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The critical stress in frictionally heated non-Newtonian plane Couette flow
Chemical Engineering Science, 1969, Vol. 24, pp. 1581-1587.
Pergamon Press.
Printed in Great Britain.
The critical stress in frictionally heated non-Newtonian plane Couette flow RAFFI M. TURIAN Department of Chemical Engineering and Metallurgy, Syracuse University, Syracuse, N.Y. 13210, U.S.A. (First received 10 February 1969; accepted26
May 1969)
Abstract-The plane Couette flow of a frictionally heated temperature dependent Ellis fluid is solved. The critical stress, above which steady-state temperature and velocity fields under prescribed boundary conditions cannot be established, is determined as a function of the Ellis number El and the parameter (Y. INTRODUCTION
and uniqueness of frictionally heated variable viscosity Newtonian Couette and Poiseuille flows was considered by Joseph [ 11. With respect to Couette flow, Joseph treated the general fluidity-temperature dependence, as well as the particular cases of the linear, the quadratic and the exponential temperature dependences. In a subsequent paper, Joseph[2] extended these results both with respect to generalizing thermal boundary conditions to include conduction-convection, and also with respect to the analysis of stability characteristics. Kearsley[3] considered solutions of the equation THE
EXISTENCE
(1) Special cases of this equation arise in frictionally heated exponentially temperature-dependent viscous flows. Kearsley obtained the general solutions of Eq. (1) in terms of tabulated funcan analysis of tions, and also presented uniqueness and existence of the solutions as a function of the parameter K. The phenomena of non-uniqueness and of critical parameters (above which steady-state solutions cannot be established under prescribed boundary conditions) arise in other physical situations, including heat generation in conducting solids [4], heat generation in buoyant viscous flows [5], and heat generation in chemically reacting systems [6].
Critical stresses in frictionally heated variable viscosity flows arise because above this value the frictionally generated heat cannot be conducted sufficiently rapidly to the surroundings. Under a steady temperature these circumstances, field, subject to prescribed boundary conditions, cannot be established. Aside from the temperature sensitivity of the viscosity and the prescribed boundary conditions the critical stress is affected by the viscosity-shear dependence. The frictionally heated Couette flow of an Ellis fluid is considered below to demonstrate this dependence. Couetteflow of Ellisfluid The viscosity-stress-temperature considered is given by ;=$[I+%]
dependence
=S[l+Y]
where TIC*, ~112, (Y, and /3 are constants,
(2)
and 8 = (T - To)/To is a dimensionless temperature referred to the wall temperature T,( T 2 To). A perturbation solution for this problem, including temperature variation of r1/2and the thermal conductivity k, was obtained by Turian[7]. When the thermal conductivity is assumed constant the equations of motion and energy
1581
d gGt=O
R. M. TURIAN
(4) can be solved easily in terms of tabulated fucntions. From Eq. (3) we note that T, must be maintained constant, rut = 7,,. But 7yz = 17(dvddy) , where the viscosity 7 decreases with increasing temperature, and with increasing shear stress (rate) for pseudoplastic liquids. In order to maintain the constant stress field when the temperature is increased, (dv,/dy) must increase, and, consequently, the viscous dissipation term in Eq. (4) increases. The calculations will be presented in terms of the following dimensionless quantities: 5 = y/b; $_I= v,/V; JI = /3 8; Br, = q,V/kT,
(5)
where b is the channel width, V is the velocity of the moving plate, and Br, is the Brinkmsin number. In terms of the dimensionless stress c Eq. (3) and (4) become:
(6) (7) where Eq. (6) is the integrated form of Eq. (3) in terms of dimensionless variables, and the reduced viscosity 7,. = &no0 is the positive real root of
when Eq. (9) is substituted in Eqs. (6) and (7) we get,
(10)
3 +Ae*=O with A = BrOPc$t{qt+p-l
(s$wVb)
(12)
Case 1: tanh { (archtanh w
(8)
In Eq. (8) El = (q,0V/bT1,2) is the Ellis number, and q1 is clearly the non-Newtonian viscosity in the absence of heat generation. From Eq. (2) and the fact T#== (vpY/b) =
(l-7,)).
