## CAVITATION AND BUBBLE DYNAMICS

by     Christopher Earls Brennen     © Oxford University Press 1995

## CHAPTER 5. TRANSLATION OF BUBBLES

5.1 INTRODUCTION

This chapter will briefly review the issues and problems involved in constructing the equations of motion for individual bubbles (or drops or solid particles) moving through a fluid and will therefore focus on the dynamics of relative motion rather than the dynamics of growth and collapse. For convenience we shall use the generic name ``particle'' when any or all of bubbles, drops, and solid particles are being considered. The analyses are implicitly confined to those circumstances in which the interactions between neighboring particles are negligible. In very dilute multiphase flows in which the particles are very small compared with the global dimensions of the flow and are very far apart compared with the particle size, it is often sufficient to solve for the velocity and pressure, ui(xi ,t) and p(xi ,t), of the continuous suspending fluid while ignoring the particles or disperse phase. Given this solution one could then solve an equation of motion for the particle to determine its trajectory. This chapter will focus on the construction of such a particle or bubble equation of motion. Interactions between particles or, more particularly, bubble, are left for later.

The body of fluid mechanical literature on the subject of flows around particles or bodies is very large indeed. Here we present a summary that focuses on a spherical particle of radius, R, and employs the following common notation. The components of the translational velocity of the center of the particle will be denoted by Vi(t). The velocity that the fluid would have had at the location of the particle center in the absence of the particle will be denoted by Ui(t). Note that such a concept is difficult to extend to the case of interactive multiphase flows. Finally, the velocity of the particle relative to the fluid is denoted by Wi(t)=Vi -Ui.

Frequently the approach used to construct equations for Vi(t) (or Wi(t)) given Ui(xi ,t) is to individually estimate all the fluid forces acting on the particle and to equate the total fluid force, Fi, to mpdVi /dt (where mp is the particle mass, assumed constant). These fluid forces may include forces due to buoyancy, added mass, drag, etc. In the absence of fluid acceleration (dUi /dt=0) such an approach can be made unambigiously; however, in the presence of fluid acceleration, this kind of heuristic approach can be misleading. Hence we concentrate in the next few sections on a fundamental fluid mechanical approach, which minimizes possible ambiguities. The classical results for a spherical particle or bubble are reviewed first. The analysis is confined to a suspending fluid that is incompressible and Newtonian so that the basic equations to be solved are the continuity equation
 ......(5.1)
and the Navier-Stokes equations
 ......(5.2)
where ρ and ν are the density and kinematic viscosity of the suspending fluid. It is assumed that the only external force is that due to gravity, g. Then the actual pressure is p′=p-ρgz where z is a coordinate measured vertically upward.

Furthermore, in order to maintain clarity we confine attention to rectilinear relative motion in a direction conveniently chosen to be the x1 direction.

For steady flows about a sphere in which dUi /dt=dVi /dt=dWi /dt=0, it is convenient to use a coordinate system, xi , fixed in the particle as well as polar coordinates (r,θ) and velocities ur,uθ as defined in Figure 5.1.

 Figure 5.1 Notation for a spherical particle.

Then Equations 5.1 and 5.2 become
 ......(5.3)
and
 ......(5.4)
 ......(5.5)
The Stokes streamfunction, ψ, is defined to satisfy continuity automatically:
 ......(5.6)
and the inviscid potential flow solution is
 ......(5.7)
 ......(5.8)
 ......(5.9)
 ......(5.10)
where, because of the boundary condition (ur)r=R=0, it follows that D=-WR3/2. In potential flow one may also define a velocity potential, φ, such that ui=∂φ/∂xi. The classic problem with such solutions is the fact that the drag is zero, a circumstance termed D'Alembert's paradox. The flow is symmetric about the x2 x3 plane through the origin and there is no wake.

The real viscous flows around a sphere at large Reynolds numbers, Re=2WR/ν>1, are well documented. In the range from about 103 to 3×105, laminar boundary layer separation occurs at θ≈84° and a large wake is formed behind the sphere (see Figure 5.2). Close to the sphere the ``near-wake'' is laminar; further downstream transition and turbulence occurring in the shear layers spreads to generate a turbulent ``far-wake.'' As the Reynolds number increases the shear layer transition moves forward until, quite abruptly, the turbulent shear layer reattaches to the body, resulting in a major change in the final position of separation (θ≈120°) and in the form of the turbulent wake (Figure 5.2). Associated with this change in flow pattern is a dramatic decrease in the drag coefficient, CD (defined as the drag force on the body in the negative x1 direction divided by ½ρW2πR2), from a value of about 0.5 in the laminar separation regime to a value of about 0.2 in the turbulent separation regime (Figure 5.3). At values of Re less than about 103 the flow becomes quite unsteady with periodic shedding of vortices from the sphere.

Figure 5.2 Smoke visualization of the nominally steady flows (from left to right) past a sphere showing, on the left, laminar separation at Re=2.8×105 and, on the right, turbulent separation at Re=3.9×105. Photographs by F.N.M.Brown, reproduced with the permission of the University of Notre Dame.

Figure 5.3 Drag coefficient on a sphere as a function of Reynolds number. Dashed curves indicate the drag crisis regime in which the drag is very sensitive to other factors such as the free stream turbulence.

