Znd. Eng. Chem. Res. 1995,34, 3185-3186
3185
Effects of Viscous Normal Stresses in Thin Draining Films Mark W.Vaughn and John C. Slattern* Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122
The hydrodynamics of film drainage is normally treated using lubrication theory. Bird et al. suggested an alternative approach. Here, we compare these two approaches and find that lubrication theory neglects an edge effect. When the surfaces of the film are mobile, there is a n additional normal force. Though typically negligible, in some cases this normal force may have a n effect.
Introduction The squeezing flow between two parallel disks provides a simplified model for film drainage during the coalescence of a bubble or drop at an interface (Edwards et al., 1991). This problem is normally analyzed using a scaling argument to neglect terms in the mass and momentum balances (Landau and Lifshitz, 1987; Bird et al., 1987). Bird et al. (1977) presented an interesting and different approach to this problem for an incompressible Newtonian fluid that avoided the use of a scaling argument. With reference t o Figure 1, they assumed creeping flow and postulated that (Bird et al. (1977) use a frame of reference that is fued with respect t o the center of the film rather than the one used here, which is fured with respect to the lower interface) u, = v,.(r,z,t>
Figure 1. Idealized film formed as a small bubble rises through a continuous liquid to aninterface between the liquid and another gas. The film is observed in a frame of reference in which the interface between the bubble and the liquid is stationary.
ferential mass balance reduces to Unfortunately, they also assumedp =p(r,t),which was inconsistent with the z-component of the differential momentum balance. They apparently recognized this later, because in the second edition of their book (Bird et al., 1987) a traditional scaling argument was used. In what follows, we will extend the core of the original Bird et al. (1977) argument both to a draining film bounded by immobile interfaces (large surface tension gradients or large surface viscosities as the result of soluble surfactants in the system) and to a draining film bounded by mobile interfaces (uniform surface tension and vanishingly small surface viscosities in the absence of soluble surfactants). As they did, we will consider the creeping flow of an incompressible Newtonian fluid, neglecting the effects of gravity. Note that these films are deformable, and their shape is specified by the normal component of the jump momentum balance (Slattery, 1990). In approximating these films as being bounded by parallel interfaces, we will follow the common practice of replacing the normal component of the jump momentum balance with an integral force balance. Experimental evidence suggests that nearly plane-parallel films are not uncommon.
and the r and z components of the differential momentum balance become
(4)
These equations are t o be solved consistent with the boundary conditions
at z = 0:
u r = 0, u, = 0
at z = h(t):
u, = 0, v, = dt
dh
(5)
(6)
at r = 0: u, = 0 (7) where h ( t ) is the position of the upper interface. Bird et al. (1977) observed that the postulated flow field and (2) implied that the derivative with respect to r in (3) is zero and that the r-component of the differential momentum balance became
Immobile Interfaces Let us follow Bird et al. (1977) in seeking a solution of the form (1). Under these circumstances the dif-
without using any scaling arguments. This and (4)
0888-588519512634-3185$09.00/00 1995 American Chemical Society
3186 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 fluid exerts on the upper interface:
imply that p = f,(z,t)
+ f2(r,t)
(9)
We conclude that
or
It is common t o impose the condition atr=R: P =Ph (11) where Ph is the hydrostatic pressure. This clearly contradicts (4), (9), and (10). If we ignore this edge effect, we find that the force which the fluid exerts on the bubble is
wherepo is the pressure in the bubble. Note that the normal stress S, on an immobile interface is always zero (Bird et al., 1987).
Mobile Interfaces When the surface is mobile (or partially mobile, as in the drainage of a thin film which is stabilized by a soluble surfactant), the situation is different. We must replace ( 5 ) and (6)with a t z = 0:
au
-I=o, a2
uz=o
a t z = h(t):
(14) (15)
Integration of (2) and (8) with these boundary conditions as well as (11)gives lrdh
ur=-2hdt zdh
uz=zdt P
=Ph
(17)
Conclusions The viscous normal stress S,, affects the force that the fluid exerts on the bubble only in the case of mobile (or partially mobile) interfaces. In the case of small bubbles, it is easy t o argue that the effect of S,, can be neglected with respect to the first term on the right in (211, as it has been common to assume (Linand Slattery, 1982). The effect of S,, is likely to be more important in the case of larger bubbles, where the first term on the right of (21) approaches zero.
Literature Cited Barber, A. D.; Hartland, S. The effects of surface viscosity on the axisymmetric drainage of planar liquid films. Can. J . Chem. Eng. 1976,54, 279. Bird, R. B.; Armstrong, R. C.; Hassager, 0. In Dynamics of Polymeric Liquids; Vol. 1, Fluid Mechanics; John Wiley & Sons: New York, 1977; Example 1.2-6. Bird, R. B.; Armstrong, R. C.; Hassager, 0. In Dynamics of Polymeric Liquids, 2nd ed.; Vol. 1,Fluid Mechanics; John Wiley & Sons: New York, 1987; pp 12,20. Edwards, D. A.; Brenner, H.; Wasan, D. T. In Interfacial Transport Processes and Rheology; Butterworth-Heinemann: Boston, 1991; Section 11.3. Landau, L. D.; Lifshitz, E. M. In Fluid Mechanics, 2nd ed.; Vol. 6 of Course of Theoretical Physics; Pergamon Press: Elmsford, NY,1987; p 66. Lin, C. Y.; Slattery, J. C. Thinning of a liquid film as a drop or bubble approaches a solid plane. AZChE J . 1982,28,147. Slattery, J. C. In Interfacial Transport Phenomena; SpringerVerlag: New York, 1990; Tables 2.4.2-2, 2.4.2-4.
r
(18) Received for review January 4, 1995 Accepted April 24, 1995@
Noting that 3% s,, = 2p a2
(19) h dt we have from (12) the z-component of the force that the
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Abstract published in Advance ACS Abstracts, August 15, 1995. @