Equations (10) and (11) are solved for the following two cases: (1). Both the stationary and moving walls at constant temperature T,,, i.e. 4(O) = 0, +( 1) = 1, and Jl(0) = JI( 1) = 0. (2) The stationary wall at constant temperature T,,, and no heat conduction through the moving wall, i.e. 4(O) = 0, 4(l) = 1, t/r(O) = 0, anddJl(l)/G=O. The solutions of Eqs. ( 10) and ( 11) are obtained in terms of I,%,,the maximum temperature. For Case 1 this occurs at the center of the channel, +,, = JI(+), and for Case 2, I,!J,, = JI( 1) and occurs at the moving wall. The integration of Eq. (11) is simple, and is effected by multiplying it by 2(d$/e) before the first integration is carried out. The solutions for the two cases posed above are :
x(28-1)
r),+qTTaEP-l= 1.
(11)
1
+1
1
) (13)
e* = eh sech2 { (arccosh de@@)(25 - 1))
(14)
Case 2:
we get,
?.@l~ = e@’(7,
x(5-1)
+ ~a-’ r),.“EEa-‘)
71
= e”‘{-++ P-l EP-l ( 1 - qr)}
(9)
I
+1
C = e@‘O sech2 { (arccosh veti)
1582
(15) (t - 1)) .
(16)
Non-Newtonian plate Couette flow
Newtonian case in the limit El = O(r), = 1), and also in the limit (Y= 1 (r), = 3). Equation (17) becomes in this limit
InEqs. (13-16)&isgivenby,
x=e@-1 a-1 EP_l = 1 arccosh deh x Veti d(eti - 1) I
(18)
(17)
which corresponds to the results obtained by Turian and Bird[8], in which x = Br,p/8 for in which x = Br&,./8 for Eqs. (13) and (14) case 1 and x,== Br&/2 for Case 2. To -obtain the relation for the power-law (Case 1), and x = Br&,./2 for Eq. (15) and (16) fluid we identify the parameter cr by s = l/n and (Case 2). Equation (17) can, therefore, be reif we let garded as giving I/J,,as a function of Br&r),, EZ m, by (r)oor;;l)s. Furthermore, vr = [mo( V/b)“-‘]/qoo in the term which inand (11. The value of q, is, of course, fixed through Eq. (8) when El and (Y are given. The cludes q.aEZ’x-l in Eq. (17), and drop the first similarity in the forms of Eq. (8) and (17) is term which includes Q, we get, evident, and this suggests the procedure for computing &,((Br&r),, El, a). The family of ted”-1) (8+1)/28(arccosh~/eh)b-0/8 (19) (@O)(S-1)/2s curves shown in Fig. 1 represent plots of r), against El with (Y as a parameter from Eq. (8). where x = BP ps/8 for Case 1 and x = BP &/2 for Case 2. We have replaced the temperature I.0 coefficient /3 by p7r, where p is the temperature --6PO coefficient of the parameter m, i.e. (m,,/m) = 3.0 0.8 exp (PO). The quantity Bi’ replaces Br,q., and, 2.0 consequently, the Brinkman number for the I.6 0.6 power-law fluid is defined in terms of the I.2 E-' I!0 I.0 “viscosity” mo( V/‘/6)‘+l. The power-law fluid 04 I.2 was considered by Gavis and Laurence[9], who I.6 presented their results in somewhat different 02 2.0 form. The function x(+~, s) is shown in Fig. 2. 30 x=
T,ss
6.
0 0.1
IO
10
loo
El
Fig. 1. The function qr + VJ,~EP-~= 1.
Given El and (Y,7, can be determined from such a plot, and, in addition, given x, Jlo can be determined by trial and error. The correct value of $,, q,(e*O- 1) will result in the point with ordinate X
and abscissa
arccosh veti0 Veti V(e*o-
line corresponding
1)>
The critical stress In Eq. (12) if we replace (l-r),.) by vr’%P-l and c by (rob/~oo~rV), where r. is the constant (dimensional) stress, we get, A = A+
where the stress parameter El are defined by
to the given value of (Y.
Equation (1 l), therefore, becomes
1583
(20)
A and the quantity
El falling on the
Newtonian andpower lawfluids Equations (8) and (17) can be reduced to the
CESVd24.10-D
~(U+l)/ZElm-1
R. M. TURIAN
=-
‘k{1+ (F)
=-
i
3
4
Fig. 2. Relation between Brinkman number and &for power-law from Eq. ( 19).