At the other end of the Reynolds number spectrum is the classic Stokes solution for flow around a sphere. In this limit the terms on the left-hand side of Equation 5.2 are neglected and the viscous term retained. This solution has the form
 ......(5.11)
 ......(5.12)
 ......(5.13)
where A and B are constants to be determined from the boundary conditions on the surface of the sphere. The force, F, on the ``particle" in the x1 direction is
 ......(5.14)
Several subcases of this solution are of interest in the present context. The first is the classic Stokes (1851) solution for a solid sphere in which the no-slip boundary condition, (uθ)r=R = 0, is applied (in addition to the kinematic condition (ur)r=R=0). This set of boundary conditions, referred to as the Stokes boundary conditions, leads to
 ......(5.15)
The second case originates with Hadamard (1911) and Rybczynski (1911) who suggested that, in the case of a bubble, a condition of zero shear stress on the sphere surface would be more appropriate than a condition of zero tangential velocity, uθ. Then it transpires that
 ......(5.16)
Real bubbles may conform to either the Stokes or Hadamard-Rybczynski solutions depending on the degree of contamination of the bubble surface, as we shall discuss in more detail in the next section. Finally, it is of interest to observe that the potential flow solution given in Equations 5.7 to 5.10 is also a subcase with
 ......(5.17)
However, another paradox, known as the Whitehead paradox, arises when the validity of these Stokes flow solutions at small (rather than zero) Reynolds numbers is considered. The nature of this paradox can be demonstrated by examining the magnitude of the neglected term, uj∂ui /∂xj, in the Navier-Stokes equations relative to the magnitude of the retained term ν∂2ui /∂xj∂xj. As is evident from Equation 5.11, far from the sphere the former is proportional to W2R/r2 whereas the latter behaves like νWR/r3. It follows that although the retained term will dominate close to the body (provided the Reynolds number Re=2WR/ν « 1), there will always be a radial position, rc, given by R/rc=Re beyond which the neglected term will exceed the retained viscous term. Hence, even if Re « 1, the Stokes solution is not uniformly valid. Recognizing this limitation, Oseen (1910) attempted to correct the Stokes solution by retaining in the basic equation an approximation to uj∂ui /∂xj that would be valid in the far field, -W∂ui /∂x1. Thus the Navier-Stokes equations are approximated by
 ......(5.18)
Oseen was able to find a closed form solution to this equation that satisfies the Stokes boundary conditions approximately:
 ......(5.19)
which yields a drag force
 ......(5.20)
It is readily shown that Equation 5.19 reduces to 5.11 as Re→0. The corresponding solution for the Hadamard-Rybczynski boundary conditions is not known to the author; its validity would be more questionable since, unlike the case of Stokes' boundary conditions, the inertial terms uj∂ui /∂xj are not identically zero on the surface of the bubble.

More recently Proudman and Pearson (1957) and Kaplun and Lagerstrom (1957) showed that Oseen's solution is, in fact, the first term obtained when the method of matched asymptotic expansions is used in an attempt to patch together consistent asymptotic solutions of the full Navier-Stokes equations for both the near field close to the sphere and the far field. They also obtained the next term in the expression for the drag force.
 ......(5.21)
The additional term leads to an error of 1% at Re=0.3 and does not, therefore, have much practical consequence.

The most notable feature of the Oseen solution is that the geometry of the streamlines depends on the Reynolds number. The downstream flow is not a mirror image of the upstream flow as in the Stokes or potential flow solutions. Indeed, closer examination of the Oseen solution reveals that, downstream of the sphere, the streamlines are further apart and the flow is slower than in the equivalent upstream location. Furthermore, this effect increases with Reynolds number. These features of the Oseen solution are entirely consistent with experimental observations and represent the initial development of a wake behind the body.

The flow past a sphere at Reynolds numbers between about 0.5 and several thousand has proven intractable to analytical methods though numerical solutions are numerous. Experimentally, it is found that a recirculating zone (or vortex ring) develops close to the rear stagnation point at about Re=30 (see Taneda 1956 and Figure 5.4). With further increase in the Reynolds number this recirculating zone or wake expands. Defining locations on the surface by the angle from the front stagnation point, the separation point moves forward from about 130° at Re=100 to about 115° at Re=300. In the process the wake reaches a diameter comparable to that of the sphere when Re≈130. At this point the flow becomes unstable and the ring vortex that makes up the wake begins to oscillate (Taneda 1956). However, it continues to be attached to the sphere until about Re=500 (Torobin and Gauvin 1959).

Figure 5.4 Streamlines of steady flow (from left to right) past a sphere at various Reynolds numbers (from Taneda 1956, reproduced by permission of the author).

At Reynolds numbers above about 500, vortices begin to be shed and then convected downstream. The frequency of vortex shedding has not been studied as extensively as in the case of a circular cylinder and seems to vary more with Reynolds number. In terms of the conventional Strouhal number, St, defined as
 ......(5.22)
the vortex shedding frequencies, f, that Moller (1938) observed correspond to a range of St varying from 0.3 at Re=1000 to about 1.8 at Re=5000. Furthermore, as Re increases above 500 the flow develops a fairly steady ``near-wake'' behind which vortex shedding forms an unsteady and increasingly turbulent ``far-wake.'' This process continues until, at a value of Re of the order of 1000, the flow around the sphere and in the near-wake again becomes quite steady. A recognizable boundary layer has developed on the front of the sphere and separation settles down to a position about 84° from the front stagnation point. Transition to turbulence occurs on the free shear layer, which defines the boundary of the near-wake and moves progessively forward as the Reynolds number increases. The flow is similar to that of the top picture in Figure 5.2. Then the events described in the previous section occur with further increase in the Reynolds number.

Since the Reynolds number range between 0.5 and several hundred can often pertain in multiphase flows, one must resort to an empirical formula for the drag force in this regime. A number of empirical results are available; for example, Klyachko (1934) recommends
 ......(5.23)
which fits the data fairly well up to Re≈1000. At Re=1 the factor in the square brackets is 1.167, whereas the same factor in Equation 5.20 is 1.187. On the other hand, at Re=1000, the two factors are respectively 17.7 and 188.5.

As a postscript to the steady, viscous flows of the last section, it is of interest to introduce and describe the forces that a bubble may experience due to gradients in the surface tension, S, over the surface. These are called Marangoni effects. The gradients in the surface tension can be caused by a number of different factors. For example, gradients in the temperature, solvent concentration, or electric potential can create gradients in the surface tension. The ``thermocapillary'' effects due to temperature gradients have been explored by a number of investigators (for example, Young, Goldstein, and Block 1959) because of their importance in several technological contexts. For most of the range of temperatures, the surface tension decreases linearly with temperature, reaching zero at the critical point. Consequently, the controlling thermophysical property, dS/dT, is readily identified and more or less constant for any given fluid. Some typical data for dS/dT is presented in Table 5.1 and reveals a remarkably uniform value for this quantity for a wide range of liquids.

