0n Stokes' Problem for a Non-Newtonian Fluid. K. R. Rajagopal, Pittsburgh, Pennsylvania, and T. Y. Na, Dearborn, Michigan. (Received August 30, 1982)

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1 Acts ]Vfeehaniea 48, (1983) ACTA MECHANCA by Spriager-Verlag n Stokes' Problem for a Non-Newtonian Fluid By K. R. Rajagopal, Pittsburgh, Pennsylvania, T. Y. Na, Dearborn, Michigan (Received August 30, 1982) 1. ntroduction The fluids of the differential type (cf. Truesdell Noll [1]) are a~nongst the many models which have been employed to describe the non-newtonian behavior exhibited by certain fluids. n this note we extend the work of Stokes [2] on the flow due to an oscillating plate for a special subclass of the fluids of the differential type, namely the incompressible fluids of grade three. The stress T in such fluids is related to the fluid motion in the following manner (cf. Truesdell Noll [1]): T = --pl ~ ~ua 1 -~- ala 2 -~ a2a12 ~- fl~a 3 fl2[a,a~ -~ A2A] -~/~3(tr A1 *) A,, where/x is the viscosity, ~1 ~2 are material moduli which are usually referred to as the normal stress moduli, p is the pressure the kinematical tensors A,, A2 As are defined recursively through A 1 = grad v -t- (grad v) T, (2.1) d A, = ~ (A,-) + A,_$(grad v) ~- (grad v) z A,. (2.2) n the above equations, v denotes the velocity d the materialtime derivative. dt 2. Equation of Motion We are interested in the flow of a fluid modeled by (1) over an infinite flat plate which is either accelerating or oscillating. Thus, we seek a solution for the 1 Fosdick Rajagopal [3] have studied the thermodynamics of fluids modeled exactly by (1) in detail. They show that the assumption that all motions of the fluid obey the Clausius-Duhem inequality the requirement that the specific Helmholtz free energy be a minimum when the fluid is locally at rest imply that >=0, ~,,>0, ~ 1+~[ < f~,~3, fll=~=0 ~3 >0. n this note, in addition to studying the problem for the above range of material parameters, we also study the problem when a x < 0, fl~ # /S3[0048]0233/$01.40

2 234 K. 1~. R~j~gopal T. Y.!~: velocity field of the form: v = u(y, t) i, (3) where u is the velocity in the x-coordinate direction i is the unit vector in the x-direction. Substituting (1) into the balance of linear momentum dv div T -~ ~ob = ~ -~, using (3) the fact that the fluid is incompressible, we obtain (when fll = 0) -- ~ (~ ~ ~ ~ ~P (4.1) a~u d- ~: d- 6(fl2 + fl~) Q ~u ~y2 ~ \ ay ] ay ~ at Ox [, (~i~ ~u ~ ] _ ~ (4.2) For the problem in question the boundary conditions are 0 = ~P. (4.3) ~z u(o, t) : U(t), Defining a modified pressure ~ through u --> O as y--> ~. [ {~/~ + 2~ ~ ~ 1 we can rewrite Eqs. (4) as ~2u + ~ -~ 6(fi2 -~ fia) ~o -- =--, (5.1) # ~y~ ~ \Sy] ~ye ~t ~x ~ ~ (5.2,3) ~y ~z Equations (5.1, 2, 3) imply that ~ is at most a function of time. The conditions ~x at y --> ~ imply ~x ~ : 0, hence, u -- -~- ~ -~- 6(fl2 -~- f13) : 0. (6) The above equation can be re-written in the following dimensionless form ~--~y~ +~,~ +s ~ bye], (7)

3 On Stokes' Problem for a Non-Newtonian Fluid 235 where Uo is a reference velocity a~uo ~ 6(fl2 + fia) ~1--, e-- ~v ~ ~v ~ u UoUt Uoy u= ~o' i=,~-- it' ~, The appropriate boundary conditions are =F(t) at t=o, ~-->0 as Tl-> oo. Let us suppose the non-dimensional velocity ~ can be exped in power series in e: ~(t, y; s) : ~0(t, y) ~- s~l(t, y) ~- e~2(t, y) ~-... (S) On substituting the expansion (8) for ~ equating like powers of s, we obtain the following equations at zeroth first powers, respectively. ~3~ o ~fi_~o = ~o + ~1_ (9) ~o= U(t) at ~=0, (10.1) Uo-~O as y-+~, (10.2) 8~ _ a2~ a~ (~~ 2 [a2~~ (11) ntroducing the similarity transformation ~1=0 at y=0, (12.1) ul --> 0 as ~ --> oo. (12.2) n = g, lo(n) = ~o ~, A(n) = --~/~, ~1 (13) we can re-write Eqsl (9) (10) in the following manner: where (1 + 7Xl) 1o" -- 7/0 =- O, o(o) = 1, /o(~)~0 as ~-->~, U(t) = e:'l (14) (15.1) (15.2)