Multiplying Eq. (22) by 2(d$/Q) once, we get, "= 2{h+~t'a+l)/2Ela-l}
(zgEj+(E!$y
and integrating
(eVo-e'&).
euNOarccosh elINo 2(eh- 1)3’2 I$,.*
(27)
Hence, no steady-state solutions, subject to the prescribed boundary conditions, exist for values of A > A,.. For A < A,, there are two solutions corresponding to &, < JIOcr.and I/J,,> &,o,,., and the two solutions coincide at the point J10= I/J~~.. The values A,. ((Y,El) can be calculated from Eqs. (24) and (25), since the left hand side terms are known and A,. are the positive real roots of the equations. Alternatively, one can calculate the Brinkman numbers from Eq. (17) corresponding to the critical condition. This equation gives with Jlo = 1.19,
I
I
(26)
~(a-1)/2 EP-~}J~=~
6
0 0
el’* arccosh el’* 2(eh- 1)3’2 Jb,
(23)
Taking the square root of Eq. (23), and integrating between the limits I/J(O) = 0 and I,!J(l/2) = $,, for Case 1, and between I/J(O)= 0 and I/J(~) =+,forCase2,wefind, e-@‘O arccosh2 v/e*0 = i{ A+ Ata+l)12 El*-‘} Case 1 (24) e-h arccosh2 q/e”@= &{A+ A((cr+1)‘2 El”-‘} Case 2. (25) For positive A and &, Eq. (24) and (25) possess stationary points at which k = dA/d& = 0. This occurs at the point I&= Jlocr. = 1.19. That A,, = Altimr. is a maximum can be demonstrated by calculating Air=,, which is given, respectively, for Cases 1 and 2, by the equations
(O-4377 El)“-l = 1. (28)
Equation (28) holds strictly at the critical condition, and represents, in essence, the relation between the Brinkman number and El at this condition. In a given physical situation, when the boundary velocity and temperatures, and the relevant physical properties of the fluid are prescribed, Br,,, r), and El are fixed. Whether the value of x(or BrJ is greater, smaller or equal to the value at the critical condition can be determined from Eqs. (28) and (8). These equations are used to generate the plots of xcr. (El, a) in Fig. 3. For (Y> 1, xcr. decreases as El increases and approaches a limit xCrernas El approachesm. The asymptotic limit xCr.mdepends on CYand can be determined from the following equation: gr., = lim c,
= lim 2.287
I
(’ - ‘Q) = 2.287. 2.287 7,. XW.
>
(29)
Equation (29) is obtained by eliminating the terms QEP-’ between Eqs. (8) and (28), and using the fact that 7,. approaches 0 as El + CQ. 1584
Non-Newtonian plane Couette flow
x 0.6 -
Fig. 3. ,yer. (E&a)
for the Ellis model from Eqs. (8) and (28).
Aside from giving the asymptotic limit for El 3 CQfor the Ellis fluid, Eq. (29) gives the relation for the power-law fluid, XC,.= (2.287) I’#
(30)
where, as was done above, the definition xcr. must be modified to account for the fact that the power-law model does not contain a material parameter which has dimensions of viscosity. Equation (30) could alternatively be obtained from Eq. (19) by substituting I,$,= I,&~.= 1.19, since at this value v(e%cr.- 1) (arccosh V(eeocr,))/V(e*~r.) = 1. It will be observed from Fig. 3 that as (Yincreases the curves for xer. begin deviating from the Newtonian limit ((w= 1) at progressively larger values of El. This is because the nonNewtonian contribution, i.e. the term including El, in Eq. (28) becomes important at progressively higher values of El as (Yincreases. For the power-law fluid, RCcr.decreases rapidly as s increases and vice versa, the behavior corresponding to that when the non-Newtonian contribution in the Ellis fluid becomes dominant. CONCLUSIONS
The Brinkman number corresponding to the critical condition decreases as the apparent vis-
cosity of the fluid decreases, i.e. as El increases. This is to be expected since the variable viscosity fluid must generate higher shear rates than the constant viscosity fluid in order to maintain a constant stress field. For pseudoplastic fluids the viscosity decreases with both increasing temperature and shear, and the temperature and non-Newtonian effects reinforce each other. For dilatant fluids, on the other hand, the temperature and non-Newtonain effects counteract each other. The system considered here was treated by Turian[7] in an earlier paper in which a perturbation scheme was- utilized. The procedure in this earlier paper was justified on the grounds that the approximate solutions would be adequate for devising corrections to viscometric data, and, at any rate, was unavoidable since a complete analytical solution could not be obtained when the temperature dependence of the thermal conductivity was also included. No analysis of the critical stress was presented in this earlier paper, and it will be useful to make a few comments regarding the results in it in light of the calculations in the present paper. The perturbation solutions for the power-law fluid were presented as expansions in the quantity, we have designated here as, x and s. Our present results indicate that even in the extreme (and physically unlikely) limit of s = m the quantity 5%. = 1. Similarly for the Ellis fluid, the expansions were made in the quantities x, q,, and cx (and temperature coefficients of properties not considered here). It is also found here that in the corresponding extreme limit for the Ellis fluid, i.e. El + CO or equivalently q, --) 0, x,... + (2.287) l/a. In addition, as expansions, these earlier solutions are inevitably limited in their ranges of applicability, and this limitation may, in many cases, predominate that imposed by the critical condition. This is certainly the case for the expansions for the Newtonian fluid presented by Turian and Bird[8]. It should be emphasized that whereas the quantity (BP p),,. for the power-law approaches 0 in the limit s = 00, the quantity (BP ps),,. approaches 1 in this limit. These extreme limits of pseudoplastic
1585 .