 TABLE 5.1 Values of the temperature gradient of the surface tension, -dS/dT, for pure liquid/vapor interfaces (in kg/s2°K). Water 2.02× 10-4 Methane 1.84× 10-4 Hydrogen 1.59× 10-4 Butane 1.06× 10-4 Helium-4 1.02× 10-4 Carbon Dioxide 1.84× 10-4 Nitrogen 1.92× 10-4 Ammonia 1.85× 10-4 Oxygen 1.92× 10-4 Toluene 0.93× 10-4 Sodium 0.90× 10-4 Freon-12 1.18× 10-4 Mercury 3.85× 10-4 Uranium Dioxide 1.11× 10-4

Surface tension gradients affect free surface flows because a gradient, dS/ds, in a direction, s, tangential to a surface clearly requires that a shear stress act in the negative s direction in order that the surface be in equilibrium. Such a shear stress would then modify the boundary conditions (for example, the Hadamard-Rybczynski conditions used in the preceding section), thus altering the flow and the forces acting on the bubble.

As an example of the Marangoni effect, we will examine the steady motion of a spherical bubble in a viscous fluid when there exists a gradient of the temperature (or other controlling physical property), dT/dx1, in the direction of motion (see Figure 5.1). We must first determine whether the temperature (or other controlling property) is affected by the flow. It is illustrative to consider two special cases from a spectrum of possibilities. The first and simplest special case, which is not so relevant to the thermocapillary phenomenon, is to assume that T=(dT/dx1)x1 throughout the flow field so that, on the surface of the bubble,
 ......(5.24)
Much more realistic is the assumption that thermal conduction dominates the heat transfer (Laplacian of T is zero) and that there is no heat transfer through the surface of the bubble. Then it follows from the solution of Laplace's equation for the conductive heat transfer problem that
 ......(5.25)
The latter is the solution presented by Young, Goldstein, and Block (1959), but it differs from Equation 5.24 only in terms of the effective value of dS/dT. Here we shall employ Equation 5.25 since we focus on thermocapillarity, but other possibilities such as Equation 5.24 should be borne in mind.

For simplicity we will continue to assume that the bubble remains spherical. This assumption implies that the surface tension differences are small compared with the absolute level of S and that the stresses normal to the surface are entirely dominated by the surface tension.

With these assumptions the tangential stress boundary condition for the spherical bubble becomes
 ......(5.26)
and this should replace the Hadamard-Rybczynski condition of zero shear stress that was used in the preceding section. Applying Equation 5.26 with Equation 5.25 and the usual kinematic condition, (ur)r=R=0, to the general solution of the preceding section leads to
 ......(5.27)
and consequently, from Equation 5.14, the force acting on the bubble becomes
 ......(5.28)
In addition to the normal Hadamard-Rybczynski drag (first term), we can identify a Marangoni force, 2πR2(dS/dx1), acting on the bubble in the direction of decreasing surface tension. Thus, for example, the presence of a uniform temperature gradient, dT/dx1, would lead to an additional force on the bubble of magnitude 2πR2(-dS/dT)(dT/dx1) in the direction of the warmer fluid since the surface tension decreases with temperature. Such thermocapillary effects have been observed and measured by Young, Goldstein, and Block (1959) and others.

Finally, we should comment on a related effect caused by surface contaminants that increase the surface tension. When a bubble is moving through liquid under the action, say, of gravity, convection may cause contaminants to accumulate on the downstream side of the bubble. This will create a positive dS/dθ gradient which, in turn, will generate an effective shear stress acting in a direction opposite to the flow. Consequently, the contaminants tend to immobilize the surface. This will cause the flow and the drag to change from the Hadamard-Rybczynski solution to the Stokes solution for zero tangential velocity. The effect is more pronounced for smaller bubbles since, for a given surface tension difference, the Marangoni force becomes larger relative to the buoyancy force as the bubble size decreases. Experimentally, this means that surface contamination usually results in Stokes drag for spherical bubbles smaller than a certain size and in Hadamard-Rybczynski drag for spherical bubbles larger than that size. Such a transition is observed in experiments measuring the rise velocity of bubbles as, for example, in the Haberman and Morton (1953) experiments discussed in more detail in Section 5.12. The effect has been analyzed in the more complex hydrodynamic case at higher Reynolds numbers by Harper, Moore, and Pearson (1967).

Though only rarely important in the context of bubbles, there are some effects that can be caused by the molecular motions in the surrounding fluid. We briefly list some of these here.

When the mean free path of the molecules in the surrounding fluid, λ, becomes comparable with the size of the particles, the flow will clearly deviate from the continuum models, which are only relevant when λ  «  R. The Knudsen number, Kn=λ/2R, is used to characterize these circumstances, and Cunningham (1910) showed that the first-order correction for small but finite Knudsen number leads to an additional factor, (1+2AKn), in the Stokes drag for a spherical particle. The numerical factor, A, is roughly a constant of order unity (see, for example, Green and Lane 1964).

When the impulse generated by the collision of a single fluid molecule with the particle is large enough to cause significant change in the particle velocity, the resulting random motions of the particle are called ``Brownian motion'' (Einstein 1956). This leads to diffusion of solid particles suspended in a fluid. Einstein showed that the diffusivity, D, of this process is given by
 ......(5.29)
where k is Boltzmann's constant. It follows that the typical rms displacement, λ, of the particle in a time, t, is given by
 ......(5.30)
Brownian motion is usually only significant for micron- and sub-micron-sized particles. The example quoted by Einstein is that of a 1μm diameter particle in water at 17°C for which the typical displacement during one second is 0.8μm.

A third, related phenomenon is the reponse of a particle to the collisions of molecules when there is a significant temperature gradient in the fluid. Then the impulses imparted to the particle by molecular collisions on the hot side of the particle will be larger than the impulses on the cold side. The particle will therefore experience a net force driving it in the direction of the colder fluid. This phenomenon is known as thermophoresis (see, for example, Davies 1966). A similar phenomenon known as photophoresis occurs when a particle is subjected to nonuniform radiation. One could, of course, include in this list the Bjerknes forces described in Section 4.10 since they constitute sonophoresis.

Having reviewed the steady motion of a particle relative to a fluid, we must now consider the consequences of unsteady relative motion in which either the particle or the fluid or both are accelerating. The complexities of fluid acceleration are delayed until the next section. First we shall consider the simpler circumstance in which the fluid is either at rest or has a steady uniform streaming motion (U=constant) far from the particle. Clearly the second case is readily reduced to the first by a simple Galilean transformation and it will be assumed that this has been accomplished.

