A note on boundary layer solutions for pseudoplastic fluids

Shorter Communications VO

Initial bubble volume, cm3.

Y Mole fraction soluble gas absorbed from diluted bubble, mole/mole.

z z’

Y' Mole fraction soluble gas absorbed from pure bubble,

mole/mole. Moles soluble gas desorbed into diluted bubble. Moles soluble gas desorbed into bubble containing 100 % inerts initially.

REFERENCES

PI PI

[31 [41 ti; ii;

DANCKWERTSP. V., Chem. Engng Sci., 1965 20 785. ALLEN H. S., Phil. Mag., 1900 50 323. CALDERBANKP. H. and JONESS. J. R., Trans. Znstn. Chem. Engrs., 1961,39 363. FFUEDLANDER S. K., A. I. Ch. E. JI., 1957 3 43. FR&SLING N., Beitr. Geophys. 1938 52 170. DAVIESR. M. and TAYLORG. I., Prof. R. Sot. 1950 AZOO375. HIGBIE R., Trans. Am. Inst. Chem. Engrs, 1935 31 365. BOOTHA. D., Numerical Methods, 2nd Ed. 1957. pp. 62, 64. Butterworths, London.

Chemical Engineerning

Science, 1966, Vol. 21, pp. 618-620 Pergamon Press Ltd., Oxford.

A note on boundary

layer solutions

for pseudoplastic

Printed in Great Britain.

fluids

(Received 23 September 1965)

THE OBJECTof the present note is to draw attention to some peculiarities arising when the equations of continuity and of momentum are solved for the two-dimensional, laminar flow of an incompressible pseudoplastic fluid, using the boundary layer approximations. The case of viscous wake flow will be used as an illustrative example. The rheological equation for the two-dimensional laminar flow of an incompressible, non-elastic pseudoplastic fluid may be reduced to (c.f. [l]), Tzy

=

e +2)

m[ao($+ $)2](n-1)‘a

Cl)

where 72y is the shear stress, XI and yl are Cartesian coordinates, ~1 and VI are the velocities in the directions x1 and yl respectively, m is a rheological constant having the dimensions of (Newtonian) viscosity, n is a rheological index and ao is a dimensional constant of magnitude unity and dimension IT21. In the following, boundary layer approximations will be introduced, so that the term au/ax will be neglected versus aujay. The boundary layer equations in dimensionless form reduce to the following, when account is taken of Eq. (1):

u!c++‘a

yd

ay

(Yd[(g)2](R-1)‘2

;]

(2)

E + -$;(Ydv) = 0. Here u is the dimensionless velocity in the direction of the main stream, u = u/U, where U is a reference velocity; x = xl/l, with I a reference length; y = (y~/l).R~l(l+n) and v = (vI/LI).R~‘(~+“) are the “stretched” coordinate and the velocity in that direction respectively. Also,

618

and 6 is an integer which takes only the values of zero (for two-dimensional flow) or unity (for axisymmetrical flow). The two-dimensional wake Consider the case of a wake arising far downstream from a two-dimensional obstacle in an otherwise uniform stream. This flow configuration has been analysed in detail for a Newtonian fluid in classical papers by GOLDSTEIN[2] and has recently been reconsidered in [3], [4]. For the configuration under consideration here one may assume that sufficiently far downstream the velocity anywhere in the wake no longer differs appreciably from that of the streaming flow outside the wake. Choosing for convenience the streaming velocity outside the wake region as reference velocity, and introducing the velocity defect, ti, one has, ic=l-u. (5) The expression (5) may be introduced into Eqs. (2) and (3) which have to be solved subject to the following boundary conditions: hijay = 0, v = 0, at y = 0, x>o (6) ii =o, aty = co, x > 0. A physically acceptable solution must also be such that li-+oasx+co. The total drag exerted by the wake must remain constant, once the wake has left the obstacle. This gives rise to the condition that Drag (p/2). us. 1 = R&

CD = --

Jorn41 - O)dy

(7)

be constant with x. Introducing (5) into (2) it is seen that, provided ci Q 1, a first order solution may be obtained by

Shorter Communications considering initially a linearized inertia term. The effect of non-zero s may be introduced later through a higher order approximation, [2]. Thus we obtain for the first order term the equation

g = g. Introducing

([(!!)2](“-1)‘zc!).

Pa)

a stream function

a*

a# u=-ax

“=-ay

the continuity equation is fulfilled identically. now the following similarity transformation, I/ = b.f(l;)

5 = a.y.x-r/(an)

Introduce (n > 4)

(9)

5

with a= b =

T R’,c’+ n) [

(1-n)‘(zn)(2*2)-1,(2n, I

y

RWl+“l.

Then Eq. (2a) reduces to, f”cf”)n-l

+ Jf” + f’ = 0,

and the drag integral normalizes I

(x > 0)

(10)

to

o*f’dc = -1.

f’@

(11)

FIG. 1.

The boundary conditions (16) become, X>O x > 0. 1

f(O) =f”(O) = 0, f’(cc) = 0

(6a)

The solution of Eq. (lo), subject to (6a) and to (ll), for the special case n = 1 (the Newtonian fluid) is of course well known, [2] f=-erf(512/2) (n= 1) (12) and, c= %

4Y’.rrX

Rl I2 . exp

(-

52/z)

(n =

1).