R. M. TURIAN
behavior are not likely to occur at any rate, since non-Newtonain fluids often possess limiting viscosities at high shear rates. Consequently, the perturbation solutions in[7] are experimentally useful. The fact that two solutions exist for each value of the stress parameter X < A,, would indicate that it is possible to develop higher temperatures within the channel at lower values of Awhen one is operating on the branch JIO> I,&. than on the branch JIO< JIW. Joseph, 1,2] considered the stability problem associated with this system for Newtonian fluids, and indicated that the branch of the solution corresponding to Jl,, > Jlocr.is unstable. This does not necessarily mean that it is not possible to generate such a flow in practice, but indicates that when such a flow is established a disturbance would probably tend to restore the temperature field corresponding to that associated with the stable branch.
T To
temperature, “K reference or boundary temperature, “K 0, velocity, cm set-’ V plate velocity, cm se& y position coordinate, cm
Greek symbols
parameter in Ellis model, dimensionless temperature coefficient of qo, dimensionless temperature coefficient in power-law model apparent viscosity, P zero-shear viscosity at temperature T, P zero-shear viscosity at temperature To, P reduced viscosity (Eq. 8), dimensionless non-Newtonian viscosity in ab~0°% sence of heat generation, P ( T - To) / To, dimensionless temperature (~02b2~/kTo~oo) dimensionless stress parameter
NOTATION
b BrO BP
E”I
El k n,
m0
s
channel width, cm r)00V2/kT0, Brinkman number, dimensionless m. ( V/b) ~-l V2/kTo, Brinkman number for power-law fluid, dimensionless TOb/qlV, dimensionless stress ( r]ooV/~l12) Ellis number, dimensionless (q00kTo/b2~~,2P)1’2, Ellis number, dimensionless thermal conductivity, cal cm-l set-l OK-l parameters in power-law model
l/n
shear stresses, dyn cm-* parameter in Ellis model, dyn cm-2 vZ/V,dimensionless velocity Bropr),/8 or Br,&/2, Eq. (17), dimensionless BP ps/8 or BP /3s/2, Eq. (19), dimensionless xcr.at El = ~0,dimensionless @I, dimensionless temperature maximum value of I/I,dimensionless Subscript cr. = quantity evaluated at critical stress condition
REFERENCES
111JOSEPH D. D., Phys. Fluids 1964 7 1761.
JOSEPH D. D.,Phys. Fluids 1965 8 2195. E. A., J. Res. natn. Bur. Stand. 1963 67B 245. 141 JOSEPH D. D., Znt. J. Heat Mass Transfer 1965 8 281. JOSEPH D. D. and WARNER W. H., Q. J. appl. Math. 1967-68 25 163. ii; LANDAU L. D. and LIFSHITZ E. M., F/aid Mechanics, p. 191. Addison-Wesley 1959. 171 TURIAN R. M.,Chem. Engng Sci. 1965 20 771. TURIAN R. M. and BIRD R. B., Chem. Engng Sci. 1963 18 689.
PI
[31 KEARSLEY
1;; GAVIS J. and LAURENCE R. L., Znd. Engng Chem. Fundls 1968 7 525. R&WIII~-On a rt%olule cas de l’boulement plan Couette d’un fluide Ellis chaufF6par frictionnement et dkpendant de la tempkature. La tension critique, audeP de laquelle il est impossible d’ktablirles
1586
Non-Newtonian
plane Couette flow
champs de v&cite et de temperature en Ctat stable dans les conditions determin6e en tant que fonction du nombte d’Ellis El et du parametre (Y.
de limite prescrites,est
Zusamme&assung-Die Gleichungen ftir die ebene Couettesche Striimung einer durch Reibung erwiirmten, temperaturabh%rgigen Ellisschen Fliissigkeit werden geliist. Die kritische Spannung, oberhalb derer sich unter gegebenen Grenzbedingungen Temperatur- und Geschwindigkeitsfelder im statiomiren Zustand nicht einstehen konnen, wird bestimmt als eine Funktion der Ellisschen Zahl El und des Parameters (Y.