In the ideal case of unsteady inviscid potential flow, it can then be shown by using the concept of the total kinetic energy of the fluid that the force on a rigid particle in an incompressible flow is given by Fi, where
 ......(5.31)
where Mij is called the added mass matrix (or tensor) though the name ``induced inertia tensor'' used by Batchelor (1967) is, perhaps, more descriptive. The reader is referred to Sarpkaya and Isaacson (1981), Yih (1969), or Batchelor (1967) for detailed descriptions of such analyses. The above mentioned methods also show that Mij for any finite particle can be obtained from knowledge of several steady potential flows. In fact,
 ......(5.32)
where the integration is performed over the entire volume of the fluid. The velocity field, uij, is the fluid velocity in the i direction caused by the steady translation of the particle with unit velocity in the j direction. Note that this means that Mij is necessarily a symmetric matrix. Furthermore, it is clear that particles with planes of symmetry will not experience a force perpendicular to that plane when the direction of acceleration is parallel to that plane. Hence if there is a plane of symmetry perpendicular to the k direction, then for i≠k, Mki=Mik=0, and the only off-diagonal matrix elements that can be nonzero are Mij, j≠k, i≠k. In the special case of the sphere all the off-diagonal terms will be zero.

Tables of some available values of the diagonal components of Mij are given by Sarpkaya and Isaacson (1981) who also summarize the experimental results, particularly for planar flows past cylinders. Other compilations of added mass results can be found in Kennard (1967), Patton (1965), and Brennen (1982). Some typical values for three-dimensional particles are listed in Table 5.2. The uniform diagonal value for a sphere (often referred to simply as the added mass of a sphere) is 2ρπR3/3 or one-half the displaced mass of fluid. This value can readily be obtained from Equation 5.32 using the steady flow results given in Equations 5.7 to 5.10. In general, of course, there is no special relation between the added mass and the displaced mass. Consider, for example, the case of the infinitely thin plate or disc with zero displaced mass which has a finite added mass in the direction normal to the surface. Finally, it should be noted that the literature contains little, if any, information on off-diagonal components of added mass matrices.

TABLE 5.2
Added masses (diagonal terms in Mij) for some three-dimensional bodies (particles):
(T) Potential flow calculations, (E) Experimental data from Patton (1965).

Particle Matrix Element Value

Sphere (T)      Mii   2ρπR3/3
Disc (T) M11 8ρR3/3
Ellipsoids (T) Mii=Kii4ρπab2/3
 a/b K11 K22(K33) 2 0.209 0.702 5 0.059 0.895 10 0.021 0.960
 Sphere near Plane Wall (T)
Mii= Kii 4ρπR3/3
 K11=½(1+3R3/8H3+....) K22=½(1+3R3/16H3+....) K33=K22
 Sphere near Free Surface (E)
Mii= Kii 4ρπR3/3
 H/R K11 8.0 0.52 4.0 0.59 2.0 0.54 1.0 0.44 0.0 0.25

Now consider the application of these potential flow results to real viscous flows at high Reynolds numbers (the case of low Reynolds number flows will be discussed in Section 5.8). Significant doubts about the applicability of the added masses calculated from potential flow analysis would be justified because of the experience of D'Alembert's paradox for steady potential flows and the substantial difference between the streamlines of the potential and actual flows. Furthermore, analyses of experimental results will require the separation of the ``added mass'' forces from the viscous drag forces. Usually this is accomplished by heuristic summation of the two forces so that
 ......(5.33)
where Cij is a lift and drag coefficient matrix and A is a typical cross-sectional area for the body. This is known as Morison's equation (see Morison et al. 1950).

Actual unsteady high Reynolds number flows are more complicated and not necessarily compatible with such simple superposition. This is reflected in the fact that the coefficients, Mij and Cij, appear from the experimental results to be not only functions of Re but also functions of the reduced time or frequency of the unsteady motion. Typically experiments involve either oscillation of a body in a fluid or acceleration from rest. The most extensively studied case involves planar flow past a cylinder (for example, Keulegan and Carpenter 1958), and a detailed review of this data is included in Sarkaya and Isaacson (1981). For oscillatory motion of the cylinder with velocity amplitude, UM , and period, t*, the coefficients are functions of both the Reynolds number, Re=2UMR/ν, and the reduced period or Keulegan-Carpenter number, Kc=UM t*/2R. When the amplitude, UM t*, is less than about 10R (Kc<5), the inertial effects dominate and Mii is only a little less than its potential flow value over a wide range of Reynolds numbers (104<Re<106). However, for larger values of Kc, Mii can be substantially smaller than this and, in some range of Re and Kc, may actually be negative. The values of Cii (the drag coefficient) that are deduced from experiments are also a complicated function of Re and Kc. The behavior of the coefficients is particularly pathological when the reduced period, Kc, is close to that of vortex shedding (Kc of the order of 10). Large transverse or ``lift'' forces can be generated under these circumstances. To the author's knowledge, detailed investigations of this kind have not been made for a spherical body, but one might expect the same qualitative phenomena to occur.

In general, a particle moving in any flow other than a steady uniform stream will experience fluid accelerations, and it is therefore necessary to consider the structure of the equation governing the particle motion under these circumstances. Of course, this will include the special case of acceleration of a particle in a fluid at rest (or with a steady streaming motion). As in the earlier sections we shall confine the detailed solutions to those for a spherical particle or bubble. Furthermore, we consider only those circumstances in which both the particle and fluid acceleration are in one direction, chosen for convenience to be the x1 direction. The effect of an external force field such as gravity will be omitted; it can readily be inserted into any of the solutions that follow by the addition of the conventional buoyancy force.

All the solutions discussed are obtained in an accelerating frame of reference fixed in the center of the fluid particle. Therefore, if the velocity of the particle in some original, noninertial coordinate system, xi*, was V(t) in the x1* direction, the Navier-Stokes equations in the new frame, xi, fixed in the particle center are
 ......(5.34)
where the pseudo-pressure, P, is related to the actual pressure, p, by
 ......(5.35)
Here the conventional time derivative of V(t) is denoted by d/dt, but it should be noted that in the original xi* frame it implies a Lagrangian derivative following the particle. As before, the fluid is assumed incompressible (so that continuity requires ∂ui /∂xi=0) and Newtonian. The velocity that the fluid would have at the xi origin in the absence of the particle is then W(t) in the x1 direction. It is also convenient to define the quantities r, θ, ur, uθ as shown in Figure 5.1 and the Stokes streamfunction as in Equations 5.6. In some cases we shall also be able to consider the unsteady effects due to growth of the bubble so the radius is denoted by R(t).