Numerical results for the functionf(Q for various values of the exponent n are shown in the graph, Fig. 1, while the corresponding numerical values of the constant Cn are given in Table 1. Table 1 ?I

(13)

1.00 0.95 090 0.85 415 314 o-70 213 315 l/2

For n < 1 the equation may be solved in closed form as well, except for a constant of integration which has to be determined numerically. Thus, -1

f’=

1-n [-lfn

.&ln.Ql+n,ln

n/U-n)

+ Cn

(14)

I

and a-b fi = - (~;fYo

e = + A.$([) 2nx The constant

Cn is determined

s F

0

1 - II _.,l/n.~clrntln [ 1+n

(15) (16)

. from,

4 + Cn

n/(1-n) =I. 1

(17)

C” 1WOOO l-01494 1.03922 1.07693 1.13489 l-22374 l-36496 1.50424 1a99385 4.33839

1 for

fi/&=o)

=

1%

3.034 3.283 3.588 3.970 4.481 5.152 6.070 6.884 9.552 > 12

Note increasing width of wake as n decreases. DISCUWON The equations of motion and continuity have been solved, using the boundary layer approximations and an Oseen type linearization, and the shear-stress to rate of deformation relationship for a pseudoplastic fluid. It was shown that a solution for the first order term can be obtained in closed form, at least for values of the rheological exponent n smaller than unity. The same solution applies for a two-

619

Shorter Communications

, dimensional weak jet-sheet at a large distance downstream from the or&e, [6]. A second approximation can easily be obtained by choosing as perturbation parameter the ratio of the velocity defect at [ = 0 (given by Eq. (15)) to the streaming velocity at intinity (i.e. unity), s = a.b.Cnnlcn-l,,-llczn,.

(18)

The resulting differential equation is, however, highly complex and cannot be solved in closed form except for it = 1. Therefore no useful purpose is served by giving the derivation here. It is well known that a non-Newtonian simple fluid should revert to Newtonian behaviour in the limit of very small rates of shear. This can be shown theoretically and is also found when these fluids are investigated experimentally. The rheological Eq. (I) does not of course conform to this requirement, but usually the discrepancy is of no great importance. We shall show, however, that in the present case this defect of Eq. (1) leads to qualitatively incorrect solutions. It is shown in the theory of laminar viscous flow of Newtonian fluids, that for flow configurations such as the one discussed here, vorticity, and hence also the velocity component I?, has to decrease at a rate faster than algebraical with distance normal to the plane of symmetry of the wake, see [7] p. 214, [3]. The pseudoplastic fluid should behave in the same manner, as (a) most of the qualitative arguments for fast decay remain valid for this case also, and (b) the fluid should revert to Newtonian behaviour at small rate of deformation. It is however quite obvious from Eq. (14) that for the present

conliauration decav is onlv alaebraical. The nresent examnle is by no means unique. To cite another, Bm~‘[8] has extended the solution of Stokes’ first problem to pseudoplastic fluids, finding that the decay of the main velocity takes place at an algebraical rate except for n = 1 (the classical case). As mentioned above, the fault lies with the model of fluid behaviour assumed, Eq. (I). This leads to solutions which are unacceptable at distances normal to the plane of symmetry of the wake which are not small. A further unexpected effect is found when it is tried to to extend the viscous wake solution to axisymmetrical geometry. The similarity transformation is, 7 = A .y.~ll(l-3~)

,

$

=

B.m$

A = [,r.(3n _ l).Bn-l]‘/(‘-36) B = Co.R210+“,/4,r

PI [31 141 :z; [71 PI

Acknowledgements-Thanks are due to the Canada National Research Council for allocation of computing time through the Computation Center of The University of British Columbia. Z. ROTEM Department of Mechanical Engineering, University of British Columbia, Vancouver 8, Canada.

ROTEMZ. and SHINNAR R. Chem. Engng. Sci. 1961 15 130 GOLDSTEINS. Proc. R. Sot. 1933, A 142 545 STEWART~~NK. J. Math. Phys 1957 36 173 WYGNANSKII. J. and ROTEMZ. to be oublished ROTEMZ. Appl. Scient. Res. 1964 A lj 353 WYGNANSKII. J. and ROTEMZ. to be published. ROSENHEADL. Ed. Laminar Boundary Layers, Oxford University Press 1963 BYRDR. B., A. I. Ch. E. JI. 1959 5 565

ERRATUM “Mass transfer from fixed surfaces to gas fluid&d Engng Sci. 21, 117. Equation (4) on page 120 should read:

Y=

(20)

which, when inserted into (2) and (3) with 6 = 1 yields a nonlinear total differential equation for which a closed form solution can only be obtained when n = 1. It is readily seen that such a similarity transformation would have meaning only for n > l/3 ! Such peculiar behaviour has been described before in a different context [5].

REFERENCES VI

(19)

where,

beds” by E. N. ZIEGLERand J. T. HOLMES,Chem.

(Ge - Gem,)

CL

[W’P>~(PP 1ii - P)PP

620