1587
Pergamon Press.
Printed in Great Britain.
The critical stress in frictionally heated non-Newtonian plane Couette flow RAFFI M. TURIAN Department of Chemical Engineering and Metallurgy, Syracuse University, Syracuse, N.Y. 13210, U.S.A. (First received 10 February 1969; accepted26
May 1969)
Abstract-The plane Couette flow of a frictionally heated temperature dependent Ellis fluid is solved. The critical stress, above which steady-state temperature and velocity fields under prescribed boundary conditions cannot be established, is determined as a function of the Ellis number El and the parameter (Y. INTRODUCTION
and uniqueness of frictionally heated variable viscosity Newtonian Couette and Poiseuille flows was considered by Joseph [ 11. With respect to Couette flow, Joseph treated the general fluidity-temperature dependence, as well as the particular cases of the linear, the quadratic and the exponential temperature dependences. In a subsequent paper, Joseph[2] extended these results both with respect to generalizing thermal boundary conditions to include conduction-convection, and also with respect to the analysis of stability characteristics. Kearsley[3] considered solutions of the equation THE
EXISTENCE
(1) Special cases of this equation arise in frictionally heated exponentially temperature-dependent viscous flows. Kearsley obtained the general solutions of Eq. (1) in terms of tabulated funcan analysis of tions, and also presented uniqueness and existence of the solutions as a function of the parameter K. The phenomena of non-uniqueness and of critical parameters (above which steady-state solutions cannot be established under prescribed boundary conditions) arise in other physical situations, including heat generation in conducting solids [4], heat generation in buoyant viscous flows [5], and heat generation in chemically reacting systems [6].
Critical stresses in frictionally heated variable viscosity flows arise because above this value the frictionally generated heat cannot be conducted sufficiently rapidly to the surroundings. Under a steady temperature these circumstances, field, subject to prescribed boundary conditions, cannot be established. Aside from the temperature sensitivity of the viscosity and the prescribed boundary conditions the critical stress is affected by the viscosity-shear dependence. The frictionally heated Couette flow of an Ellis fluid is considered below to demonstrate this dependence. Couetteflow of Ellisfluid The viscosity-stress-temperature considered is given by ;=$[I+%]
dependence
=S[l+Y]
where TIC*, ~112, (Y, and /3 are constants,
(2)
and 8 = (T - To)/To is a dimensionless temperature referred to the wall temperature T,( T 2 To). A perturbation solution for this problem, including temperature variation of r1/2and the thermal conductivity k, was obtained by Turian[7]. When the thermal conductivity is assumed constant the equations of motion and energy
1581
d gGt=O
R. M. TURIAN
(4) can be solved easily in terms of tabulated fucntions. From Eq. (3) we note that T, must be maintained constant, rut = 7,,. But 7yz = 17(dvddy) , where the viscosity 7 decreases with increasing temperature, and with increasing shear stress (rate) for pseudoplastic liquids. In order to maintain the constant stress field when the temperature is increased, (dv,/dy) must increase, and, consequently, the viscous dissipation term in Eq. (4) increases. The calculations will be presented in terms of the following dimensionless quantities: 5 = y/b; $_I= v,/V; JI = /3 8; Br, = q,V/kT,
(5)
where b is the channel width, V is the velocity of the moving plate, and Br, is the Brinkmsin number. In terms of the dimensionless stress c Eq. (3) and (4) become:
(6) (7) where Eq. (6) is the integrated form of Eq. (3) in terms of dimensionless variables, and the reduced viscosity 7,. = &no0 is the positive real root of
when Eq. (9) is substituted in Eqs. (6) and (7) we get,
(10)
3 +Ae*=O with A = BrOPc$t{qt+p-l
(s$wVb)
(12)
Case 1: tanh { (archtanh w
(8)
In Eq. (8) El = (q,0V/bT1,2) is the Ellis number, and q1 is clearly the non-Newtonian viscosity in the absence of heat generation. From Eq. (2) and the fact T#== (vpY/b) =
(l-7,)).
Equations (10) and (11) are solved for the following two cases: (1). Both the stationary and moving walls at constant temperature T,,, i.e. 4(O) = 0, +( 1) = 1, and Jl(0) = JI( 1) = 0. (2) The stationary wall at constant temperature T,,, and no heat conduction through the moving wall, i.e. 4(O) = 0, 4(l) = 1, t/r(O) = 0, anddJl(l)/G=O. The solutions of Eqs. ( 10) and ( 11) are obtained in terms of I,%,,the maximum temperature. For Case 1 this occurs at the center of the channel, +,, = JI(+), and for Case 2, I,!J,, = JI( 1) and occurs at the moving wall. The integration of Eq. (11) is simple, and is effected by multiplying it by 2(d$/e) before the first integration is carried out. The solutions for the two cases posed above are :
x(28-1)
r),+qTTaEP-l= 1.