First consider inviscid potential flow for which Equations 5.34 may be integrated to obtain the Bernoulli equation
 ......(5.36)
where φ is a velocity potential (ui=∂φ/∂xi) and ψ must satisfy the equation
 ......(5.37)
This is of course the same equation as in steady flow and has harmonic solutions, only five of which are necessary for present purposes:
 ......(5.38)
 ......(5.39)
 ......(5.40)
 ......(5.41)
The first part, which involves W and D, is identical to that for steady translation. The second, involving A and B, will provide the fluid velocity gradient in the x1 direction, and the third, involving E, permits a time-dependent particle (bubble) radius. The W and A terms represent the fluid flow in the absence of the particle, and the D, B ,and E terms allow the boundary condition
 ......(5.42)
to be satisfied provided
 ......(5.43)
In the absence of the particle the velocity of the fluid at the origin, r=0, is simply -W in the x1 direction and the gradient of the velocity ∂u1/∂x1=4A/3. Hence A is determined from the fluid velocity gradient in the original frame as
 ......(5.44)
Now the force, F1, on the bubble in the x1 direction is given by
 ......(5.45)
which upon using Equations 5.35, 5.36, and 5.39 to 5.41 can be integrated to yield
 ......(5.46)
Reverting to the original coordinate system and using τ as the sphere volume for convenience (τ=4πR3/3), one obtains
 ......(5.47)
where the two Lagrangian time derivatives are defined by
 ......(5.48)
 ......(5.49)
Equation 5.47 is an important result, and care must be taken not to confuse the different time derivatives contained in it. Note that in the absence of bubble growth, of viscous drag, and of body forces, the equation of motion that results from setting F1=mp dV/dt* is
 ......(5.50)
where mp is the mass of the ``particle.'' Thus for a massless bubble the acceleration of the bubble is three times the fluid acceleration.

In a more comprehensive study of unsteady potential flows Symington (1978) has shown that the result for more general (i.e., noncolinear) accelerations of the fluid and particle is merely the vector equivalent of Equation 5.47:
 ......(5.51)
where
 ......(5.52)
The first term in Equation 5.51 represents the conventional added mass effect due to the particle acceleration. The factor 3/2 in the second term due to the fluid acceleration may initially seem surprising. However, it is made up of two components:

1. ½ρτdVi /dt*, which is the added mass effect of the fluid acceleration
2. ρτDUi /Dt*, which is a ``buoyancy''-like force due to the pressure gradient associated with the fluid acceleration.
The last term in Equation 5.51 is caused by particle (bubble) volumetric growth, dτ/dt*, and is similar in form to the force on a source in a uniform stream.

Now it is necessary to ask how this force given by Equation 5.51 should be used in the practical construction of an equation of motion for a particle. Frequently, a viscous drag force FiD, is quite arbitrarily added to Fi to obtain some total ``effective" force on the particle. Drag forces, FiD, with the conventional forms
 ......(5.53)
 ......(5.54)
have both been employed in the literature. It is, however, important to recognize that there is no fundamental analytical justification for such superposition of these forces. At high Reynolds numbers, we noted in the last section that experimentally observed added masses are indeed quite close to those predicted by potential flow within certain parametric regimes, and hence the superposition has some experimental justification. At low Reynolds numbers, it is improper to use the results of the potential flow analysis. The appropriate analysis under these circumstances is examined in the next section.

In order to elucidate some of the issues raised in the last section, it is instructive to examine solutions for the unsteady flow past a sphere in low Reynolds number Stokes flow. In the asymptotic case of zero Reynolds number, the solution of Section 5.3 is unchanged by unsteadiness, and hence the solution at any instant in time is identical to the steady-flow solution for the same particle velocity. In other words, since the fluid has no inertia, it is always in static equilibrium. Thus the instantaneous force is identical to that for the steady flow with the same Vi(t).

The next step is therefore to investigate the effects of small but nonzero inertial contributions. The Oseen solution provides some indication of the effect of the convective inertial terms, uj∂ui /∂xj, in steady flow. Here we investigate the effects of the unsteady inertial term, ∂ui /∂t. Ideally it would be best to include both the ∂ui /∂t term and the Oseen approximation to the convective term, U∂ui /∂x. However, the resulting unsteady Oseen flow is sufficiently difficult that only small-time expansions for the impulsively started motions of droplets and bubbles exist in the literature (Pearcey and Hill 1956).

Consider, therefore the unsteady Stokes equations in the absence of the convective inertial terms:
 ......(5.55)
Since both the equations and the boundary conditions used below are linear in ui, we need only consider colinear particle and fluid velocities in one direction, say x1. The solution to the general case of noncolinear particle and fluid velocities and accelerations may then be obtained by superposition. As in Section 5.7 the colinear problem is solved by first transforming to an accelerating coordinate frame, xi, fixed in the center of the particle so that P=p+ρx1dV/dt. Elimination of P by taking the curl of Equation 5.55 leads to
 ......(5.56)
where L is the same operator as defined in Equation 5.37. Guided by both the steady Stokes flow and the unsteady potential flow solution, one can anticipate a solution of the form
 ......(5.57)
plus other spherical harmonic functions. The first term has the form of the steady Stokes flow solution; the last term would be required if the particle were a growing spherical bubble. After substituting Equation 5.57 into Equation 5.56, the equations for f, g, h are
 ......(5.58)
 ......(5.59)
 ......(5.60)
Moreover, the form of the expression for the force, F1, on the spherical particle (or bubble) obtained by evaluating the stresses on the surface and integrating is
 ......(5.61)
It transpires that this is independent of g or h. Hence only the solution to Equation 5.58 for f(r,t) need be sought in order to find the force on a spherical particle, and the other spherical harmonics that might have been included in Equation 5.58 are now seen to be unnecessary.

