y oniy. The equations of continuity, momentum and energy therefore are: ~y(pV) = O, (1 a) of pv = constant, (1 b) pv du~ = dy(/Z~)d , (2) 4 J.L. BANSAL AND N. C. JAIN p~~=-~+~d~ dp d(43~'~ dv) , (3) 1 d(dH) 4 (dv~' (du\ 2 dH dp (4) Pr dy tZ -dy +3/z\~/ +q = PV -df -- v -- and the equation of state is 1 p = p Y- H, (5) where u and v ate the velocity components along x, y directions respectively, H = cpT and the other symbols have their usual meaning.. The boundary conditions are y =0: u =0, v =yo, H =Ho, p =po, .) (6) y=b: u=U, v--v~~176H =Hz, p =p~.~ Pz The normal velocities v0 and vopo/px are prescribed in such a manner so that the equation of continuity (1) is satis¡ i.e. pv = constant = Poro, (7) Let us introduce the following non-dimensional quantities: u v* = v ~ H p, = u*=o' O' '7= , H*=~, m' tz* = t~ p* = P ;t -- P~176 (injection parameter), /zz ' P~ ' /- Re- pzUb (Reynolds number), q where bis the distance between the plates, U the velocity of the moving plate and the subscripts 1 and 2 denote the values of the variables at the station- ary plate and at the moving plate respectively. Equations (1) to (5) in non-dimensional form are: p* v* = R--e' (8) du* d du*'~ (9) a --g~-~ = it~ ( t~ * ~d~ s ' Plane Couette Flow of a Compressible Fluid 5 ~~~d,/ =--Re--+~%" ~ (~~.~~'~a~71' oo> d [Pr 1 aH*] 4/~, (dv*'~ 2 (au*'~ ~ (r -- 1) M., 2#* -d~-~ .! 4- ~ k--d~-] +/~* \-~/ _ ,~ dH* Re v* dl)_* (11) (~, -- 1) M,** d~/ d~/ ' and p, = p* H* 9,M** z , (12) where U ~ = O' -- 1) M. ~" (13) The boundary conditions (6) reduce to =0: u*=0, v*=vo*, H*=Ho*,) (14) =1" u* =1, v* R-~'~ I-I* 1.1 4. ANALYSIS Equation (9), on integration, with proper boundary conditions yields i~ , du*d~7' = ~u* + ~'o*, (15) where du*~ o* = (~* --~ ]~0' (16) is the dimensionless shearing stress on the stationary plate, to be deter- mined. Now eliminating p* between equations (10) and (11), we get d 1 /~, dH*] tdv *'2 (du*'~ ~ d~- [Pr (7, -- 1) M| ~ --~-~ .] + ~ ~*t,~) + /~* ~~/ -- ~v* k-[d~*~ ] ffi ~ di'I* -- v* d (4 dv*'~ (17) (7'-- 1)M"~ ~ ~ k3#* -~ J" 6 J.L. BANSAL AND N. C. JAIN Taking the help of (1'5), the equation (17) can be written as d[ 1 dH* 4 , ,dv*] Pr(9"- 1)M. '~* d~ + ~~ v ~j [ H* v**1 -- a Cd (V -- 1) M** 2 + -2-J *'~ du ~ =--(au*+-o ) ~. (18) Integrating equation (18), we get ~ 4Pr(9,--1) M**~ + --A (y_ 1) M**2q- ;~u*Z = A 2 %* u*, (19) where A is a constant of integration. Case I.~For ;~ = 0 (i.e., in the absence of perosity) and for arbitrary values of the Prandtl number Pr the equations (15) and (19) reduce to tz , du* -d-~ = ~~ (20) and ,dH* /z -~-~- = Pr (~, -- 1) M| ~ (A -- r0* u*). (21) Equafion (21) with the help of (20), becomes dH* Pr (V 1) M*** (A' -- u*), (22) du ~ ~- where A' is now the unknown constant. Integrating the equation (22), aad applying the boundary conditions on u* and H* from (1:,4), we get Pr H* -- Ho* = (1 -- Ho*) u* q- ~- (9' -- 1) M.. ~ u* (1 -- u*), (23) Plane Couette Flow of a Compressible Fluid 7 or in terms of temperature T To -- u* + ~ u* (1 -- u*), (24) T1 -- To g., where E = (r -- 1) M,,2 TI (Eckert number). (25) Ti -- To One can easily see that for ah incompressible fluid tx* = 1 and therefore from (20), with proper boundary conditions, we get u* = ~/ and ~'o* = 1. (26) In this case the expression in (24) reduces to the well known result for the temperature distribution, Schlichting, ~ when the frictional hea-t is taken into account. Fora compressible fluid let us considera linear relationship between t~ and T, i.e. tz * ----- H*. (27) Therefore from (20), using (23), on integration, we find u *~ Pr /'u .2 ~~'o* = He*u* +(1- Ho*) ~- -4 -2-- (y-- 1) M**~k -~ ~~), (28) since at ~7 =0, u* =0. Now, in order to find out ~'0" we apply the condition ~ = 1, u* = 1 in (28) and find Pr "ro* = 1[1 + H0*+ ~ (,-- 1) M.. ~] , (29) Further details may be fourtd in Liepmann and Roshko. lo 8 J.L. BANSAL AND N. C. JAIN Case H.--A q: O. In order to get a compact solution of the equation (19) we make an assumption that Pr = 3/4 which is an important special case, sinee it is very near to the value for air. Let H* V*~ = h*. (30) (y--1) M.~ 2 + -~- Hence equation (19), with the help of (15), becomes dh* 3 A* 3 (C -- A* u .2 --2 u*) (31) du* 4 A* u* +~1 h* ----" 74 (A* u* + 1) " wh~e A A* ~" **'0 (32) and C is now an unknown constant. Equation (31) is a linear differential equation and its solution with the boundary conditions (14) is h* = aa (A* u* + 1)s/4 _ a~ (A* u* V 1)2 _ ax ,... (33) where __ hi* -- ho* (A* + l)S/4 __ a~ {(A* + 1)8/4 _ (A* + 1)*} -1 -- (A* + 1)s/,_ 1 3 as =ho*+al+al ~-~o~ ~o ~~ ho*--(?_l) M.l+ 2 ' and 1 hx* = (7'-- 1) M.
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