(11)
1
+1
1
) (13)
e* = eh sech2 { (arccosh de@@)(25 - 1))
(14)
Case 2:
we get,
?.@l~ = e@’(7,
x(5-1)
+ ~a-’ r),.“EEa-‘)
71
= e”‘{-++ P-l EP-l ( 1 - qr)}
(9)
I
+1
C = e@‘O sech2 { (arccosh veti)
1582
(15) (t - 1)) .
(16)
Non-Newtonian plate Couette flow
Newtonian case in the limit El = O(r), = 1), and also in the limit (Y= 1 (r), = 3). Equation (17) becomes in this limit
InEqs. (13-16)&isgivenby,
x=e@-1 a-1 EP_l = 1 arccosh deh x Veti d(eti - 1) I
(18)
(17)
which corresponds to the results obtained by Turian and Bird[8], in which x = Br,p/8 for in which x = Br&,./8 for Eqs. (13) and (14) case 1 and x,== Br&/2 for Case 2. To -obtain the relation for the power-law (Case 1), and x = Br&,./2 for Eq. (15) and (16) fluid we identify the parameter cr by s = l/n and (Case 2). Equation (17) can, therefore, be reif we let garded as giving I/J,,as a function of Br&r),, EZ m, by (r)oor;;l)s. Furthermore, vr = [mo( V/b)“-‘]/qoo in the term which inand (11. The value of q, is, of course, fixed through Eq. (8) when El and (Y are given. The cludes q.aEZ’x-l in Eq. (17), and drop the first similarity in the forms of Eq. (8) and (17) is term which includes Q, we get, evident, and this suggests the procedure for computing &,((Br&r),, El, a). The family of ted”-1) (8+1)/28(arccosh~/eh)b-0/8 (19) (@O)(S-1)/2s curves shown in Fig. 1 represent plots of r), against El with (Y as a parameter from Eq. (8). where x = BP ps/8 for Case 1 and x = BP &/2 for Case 2. We have replaced the temperature I.0 coefficient /3 by p7r, where p is the temperature --6PO coefficient of the parameter m, i.e. (m,,/m) = 3.0 0.8 exp (PO). The quantity Bi’ replaces Br,q., and, 2.0 consequently, the Brinkman number for the I.6 0.6 power-law fluid is defined in terms of the I.2 E-' I!0 I.0 “viscosity” mo( V/‘/6)‘+l. The power-law fluid 04 I.2 was considered by Gavis and Laurence[9], who I.6 presented their results in somewhat different 02 2.0 form. The function x(+~, s) is shown in Fig. 2. 30 x=
T,ss
6.
0 0.1
IO
10
loo
El
Fig. 1. The function qr + VJ,~EP-~= 1.
Given El and (Y,7, can be determined from such a plot, and, in addition, given x, Jlo can be determined by trial and error. The correct value of $,, q,(e*O- 1) will result in the point with ordinate X
and abscissa
arccosh veti0 Veti V(e*o-
line corresponding
1)>
The critical stress In Eq. (12) if we replace (l-r),.) by vr’%P-l and c by (rob/~oo~rV), where r. is the constant (dimensional) stress, we get, A = A+
where the stress parameter El are defined by
to the given value of (Y.
Equation (1 l), therefore, becomes
1583
(20)
A and the quantity
El falling on the
Newtonian andpower lawfluids Equations (8) and (17) can be reduced to the
CESVd24.10-D
~(U+l)/ZElm-1
R. M. TURIAN
=-
‘k{1+ (F)
=-
i
3
4
Fig. 2. Relation between Brinkman number and &for power-law from Eq. ( 19).