Fourier or Laplace transform methods may be used to solve Equation 5.58 for f(r,t), and we choose Laplace transforms. The Laplace transforms for the relative velocity W(t), and the function f(r,t) are denoted by (s) and (r,s):
 ......(5.62)
Then Equation 5.58 becomes
 ......(5.63)
where α2=s/ν, and the solution after application of the condition that 1(s,t) far from the particle be equal to (s) is
 ......(5.64)
where α=(s/ν)½ and A and B are as yet undetermined functions of s. Their determination requires application of the boundary conditions on r=R. In terms of A and B the Laplace transform of the force 1(s) is
 ......(5.65)
where
 ......(5.66)
The classical solution (see Landau and Lifshitz 1959) is for a solid sphere (i.e., constant R) using the no-slip (Stokes) boundary condition for which
 ......(5.67)
and hence
 ......(5.68)
so that
 ......(5.69)
For a motion starting at rest at t=0 the inverse Laplace transform of this yields
 ......(5.70)
where is a dummy time variable. This result must then be written in the original coordinate framework with W=V-U and can be generalized to the noncolinear case by superposition so that
 ......(5.71)
where d/dt* is the Lagrangian time derivative following the particle. This is then the general force on the particle or bubble in unsteady Stokes flow when the Stokes boundary conditions are applied.

Compare this result with that obtained from the potential flow analysis, Equation 5.51 with τ taken as constant. It is striking to observe that the coefficients of the added mass terms involving dVi /dt* and dUi /dt* are identical to those of the potential flow solution. On superficial examination it might be noted that dUi /dt* appears in Equation 5.71 whereas DUi /Dt* appears in 5.51; the difference is, however, of order Wj∂Ui /dxj and terms of this order have already been dropped from the equation of motion on the basis that they were negligible compared with the temporal derivatives like ∂Wi /∂t. Hence it is inconsistent with the initial assumption to distinguish between d/dt* and D/Dt* in the present unsteady Stokes flow solution.

The term 9νW/2R2 in Equation 5.71 is, of course, the steady Stokes drag. The new phenomenon introduced by this analysis is contained in the last term of Equation 5.71. This is a fading memory term that is often named the Basset term after one of its identifiers (Basset 1888). It results from the fact that additional vorticity created at the solid particle surface due to relative acceleration diffuses into the flow and creates a temporary perturbation in the flow field. Like all diffusive effects it produces an ω½ term in the equation for oscillatory motion.

Before we conclude this section, comment should be included on two other analytical results. Morrison and Stewart (1976) have considered the case of a spherical bubble for which the Hadamard-Rybczynski boundary conditions rather than the Stokes conditions are applied. Then, instead of the conditions of Equation 5.67, the conditions for zero normal velocity and zero shear stress on the surface require that
 ......(5.72)
and hence in this case (see Morrison and Stewart 1976)
 ......(5.73)
so that
 ......(5.74)
The inverse Laplace transform of this for motion starting at rest at t=0 is
 ......(5.75)
Comparing this with the solution for the Stokes conditions, we note that the first two terms are unchanged and the third term is the expected Hadamard-Rybczynski steady drag term (see Equation 5.16). The last term is significantly different from the Basset term in Equation 5.71 but still represents a receding memory.

The second interesting case is that for unsteady Oseen flow, which essentially consists of attempting to solve the Navier-Stokes equations with the convective initial terms approximated by Uj∂ui /∂xj. Pearcey and Hill (1956) have examined the small-time behavior of droplets and bubbles started from rest when this term is included in the equations.

We now return to the discussion of higher Re flow and specifically address the effects due to bubble growth or collapse. A bubble that grows or collapses close to a boundary may undergo translation due to the asymmetry induced by that boundary. A relatively simple example of the analysis of this class of flows is the case of the growth or collapse of a spherical bubble near a plane boundary, a problem first solved by Herring (1941) (see also Davies and Taylor 1942, 1943). Assuming that the only translational motion of the bubble (with velocity, W) is perpendicular to the plane boundary, the geometry of the bubble and its image in the boundary will be as shown in Figure 5.5. For convenience, we define additional polar cooordinates, (, ), with origin at the center of the image bubble. Assuming inviscid, irrotational flow, Herring (1941) and Davies and Taylor (1943) constructed the velocity potential, φ, near the bubble by considering an expansion in terms of R/h where h is the distance of the bubble center from the boundary. Neglecting all terms that are of order R3/h3 or higher, the velocity potential can be obtained by superposing the individual contributions from the bubble source/sink, the image source/sink, the bubble translation dipole, the image dipole, and one correction factor described below. This combination yields
 ......(5.76)
The first and third terms are the source/sink contributions from the bubble and the image respectively. The second and fourth terms are the dipole contributions due to the translation of the bubble and the image. The last term arises because the source/sink in the bubble needs to be displaced from the bubble center by an amount R3/8h2 normal to the wall in order to satisfy the boundary condition on the surface of the bubble to order R2/h2. All other terms of order R3/h3 or higher are neglected in this analysis assuming that the bubble is sufficiently far from the boundary so that h » R. Finally, the sign choice on the last three terms of Equation 5.76 is as follows: the upper, positive sign pertains to the case of a solid boundary and the lower, negative sign provides an approximate solution for a free surface boundary.

Figure 5.5 Schematic of a bubble undergoing growth or collapse close to a plane boundary. The associated translational velocity is denoted by W.

It remains to use this solution to determine the translational motion, W(t), normal to the boundary. This is accomplished by invoking the condition that there is no net force on the bubble. Using the unsteady Bernoulli equation and the velocity potential and fluid velocities obtained from Equation 5.76, Davies and Taylor (1943) evaluate the pressure at the bubble surface and thereby obtain an expression for the force, Fx, on the bubble in the x direction:
 ......(5.77)
Adding the effect of buoyancy due to a component, gx, of the gravitational acceleration in the x direction, Davies and Taylor then set the total force equal to zero and obtain the following equation of motion for W(t):
 ......(5.78)
In the absence of gravity this corresponds to the equation of motion first obtained by Herring (1941).

Many of the studies of growing and collapsing bubbles near boundaries have been carried out in the context of underwater explosions (see Cole 1948). An example illustrating the solution of Equation 5.78 and the comparison with experimental data is included in Figure 5.6 taken from Davies and Taylor (1943).

Figure 5.6 Data from Davies and Taylor (1943) on the mean radius and central elevation of a bubble in oil generated by a spark-initiated explosion of 1.32×106ergs situated 6.05cm below the free surface. The two measures of the bubble radius are one half of the horizontal span (triangles) and one quarter of the sum of the horizontal and vertical spans (circles). Theoretical calculations using Equation 5.78 indicated by the solid lines.

