SHAPE OF A DROP OR BUBBLE A T LOW REYNOLDS NUMBER F R A N K Y . P A N 1 A N D A N D R E A S A C R I V O S Department of Chemical Engineering, Stanford Un.versity, Stan ford, Calif. 94305
The shape of a gas bubble or liquid drop in steady translation through an unbounded, quiescent, and viscous fluid medium i s determined for the case where the Reynolds number pertaining to the continuous phase and the Weber number are both small and the interface is clean. The inertia forces of the internal circulation do not affect the shape o f a gas bubble, and contribute in a minor, but significant, way to the deformation of a liquid drop.
H E N moving through ,an unbounded, quiescent fluid, a gas bubble or a liquid drop can assume one of several shapesfor example, that of a sphere, an ellipsoid, or a spherical cap. And yet, despite the numerous experimental investigations on the subject (Haberman and Morton, 1953; H u and Kintner, 1955; Hughes and Gillilartd, 1952; Klee and Treybal, 1956; Licht and Narasimhamurty, 1955; Warshay et al., 1959; Wellek et al., 1962) attempts to correlate the various observed geometries with the many physical parameters of any given system have not proved too successful. This is owing partly, to the difficulty of obtaining accurate and reliable data, either because of droplet oscillations or breakup, or because of the spurious effects of surface x t i v e agents, which, especially for systems containing an aqueous phase, can sometimes alter drastically the character of the flow even if present in minute quantities (Davis and Acrivos, 1966; Horton et al., 1965; Savic, 1953). Perhaps, though, the primary reason for this lack of success is that the pertinent physical parameters are generally much too numerous to permit a meaningful description of the phenomenon by using dimensional analysis alone. On the theoretical side, progress has also been hampered by the difficulty of the problem, primarily because of the nonlinearities in the equations of motion and the a priorz unknowm geometry of the bubble surface. Hence, past theoretical analyses have been, as a rule, approximate (Lane and Green, 1956). Nevertheless, since small bubbles or drops are, generally, almost spherical in shape, Taylor and Acrivos (1964) were able to show by a perturbation technique that, to a first approximation in creeping flow, these will deform into oblate spheroids. This andysis is, however, restricted to cases where the Weber number and the Reynolds number pertaining ‘to both phases are all small-Le., 0 ( 1 ) or less-and where any effects due to surface active agents can safely be ignored. Furthermore, for the purpose of testing this theoretical result experimentally it is necessary that neither the bubble radius nor the Weber number be too small; otherwise the deformation itself would be too small to be detectable. As a consequence, there are practically no systems to which this solution by Taylor and Acrivos could be applied directly and without ambiguity except, possibly, for the case of a gas bubble rising in a very viscous oil. Here, with a bubble of reasonable size-i.e., having a diameter of about 1 cm.the Reynolds number in the continuous phase could be made 0 ( 1 ) or less, the Weber number would be 0 ( 1 ) , and surface effects due to the presence of impurities would be negligible;
in fact, it appears that all the restrictions imposed in the analysis could be satisfied except for the magnitude of the gas-phase Reynolds number which, typically, would be 0 (lo2). I n view of the importance of the subject it seemed desirable, therefore, to extend the Taylor and Acrivos solution to the more general case where the Reynolds number pertaining to the motion inside of the bubble is unrestricted and then to compare the theoretical predictions with appropriate experimental results. Analysis
Let us consider a bubble rising under buoyancy in a viscous unbounded, and quiescent fluid. Assuming axial symmetry, we can express the dimensionless equations of motion in spherical coordinates ( r , p = cos e) as
for the bulk, and, using a carat to distinguish the variables pertaining to the interior of the bubble,
inside the bubble, where y and K are, respectively, the ratio of the density and viscosity of the bubble to that of the exterior fluid, # is the stream function and
(3) The origin of the coordinates will be located at the center of mass of the bubble. I n addition, we select as the characteristic length the radius, a , of a sphere having an equivalent volume, and represent the interface by W P ) =
1
+
(4)
where { ( p ) will measure the extent of the deformation, I n this analysis we assume that Max { ( p ) >
+ 1)' ] + O(Re):!
Pz(d
12(K
e,
(n
(26)
+ 2m - 2 ) ( n + 2m - 3)+ n(n + I ) ( n + Zm - I) VOL 7
NO. 2
M A Y 1968
229
where n is an integration constant. However, since from an over-all force balance,
in which the drag coefficient, FD, is given by Brenner and Cox (1963) as
it follows that
such that Re < 1 and We is at most O(1). In practice, however, the latter is almost always met whenever the former is true, since We is equal to (pY2/au)Re2, with (pv2/au) being O(10) or less for most liquids. For a gas bubble rising in a liquid, the magnitudes of K and y are usually 0(10+) to O(lO-&),and O(10e3),respectively, and thus, within the limits of the present analysis, k e should be at most O(102). In this range, however, the contributions of the last term in Equation 32 to { ( p ) are rather insignificant (less than l%), so that by setting K and y both equal to zero we obtain for the first-order deformation {(p) =
-0.21 WeP,(p)
(34)
which is identical to the result previously derived by Taylor and Acrivos (1964). In other words, the inertia effect due to the gas motion inside the bubble is so small that, for the purpose of calculating the deformation, it may be neglected. We conclude, therefore, that the result of Taylor and Acrivos (1964) is applicable to bubbles even when B e 1 ; thus, according to Taylor and Acrivos (1964), if the perturbation due to the deformation is taken into account, the surface of the bubble is described by R(p) = 1 - 0.21 WeP2(p) - 0.09 We2P&) (35) Re
where
>>
where the last term presupposes that
PV2
> O(1).
au
Equation 30 can easily be solved in terms of Legendre polynomials, subject to the requirements (5 and 6), to yield
Clearly, the assumption that the shape of the drop deviates only slightly from a sphere irrespective of the value of K and y is met as long as We is sufficiently small. Also, the term containing Pl(p), which was present in the expressions for both pressures and stream functions, is seen to cancel out identically in the inhomogeneous part of Equation 30. This is as should be, because P l ( p ) is a solution to the associated homogeneous equation of 30, so that its presence in the inhomogeneous term would exclude solutions finite in the range -1 p 1 and thereby invalidate the small deformation assumption regard0(1), less of how small the value of We. In addition, if Be we may use instead the stream function representations 23 and 24 and obtain
< <