Multiplying Eq. (22) by 2(d$/Q) once, we get, "= 2{h+~t'a+l)/2Ela-l}
(zgEj+(E!$y
and integrating
(eVo-e'&).
euNOarccosh elINo 2(eh- 1)3’2 I$,.*
(27)
Hence, no steady-state solutions, subject to the prescribed boundary conditions, exist for values of A > A,.. For A < A,, there are two solutions corresponding to &, < JIOcr.and I/J,,> &,o,,., and the two solutions coincide at the point J10= I/J~~.. The values A,. ((Y,El) can be calculated from Eqs. (24) and (25), since the left hand side terms are known and A,. are the positive real roots of the equations. Alternatively, one can calculate the Brinkman numbers from Eq. (17) corresponding to the critical condition. This equation gives with Jlo = 1.19,
I
I
(26)
~(a-1)/2 EP-~}J~=~
6
0 0
el’* arccosh el’* 2(eh- 1)3’2 Jb,
(23)
Taking the square root of Eq. (23), and integrating between the limits I/J(O) = 0 and I,!J(l/2) = $,, for Case 1, and between I/J(O)= 0 and I/J(~) =+,forCase2,wefind, e-@‘O arccosh2 v/e*0 = i{ A+ Ata+l)12 El*-‘} Case 1 (24) e-h arccosh2 q/e”@= &{A+ A((cr+1)‘2 El”-‘} Case 2. (25) For positive A and &, Eq. (24) and (25) possess stationary points at which k = dA/d& = 0. This occurs at the point I&= Jlocr. = 1.19. That A,, = Altimr. is a maximum can be demonstrated by calculating Air=,, which is given, respectively, for Cases 1 and 2, by the equations
(O-4377 El)“-l = 1. (28)
Equation (28) holds strictly at the critical condition, and represents, in essence, the relation between the Brinkman number and El at this condition. In a given physical situation, when the boundary velocity and temperatures, and the relevant physical properties of the fluid are prescribed, Br,,, r), and El are fixed. Whether the value of x(or BrJ is greater, smaller or equal to the value at the critical condition can be determined from Eqs. (28) and (8). These equations are used to generate the plots of xcr. (El, a) in Fig. 3. For (Y> 1, xcr. decreases as El increases and approaches a limit xCrernas El approachesm. The asymptotic limit xCr.mdepends on CYand can be determined from the following equation: gr., = lim c,
= lim 2.287
I
(’ - ‘Q) = 2.287. 2.287 7,. XW.
>
(29)
Equation (29) is obtained by eliminating the terms QEP-’ between Eqs. (8) and (28), and using the fact that 7,. approaches 0 as El + CQ. 1584
Non-Newtonian plane Couette flow
x 0.6 -
Fig. 3. ,yer. (E&a)
for the Ellis model from Eqs. (8) and (28).
Aside from giving the asymptotic limit for El 3 CQfor the Ellis fluid, Eq. (29) gives the relation for the power-law fluid, XC,.= (2.287) I’#
(30)
where, as was done above, the definition xcr. must be modified to account for the fact that the power-law model does not contain a material parameter which has dimensions of viscosity. Equation (30) could alternatively be obtained from Eq. (19) by substituting I,$,= I,&~.= 1.19, since at this value v(e%cr.- 1) (arccosh V(eeocr,))/V(e*~r.) = 1. It will be observed from Fig. 3 that as (Yincreases the curves for xer. begin deviating from the Newtonian limit ((w= 1) at progressively larger values of El. This is because the nonNewtonian contribution, i.e. the term including El, in Eq. (28) becomes important at progressively higher values of El as (Yincreases. For the power-law fluid, RCcr.decreases rapidly as s increases and vice versa, the behavior corresponding to that when the non-Newtonian contribution in the Ellis fluid becomes dominant. CONCLUSIONS
The Brinkman number corresponding to the critical condition decreases as the apparent vis-
cosity of the fluid decreases, i.e. as El increases. This is to be expected since the variable viscosity fluid must generate higher shear rates than the constant viscosity fluid in order to maintain a constant stress field. For pseudoplastic fluids the viscosity decreases with both increasing temperature and shear, and the temperature and non-Newtonian effects reinforce each other. For dilatant fluids, on the other hand, the temperature and non-Newtonain effects counteract each other. The system considered here was treated by Turian[7] in an earlier paper in which a perturbation scheme was- utilized. The procedure in this earlier paper was justified on the grounds that the approximate solutions would be adequate for devising corrections to viscometric data, and, at any rate, was unavoidable since a complete analytical solution could not be obtained when the temperature dependence of the thermal conductivity was also included. No analysis of the critical stress was presented in this earlier paper, and it will be useful to make a few comments regarding the results in it in light of the calculations in the present paper. The perturbation solutions for the power-law fluid were presented as expansions in the quantity, we have designated here as, x and s. Our present results indicate that even in the extreme (and physically unlikely) limit of s = m the quantity 5%. = 1. Similarly for the Ellis fluid, the expansions were made in the quantities x, q,, and cx (and temperature coefficients of properties not considered here). It is also found here that in the corresponding extreme limit for the Ellis fluid, i.e. El + CO or equivalently q, --) 0, x,... + (2.287) l/a. In addition, as expansions, these earlier solutions are inevitably limited in their ranges of applicability, and this limitation may, in many cases, predominate that imposed by the critical condition. This is certainly the case for the expansions for the Newtonian fluid presented by Turian and Bird[8]. It should be emphasized that whereas the quantity (BP p),,. for the power-law approaches 0 in the limit s = 00, the quantity (BP ps),,. approaches 1 in this limit. These extreme limits of pseudoplastic
1585 .