Another application of this analysis is to the translation of cavitation bubbles near walls. Here the motivation is to understand the development of impulsive loads on the solid surface (see Section 3.6), and therefore the primary focus is on bubbles close to the wall so that the solution described above is of limited value since it requires h » R. However, as discussed in Section 3.5, considerable progress has been made in recent years in developing analytical methods for the solution of the inviscid free surface flows of bubbles near boundaries. One of the concepts that is particularly useful in determining the direction of bubble translation is based on a property of the flow first introduced by Kelvin (see Lamb 1932) and called the Kelvin impulse. This vector property applies to the flow generated by a finite particle or bubble in a fluid; it is denoted by IKi and defined by
 ......(5.79)
where φ is the velocity potential of the irrotational flow, SB is the surface of the bubble, and ni is the outward normal at that surface (defined as positive into the bubble). If one visualizes a bubble in a fluid at rest, then the Kelvin impulse is the impulse that would have to be applied to the bubble in order to generate the motions of the fluid related to the bubble motion. Benjamin and Ellis (1966) were the first to demonstrate the value of this property in determining the interaction between a growing or collapsing bubble and a nearby boundary (see also Blake and Gibson 1987).

5.10 EQUATION OF MOTION

In a multiphase flow with a very dilute discrete phase the fluid forces discussed in Sections 5.1 to 5.8 will determine the motion of the particles that constitute that discrete phase. In this section we discuss the implications of some of the fluid force terms. The equation that determines the particle velocity, Vi, is generated by equating the total force, FiT, on the particle to mpdVi /dt*. Consider the motion of a spherical particle or (bubble) of mass mp and volume τ (radius R) in a uniformly accelerating fluid. The simplest example of this is the vertical motion of a particle under gravity, g, in a pool of otherwise quiescent fluid. Thus the results will be written in terms of the buoyancy force. However, the same results apply to motion generated by any uniform acceleration of the fluid, and hence g can be interpreted as a general uniform fluid acceleration (dU/dt). This will also allow some tentative conclusions to be drawn concerning the relative motion of a particle in the nonuniformly accelerating fluid situations that can occur in general multiphase flow. For the motion of a sphere at small relative Reynolds number, ReW « 1 (where ReW=2WR/ν and W is the typical magnitude of the relative velocity), only the forces due to buoyancy and the weight of the particle need be added to Fi as given by Equations 5.71 or 5.75 in order to obtain FiT. This addition is simply given by (ρτ-mp)gi where g is a vector in the vertically upward direction with magnitude equal to the acceleration due to gravity. On the other hand, at high relative Reynolds numbers, ReW » 1, one must resort to a more heuristic approach in which the fluid forces given by Equation 5.51 are supplemented by drag (and lift) forces given by ½ρACij|Wj|Wj as in Equation 5.33. In either case it is useful to nondimensionalize the resulting equation of motion so that the pertinent nondimensional parameters can be identified.

Examine first the case in which the relative velocity, W (defined as positive in the direction of the acceleration, g, and therefore positive in the vertically upward direction of the rising bubble or sedimenting particle), is sufficiently small so that the relative Reynolds number is much less than unity. Then, using the Stokes boundary conditions, the equation governing W may be obtained from Equation 5.70 as
 ......(5.80)
where the dimensionless time
 ......(5.81)
and w=W/W where W is the steady terminal velocity given by
 ......(5.82)
In the absence of the Basset term the solution of Equation 5.80 is simply
 ......(5.83)
and the typical response time, tr, is called the relaxation time for particle velocity (see, for example, Rudinger 1969). In the general case that includes the Basset term the dimensionless solution, w(t*), of Equation 5.80 depends only on the parameter mp/ρτ (particle mass/displaced fluid mass) appearing in the Basset term. Indeed, the dimensionless Equation 5.80 clearly illustrates the fact that the Basset term is much less important for solid particles in a gas where mp/ρτ » 1 than it is for bubbles in a liquid where mp/ρτ « 1. Note also that for initial conditions of zero relative velocity (w(0)=0) the small-time solution of Equation 5.80 takes the form
 ......(5.84)
Hence the initial acceleration at t=0 is given dimensionally by 2g(1-mp/ρτ)/(1+2mp/ρτ) or 2g in the case of a massless bubble and -g in the case of a heavy solid particle in a gas where mp » ρτ. Note also that the effect of the Basset term is to reduce the acceleration of the relative motion, thus increasing the time required to achieve terminal velocity.

Numerical solutions of the form of w(t*) for various mp/ρτ are shown in Figure 5.7 where the delay caused by the Basset term can be clearly seen. In fact in the later stages of approach to the terminal velocity the Basset term dominates over the added mass term, (dw/dt*). The integral in the Basset term becomes approximately 2t*½dw/dt* so that the final approach to w=1 can be approximated by
 ......(5.85)
where C is a constant. As can be seen in Figure 5.7, the result is a much slower approach to W for small mp/ρτ than for larger values of this quantity.

Figure 5.7 The velocity, W, of a particle released from rest at t*=0 in a quiescent fluid and its approach to terminal velocity, W. Horizontal axis is a dimensionless time defined in text. Solid lines represent the low Reynolds number solutions for various particle mass/displaced mass ratios, mp/ρτ, and the Stokes boundary condition. The dashed line is for the Hadamard-Rybczynski boundary condition and mp/ρτ=0. The dash-dot line is the high Reynolds number result; note that t* is nondimensionalized differently in that case.

The case of a bubble with Hadamard-Rybczynski boundary conditions is very similar except that
 ......(5.86)
and the equation for w(t*) is
 ......(5.87)
where the function, Γ(ξ), is given by
 ......(5.88)
For the purposes of comparison the form of w(t*) for the Hadamard-Rybczynski boundary condition with mp/ρτ=0 is also shown in Figure 5.7. Though the altered Basset term leads to a more rapid approach to terminal velocity than occurs for the Stokes boundary condition, the difference is not qualitatively significant.