4 236 K. 1%. l~ajagop~l T. Y. Na: Also, Eqs. (11) ~nd (12) can be written as (1 + 3r~1)/1" -- 3y/1 = --(/o') 2 s (16) /i(0) = O, (17.1) 5(~) --> 0 as 7 --> vr (17.2) t is straightforward to verify that the solutions to (14)--(15) (16)--(17) are, respectively /1(7) = 6(1 + 7~i) (1 + 4~ml) 2 + 3~,~1 (19) Since y may be both real or imaginary, both eases are considered. Case 1:7 is real n this ease Eq. (8) yields (y, t; e) = e~/o(7) + ~e~/1(7 ) (20) The numerical values of/0(7) /(V), for various values of 7 51 are provided in Table 1. Thus, one can obtain the numerical value of ~ for various values of s. Of course, when e ~- 0, (18) implies that there is an exact solution for the problem. The skin friction on the plate r~ for the problem is given by t follows from (18) (19) that v~ = eu02{e~%'(0),-]- ee3~h'(0) +.--}. (21) /0'(7) = -- ~/~/0(7), (22) /'(7) = 6(1 + ~,~/(1 + 4~,~1) i + ~,~1 1 + ~,~ (23) -- "1 +37~ ~ '1 +37~ ~ 7 9 The numerical values of ]0'(7) ]1'(7) for values of 7 Xa, are provided in Table 1. Case 2:7 is imaginary n the case of an oscillating plate, i.e., when 7 is imaginary, say 7 = ieo, a straightforward computation verifies g(i, ~; e) -~ {/or cos ~oi --/0x sin col} + e{/,r cos 3~i --/ sin 3e)t} +..-, (24) where /or = e -~ cos ~7 (25)

5 On Stoke's Problem for a Non-Newtonian Fluid 237 Table 1 /o(v) /o'(n) 1~(~) h'(v) ~1 ~ =.5 ~1 = --2?=.5 7 = J{ ~) O /ox : --e-~" sin Ox# (26) are the real imaginary parts of/o- Also, /1R : --{An cos ml# + At sin mi# } e -m~' + {AR cos 3~i~ + A sin 3~i~} e -3~, /11 : --{A cos mi~ -- AR sin mx~ } e -m~n + {At cos 3~z~ -- ARsin 3~i~} e -3~", (27> (2s) where /1R ]11 are the real imaginary parts of/1, respectively. n the above equations ~R={'al+(al~+a22)l/2} 1/2, ~= a2 (29.1,2) 2 {2[a x -~ (al 2 -]- a22)1/2]} 112 ' mr = bl -4- (b12 + b2~) 11~. 112, m~ = (30.1, 2) 2 {2[bi + (bl 2 -{- b~2)112]} 112 '

6 238 K.R. Rajagopal T. Y. :Na: 11 1 R~RR~R~ ~[! l t l li i i 1 1 [,r 11 r

7 On Stokes' Problem for a Non-Newtonian Fluid 239 with ~(D 2 (D -- a2-- bl )2 be ~- 30) J. -}- 9~120) 2 ' 1 -}- 9~120) 2 ' Aa = --50)2~ 6[( )~12) ~x20)~] ' At = -(1-40)2cr 0) 6[(1-40)2~12) 2-25~120)2]" (31.1, 2) (32.1, 2) (33) (34) Again, when s : 0 one can establish an exact solution (cf. Rajagopal [4]). The skin friction on the plate is given by the real part of -r ~- euo2{e~'t[/on(o) -k /10t(0)] -k se'3'~ ~- q~1(0)] -k.. "}. (35) t follows from Eqs. (25)--(34) that /or(o) a, (36) /or(0) = --6t, (37) /~t(o) = --Aimt -k Anm~ -- 3A~ -- 3AR~R, (38) /~(0) = ARm1 +Aimn -- 3AR~ -- 3A~R. (39) n Table 2 the numerical values of Uo'(V) =/or(q) cos ~oi --/o1(v) sin r t u1'(v) =/1R(V) cos 3~oi --/it(v) sin 3col (40) (41) are Provided for various values of toi ~1. References [1] Truesdell, C., Nell, W.: The non-linear field theories of mechanics. Hbuch der Physik, /3. Berlin--Heidelberg--New York: Springer [2] Stokes, G. G. : On the effect of the internal friction of fluids on the notion of pendulums. Cambr. Phil. Trans. 9, 8 (1851). [3] Fosdiek, R. L., Rajagopal, K. R. : Thermodynamics stability of fluids of third grade. Prec. R. See. Lend. A 889, (1980). [4] Rajagopal, K. R.: Unsteady unidirectional flows of a non-newtonian fluid. nternational Journal of Non-Linear Mechanics (in press). Dr. K. R. Ra]agopal Departmewt of Mechanical Engineering University o/pittsburgh Pittsburgh, PA 15261, U.S.A. Dr. T. Y. Na Department o] Mechanical Engineering University of Michigan Dearborn, Michigan, U.S.A. 16 Acta Mech. 48/3--3