CHUN, F (reprint author), ACAD SINICA,INST MECH,BEIJING,PEOPLES R CHINA.

Fan C(范椿)

Institute Of Mechanics,Chinese Academy of Sciences

Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/82/002181-08 $ 3.10/0 Voi. 33, March 1982 9 Birkh~iuser Verlag Basel, 1982 Stability of flow of a generalized Newtonian fluid down an inclined plane By Fan Chun, Institute of Mechanics, Academia Sinica, Beijing, China Introduction The stability of a layer of triply non-linear fluid, power law fluid, and a second order fluid flowing down an inclined plane has been consid~ed by Yih [1], Fan [2], and Gupta [3] respectively. Gupta found that the critical R@nolds number (Re)or depend on the elastic properties of the fluid, but" the second order fluid had a constant viscosity and hence the effect of variation of apparent on (Re)or was not considered. In this paper, the same flow problem is considered for a generalized Newtonian fluid. Yih's [4] perturbation technique is used in the following analysis. Differential system governing stability A layer of a generalized Newtonian fluid of thickness d flows down a plane (Fig. 1) inclined at an angle fl to the horizontal. The steady primary flow is taken parallel to the xl-axis with the x2-axis normal to the plate downwards, the origin being taken at the undisturbed free surface. The equation of momentum and continuity are ~(~u~/~t~ + uj 8uJ~xj) = ~z~j/~x~ + ~ X~ , (1) ~u/~x~ =0, (2) Figure 1 Definition sketch x2 xl I82 Fan Churl ZAMP where 0 is density, t 1 is time and X~ are the components of force due to gravity. For a generalized Newtonian fluid, the relationship between the stress tensor o and the rates of deformation ~j is (3) where pij = ~u~l~xi + ~uil~x~, and 6~j is Kronecker delta, pl is pressure, f is an arbitrary function with continuous derivatives. (Several examples are given in Fig. 2.) The primary flow is steady and unidirectional and the velocity depends on xz only. Using a bar to denote various quantities for this flow, (1) gives f(~,2) ~, = _ 0 9 sin fl x2, (4) - ~o g cos 1~ = P. (5) We now superimpose small disturbances on the main flow and write ul = ~ + ~2, u2 = 0, Pl = P + P ,  (6) where the tilde denotes various perturbed quantities. Substituting (6) into (3), expanding f in a Taylor series about z7 '~, and neglecting quadratic terms in the perturbed quantities, we obtain r~ = - p~ + 2f~6/Sx, ; rz2 = - p~ + 2f~f~/~xz;  z~z = ( f  + 2 fi,2 f , )  (afJ/Sx~ + OtT/Ox2) + ~' f , (7) where f f  are respectively.f(zT'~), df(~'2)/d (a'2). Figure 2 The relationship between the stress z and the rates of deformation ~for the power-law fluid and the power series fluid. 1.0 / 0.8 0.6 n.1.03~i~1.0.8 0.4 n v-1.15~ ///r 0.2 ~.3  014~"  I I I I I 0 0.2 OA 0.6 0.8 1.0 Vol. 33, 1982 Stability of flow of a generalized Newtonian fluid 183 Substituting (6), (7) into (1), cancelling out the terms corresponding entirely to the primary flow, and neglecting quadratic terms in the perturbed quantities, we obtain ~(.~_ ~ ) 8fi ~2~ - ,2  , + - -  + ( f+2u f) -7--7o+ ( f+2~'2 f ' ) ,  (8) ax~ +u = ( f -  2 ~'~.f') - -  Ox2 8xl ~x2 -,z , a~ df  a~ + ( f+2u f )~-74  - -  dx2 ~x2 cx~ (9) We introduce the following dimensionless variables x = x Jd ;  y = xz/d; t = t 1 ~' (d) ; u = ff/~'(d) d; v =O/~'(d) d; p =fi/~[~'(d)] 2 d 2 (10a) and the following dimensionless quantities Re = ~ fd(d) d2/f([~'(d)]2), U= ~/~'(d)  d.  (10b) By substituting (10a), (10b) into (4), we obtain f=f ( [~t ' (d) ]O(y /U ' ) ,  f ( [~'(d)]0 = ( f+  2~'2f  ') U" ,  (11) where the primes of U', U" denote differentiation with respect o y. Further, (2) reduces to ~u/Ox + ~v/~y = 0. (12) We now introduce the stream function ~v and write u = O~vlay ;  v = - ag / lax  (13) and write ~v= ~0(y) exp[i e(x - c t)], (14) where e (= 2zr d/2) is perturbation wave number, 2 is perturbation wave- length, e (= cr + i c 0 is wave velocity, i is imaginary number. Substituting (10a), (10b)-(14) into (8), (9), and eliminating p from (8), (9), we obtain finally i ot Re[ (U-  c) ( ( '  - ~2 ~o) - U" ~o] = (~o"/U")" + 2e2[( - 2y/U '  + 1/U")  ~o'1' + (1/U") ~' ~ + (1/U")" ~2(p. (15) 184 Fan Chun ZAMP This equation is true for all f and hence is valid for any generalized Newtonian fluid. The boundary conditions at the bottom of the layer are: or  u=0;  v=0 at y=l  ~o'(1) = 0, ~0(1) =0.  (16) The boundary conditions on the free surface are more complicated, since they must be applied on the free surface, not merely at y = 0. Let r/d be the dimensional deviation of the free surface from its mean position, so that ~q/~t + U(O) ~ l~x = v = - i ~ ~0(0) exp[i 0~(x - c t)] or  r/= [~0(0)/{c - U(0)) exp[i ~(x - c t)]. (17) At the free surface the shear stress must vanish, and the normal stress must balance with the normal stress induced by surface tension, thus we have [ ( f+ 2fi'zf ') (~f)/~xl + ~a/~xz) + ~'f]~-,  = 0 (18 a) and p f ~v ~2r/ e[a,(a)]2 d2 p+ + =0 (lSb) f ( [a '  (d)] 2 Re ~y S ~ , where S = Td/o[~t' (d)]", T being the surface tension. Equation (18 a) and (18 b) may be written in term of ~o as r + ~z ~0(r/) + U"(r/) ~0(0)/[c - U(0)] = 0 (19a) and {[~(cot fl + ~z SRe)] ~0(0)/[c- U(0) ] .  ~ Re(U-  c) ~o' + ~Re U'~o - i[(~o"/U")' + (1/U")' o~ 2~0 - (4y/U' - 1/U") ~2 r = 0. (19b) Solution for long wave Because we only consider the case of long waves, it is convenient to use Yih's method [4] and expand the eigenfunction ~0 and the eigenvalue c in power series of the small parameter ~, thus c= Co + ~ q + ~2 c2+ .. . .  (20) Substituting the foregoing series into equation (15) and boundary conditions at the free surface (19a), (19b) and collecting terms to the zeroth- Vol. 33, 1982 Stability of flow of a generalized Newtonian fluid 185 order in ~, we have the following differential equation (~o'/U")" = 0 (21) and the boundary conditions at the free surface are + + u ' (1 ) ] / [Co  - u(0)] = 0 or co =-  U'(1) (22) and [q~o' (t/)/U"]' = 0. (23) The final result of equation (21) and the boundary conditions (16), (23) is ~Oo(Y ) = U(y) - U' (1)y  + U' (1) ,  (24) where a multiplicative constant can be chosen to be unity without loss of generality. The first-order approximation is obtained by collecting terms of  order e, which yields the following governing differential equation (~o;'/U")" = i Re[(U - Co) ~0o" - U" ~0o]. (25) Since only terms of first order in e are retained in the differential system, boundary condition (19 b) becomes cos fl ~0 o(0)/[c 0 - U(0)] - Re (U - co) ~o + Re U' ~o + i(~o~'/U")" = 0. (26) As to boundary condition (19a) care must be taken that c suffers a change in the second approximation, so that the proper form of (19 a) is now ~o ' (t/1) + ~ ~o' 1'(r/1) + U" (q0 [q~o (0) + 9 ~ol (0)]/[c o - U (0) + e cd = 0. (27) For a first order approximation t/1 ~ 1 the terms of order t h are negligible, and so are the terms of  order e~ in boundary condition (27). The final result of equation (25) with its boundary conditions (26), (27), (16) is I 11 cl = i {cot fl - Re U' (1) [U' (1) + U(0)]} U' (1) + 2  U dy 0 where + i Re[G'(1)  - G(1)],  (28) 1 G(y)=U' (1 )S  I U" I fyU"dydy  dydy .  O0 O0 The critical Reynolds number  is [l+2iUdy/U'(1)]cotfl (Re)er  - ~ ~ a ~,  (29)  [U'(1)]z+4U'(1) I Udy+2 ~ UZdy+ I U'Zydy-~  U'2ydydy 0 0 0 O0 186 Fan Chun ZAMP If Re > (Re)e~, ci is positive, and the flow is unstable. If Re < (Re)er, c~ is negative, and the flow is stable. Calculation and disnussion In this paper, two kinds of generalized Newtonian fluid are considered. 1. Power series f i s  given by f=/z  +/q  ~z +/% ~, +/z, ~o. The dimensionless velocity gradient is U" + M~(U') 3 + M2(U')  ~ + M3 (g') 7 = -y ,  where M1 =/z3 (0 g sin ,8) 2 dV/.z3; M2 =/Zs(O g sin ,8)4 d4/,us; M3 = ~, (O g sin ,8)o d6//.z7. U(y)  has not, in general, analytical expression, numerical computation isneeded. For ]Mll < 0.2; M2 = M3 = O, we obtain an approximate value as follows (Re)cr = ( -  1 + M1 - 3M~)2 (5/2 + 46MJ7  - 2399MxV294) cot ,8. The result agrees with that of Yih [1]. Numerical computation has been carried out. The critical values of (Re)cr/Cot/~ for various values of M~, M2, M3 are given in Table 1 and illustrated in Fig. 3. Fig. 2 shows that constitutive quation of power law fluid of n = 1 - 1.3 bear similarity to that of power series fluid of MI = 0-0.8; M2 = M3 = 0. It 10 3.0 Figure 3 2.6 The critical Reynolds number (Re) JcotB  as e function of dimensionless parameter M. M1, M2, M3 are respectively 9 0.1, 0.05, 0.05; @ 0.3, Z2 0.3, 0; | 0.6, 0.6, 0; @ 0.8, 0.4, 0.2; | 0.8, - 0.2, 0. . . . .  