R. M. TURIAN
behavior are not likely to occur at any rate, since non-Newtonain fluids often possess limiting viscosities at high shear rates. Consequently, the perturbation solutions in[7] are experimentally useful. The fact that two solutions exist for each value of the stress parameter X < A,, would indicate that it is possible to develop higher temperatures within the channel at lower values of Awhen one is operating on the branch JIO> I,&. than on the branch JIO< JIW. Joseph, 1,2] considered the stability problem associated with this system for Newtonian fluids, and indicated that the branch of the solution corresponding to Jl,, > Jlocr.is unstable. This does not necessarily mean that it is not possible to generate such a flow in practice, but indicates that when such a flow is established a disturbance would probably tend to restore the temperature field corresponding to that associated with the stable branch.
T To
temperature, “K reference or boundary temperature, “K 0, velocity, cm set-’ V plate velocity, cm se& y position coordinate, cm
Greek symbols
parameter in Ellis model, dimensionless temperature coefficient of qo, dimensionless temperature coefficient in power-law model apparent viscosity, P zero-shear viscosity at temperature T, P zero-shear viscosity at temperature To, P reduced viscosity (Eq. 8), dimensionless non-Newtonian viscosity in ab~0°% sence of heat generation, P ( T - To) / To, dimensionless temperature (~02b2~/kTo~oo) dimensionless stress parameter
NOTATION
b BrO BP
E”I
El k n,
m0
s
channel width, cm r)00V2/kT0, Brinkman number, dimensionless m. ( V/b) ~-l V2/kTo, Brinkman number for power-law fluid, dimensionless TOb/qlV, dimensionless stress ( r]ooV/~l12) Ellis number, dimensionless (q00kTo/b2~~,2P)1’2, Ellis number, dimensionless thermal conductivity, cal cm-l set-l OK-l parameters in power-law model
l/n
shear stresses, dyn cm-* parameter in Ellis model, dyn cm-2 vZ/V,dimensionless velocity Bropr),/8 or Br,&/2, Eq. (17), dimensionless BP ps/8 or BP /3s/2, Eq. (19), dimensionless xcr.at El = ~0,dimensionless @I, dimensionless temperature maximum value of I/I,dimensionless Subscript cr. = quantity evaluated at critical stress condition
REFERENCES
111JOSEPH D. D., Phys. Fluids 1964 7 1761.
JOSEPH D. D.,Phys. Fluids 1965 8 2195. E. A., J. Res. natn. Bur. Stand. 1963 67B 245. 141 JOSEPH D. D., Znt. J. Heat Mass Transfer 1965 8 281. JOSEPH D. D. and WARNER W. H., Q. J. appl. Math. 1967-68 25 163. ii; LANDAU L. D. and LIFSHITZ E. M., F/aid Mechanics, p. 191. Addison-Wesley 1959. 171 TURIAN R. M.,Chem. Engng Sci. 1965 20 771. TURIAN R. M. and BIRD R. B., Chem. Engng Sci. 1963 18 689.
PI
[31 KEARSLEY
1;; GAVIS J. and LAURENCE R. L., Znd. Engng Chem. Fundls 1968 7 525. R&WIII~-On a rt%olule cas de l’boulement plan Couette d’un fluide Ellis chaufF6par frictionnement et dkpendant de la tempkature. La tension critique, audeP de laquelle il est impossible d’ktablirles
1586
Non-Newtonian
plane Couette flow
champs de v&cite et de temperature en Ctat stable dans les conditions determin6e en tant que fonction du nombte d’Ellis El et du parametre (Y.
de limite prescrites,est
Zusamme&assung-Die Gleichungen ftir die ebene Couettesche Striimung einer durch Reibung erwiirmten, temperaturabh%rgigen Ellisschen Fliissigkeit werden geliist. Die kritische Spannung, oberhalb derer sich unter gegebenen Grenzbedingungen Temperatur- und Geschwindigkeitsfelder im statiomiren Zustand nicht einstehen konnen, wird bestimmt als eine Funktion der Ellisschen Zahl El und des Parameters (Y.
1587