values of power-law fluid, . . . . .  approximate values of Yih, 1.~ various values of M1, with M2 = M3 = 0. 1.1 1.2 1.3n | | i I I I 0 2 0.4 0.6 0.8 M 1 Vol. 33, 1982 Stability of flow of a generalized Newtonian fluid Table 1 Values of (Re)Jcot ~ for various values of M~, M2, M3. 187 M 1 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.10 0.30 0.60 0.80 0.80 Mz 0 0 0 0 0 0 0 0 0.05 0.30 0.60 0.40 -0.20 M~ " 0 0 0 0 0 0 0 0 0.05 0 0 0.20 0 (Re) e~ 2.55 2.63 2.69 2.75 2.79 2.83 2.86 2.90 2.63 2.83 2.98 3.00 2.84 cot ]? can also be seen from Fig. 3 that there exists a similarity between the stability characteristic of  power law fluid of n = 1 -  1.3 and that of  power series fluid o fM 1 = 0 -0 .8 ;  Ms = M3 = 0. 2. Solution and calculation fo r  viscoplastic f lu id  We write the constitutive quation of  a viscoplastic fluid as for 1 > -g Zst zts = z~ , (31 a) ~ij = 0 for -g zst rt~ = ~.  (31 b) where m is modulus index, n is power law index, z0 is the yield stress. Equation (31) has been found to describe adequately the rheological behavior  of emulsion used in film coating and thus the present work might be of  interest o the film industry. As is usual in solving viscoplastic flow problems we need to consider two layers one where (31 a) is applicable and the other where (31 b) is applicable. 1 The interface is where 2 z,t rt, = zo ~ 9 On non-dimensionalising all quantities, as above, we define the Reynolds number to be Re=[0gs in f ld (1 -  O)/m]Z/nd/9s in f l ,  where O= ro/Ogsinf ld.  The steady velocity distr ibution is n U(y)= l+n[1 -O- (y -O)a /n+l / (1 -O)  w] at y>-O,  n - - (1 -O)  at y~O U(y)  = 1 + n - " Superimposing small disturbances of  the form given above, using (31 b) we find that the interface is given by y = O + r/, where r/ is defined by kinematic ondit ion to be r /= [~0 (O) /{c  - U(O)}] exp [i e (x - c t)]. At the interface we assume: (a) The total velocity components must be continuous; (b) The stresses must be continuous. 188 Figure 4 The critical Reynolds number (Re)or/cot fl as a function of power law index n for various values of dimensionless yield stress O. 10 (Re).c.r cot Fan Chun ZAMP 01012 014 d6 d8 ' l.On Using foregoing interface condition and bottom condition (16), proceed- ing as in previous ection we obtain the critical Reynolds number to be (Re)er = (1 + n)(3n + 2)(1 + n + nO) / ( l  - O)[2(1 + n) 2 + (4n + 3) nO] .  Fig. 4 shows the critical Reynolds number as a function of power law index n for various values of dimensionless yield stress O. It can be seen from Fig. 4 that (Re)er/cot fl increases (more stable) with increasing O for any fixed n. Thus the effect of plasticity is to stabilize the flow. The analysis of this paper is restricted to the case of two dimensional long wave (wave number e ~ 1), and small perturbation amplitude ( r /~ 1). Acknowledgement The author is deeply indebted to Professor C. F. Chan Man Fong and to the referee for their comments. Reference [1] C. S. Yih, Phys. Fluids 8, 1257 (1965). [2] Fan Chun, to be published in Acta Mechanica Sinica No. 2 (1982). [3] A~ S. Gupta, J. Fluid Mech. 28, 17 (1967). [4] C. S. Yih, Phys. Fluids 6, 321 (1963). Summary The analogue of Orr-Sommerfeld equation is derived for a generalized Newtonian fluid. Based on this equation, the stability of such fluid flowing down an inclined plane under gravity is studied. The critical Reynolds number is given as a function of dimensionless teady flow velocity U(y) and the slope of the plane, and is computed for several f uids. R~sum~ On a Ctudir le probl~me de stabilit6 de l'~coulement d'un fluide Newtonien gCnCralisr sur un plan indinr. (Received: May 8, 1981; revised: December 9,1981.) 

Stability of flow of a generalized newtonian fluid down an inclined plane

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Stability of flow of a generalized newtonian fluid down an inclined plane

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Institute Of Mechanics,Chinese Academy of Sciences