If the terminal Reynolds number is much greater than unity then, in the absence of particle growth, Equation 5.51 heuristically supplemented with a drag force of the form of Equation 5.53 leads to the following equation of motion for unidirectional motion:
 ......(5.89)
where w=W/W,t*=t/tr,
 ......(5.90)
and
 ......(5.91)
The solution to Equation 5.89 for w(0)=0,
 ......(5.92)
is also shown in Figure 5.7 though, of course, t* has a different definition in this case.

For the purposes of reference in Section 5.12 note that, if we define a Reynolds number, Re, Froude number, Fr, and drag coefficient, CD, by
 ......(5.93)
then the expressions for the terminal velocities, W, given by Equations 5.82, 5.86, and 5.91 can be written as
 ......(5.94)
respectively. Indeed, dimensional analysis of the governing Navier-Stokes equations requires that the general expression for the terminal velocity can be written as
 ......(5.95)
or, alternatively, if CD is defined as 4/3Fr2, then it could be written as
 ......(5.96)

Qualitative estimates of the magnitude of the relative motion in multiphase flows can be made from the analyses of the last section. Consider a general steady fluid flow characterized by a velocity, U, and a typical dimension, ; it may, for example, be useful to visualize the flow in a converging nozzle of length, , and mean axial velocity, U. A particle in this flow will experience a typical fluid acceleration (or effective g) of U2/ℓ for a typical time given by ℓ/U and hence will develop a velocity, W, relative to the fluid. In many practical flows it is necessary to determine the maximum value of W (denoted by WM) that could develop under these circumstances. To do so, one must first consider whether the available time, ℓ/U, is large or small compared with the typical time, tr, required for the particle to reach its terminal velocity as given by Equation 5.81 or 5.90. If tr « ℓ/U then WM is given by Equation 5.82, 5.86, or 5.91 for W and qualitative estimates for WM/U would be
 ......(5.97)
when WR/ν « 1 and WR/ν » 1 respectively. We refer to this as the quasistatic regime. On the other hand, if tT » ℓ/U, WM can be estimated as Wℓ/Utr so that WM/U is of the order of
 ......(5.98)
for all WR/ν. This is termed the transient regime.

In practice, WR/ν will not be known in advance. The most meaningful quantities that can be evaluated prior to any analysis are a Reynolds number, UR/ν, based on flow velocity and particle size, a size parameter
 ......(5.99)
and the parameter
 ......(5.100)
The resulting regimes of relative motion are displayed graphically in Figure 5.8. The transient regime in the upper right-hand sector of the graph is characterized by large relative motion, as suggested by Equation 5.98. The quasistatic regimes for WR/ν » 1 and WR/ν « 1 are in the lower right- and left-hand sectors respectively. The shaded boundaries between these regimes are, of course, approximate and are functions of the parameter Y, which must have a value in the range 0<Y<1. As one proceeds deeper into either of the quasistatic regimes, the magnitude of the relative velocity, WM/U, becomes smaller and smaller. Thus, homogeneous flows (see Chapter 6) in which the relative motion is neglected require that either X« Y2 or X « Y/(UR/ν). Conversely, if either of these conditions is violated, relative motion must be included in the analysis.

Figure 5.8 Schematic of the various regimes of relative motion between a particle and the surrounding flow.

In the case of bubbles, drops, or deformable particles it has thus far been tacitly assumed that their shape is known and constant. Since the fluid stresses due to translation may deform such a particle, we must now consider not only the parameters governing the deformation but also the consequences in terms of the translation velocity and the shape. We concentrate here on bubbles and drops in which surface tension, S, acts as the force restraining deformation. However, the reader will realize that there would exist a similar analysis for deformable elastic particles. Furthermore, the discussion will be limited to the case of steady translation, caused by gravity, g. Clearly the results could be extended to cover translation due to fluid acceleration by using an effective value of g as indicated in the last section.

The characteristic force maintaining the sphericity of the bubble or drop is given by SR. Deformation will occur when the characteristic anisotropy in the fluid forces approaches SR; the magnitude of the anisotropic fluid force will be given by μWR for WR/ν « 1 or by ρW2R2 for WR/ν » 1. Thus defining a Weber number, We=2ρW2R/S, deformation will occur when We/Re approaches unity for Re « 1 or when We approaches unity for Re » 1. But evaluation of these parameters requires knowledge of the terminal velocity, W, and this may also be a function of the shape. Thus one must start by expanding the functional relation of Equation 5.95 which determines W to include the Weber number:
 ......(5.101)
This relation determines W where Fr is given by Equation 5.93. Since all three dimensionless coefficients in this functional relation include both W and R, it is simpler to rearrange the arguments by defining another nondimensional parameter known as the Haberman-Morton number, Hm, which is a combination of We, Re, and Fr but does not involve W. The Haberman-Morton number is defined as
 ......(5.102)
In the case of a bubble, mp « ρτ and therefore the factor in parenthesis is usually omitted. Then Hm becomes independent of the bubble size. It follows that the terminal velocity of a bubble or drop can be represented by functional relation
 ......(5.103)
and we shall confine the following discussion to the nature of this relation for bubbles (mp « ρτ).

Figure 5.9 Values of the Haberman-Morton parameter, Hm, for various pure substances as a function of reduced temperature.

Some values for the Haberman-Morton number (with mp/ρτ=0) for various saturated liquids are shown in Figure 5.9; other values are listed in Table 5.3. Note that for all but the most viscous liquids, Hm is much less than unity. It is, of course, possible to have fluid accelerations much larger than g; however, this is unlikely to cause Hm values greater than unity in practical multiphase flows of most liquids.

 TABLE 5.3 Values of the Haberman-Morton numbers, Hm=gμ4/ρS3, for various liquids at normal temperatures. Filtered Water 0.25× 10-10 Turpentine 2.41× 10-9 Methyl Alcohol 0.89× 10-10 Olive Oil 7.16× 10-3 Mineral Oil 1.45× 10-2 Syrup 0.92× 106

Having introduced the Haberman-Morton number, we can now identify the conditions for departure from sphericity. For low Reynolds numbers (Re « 1) the terminal velocity will be given by the equation Re=C Fr2 where C is some constant. Then the shape will deviate from spherical when We≥Re or, using Re=C Fr2 and Hm=We3Fr-2Re-4, when
 ......(5.104)
Thus if Hm<1 all bubbles for which Re « 1 will remain spherical. However, there are some unusual circumstances in which Hm>1 and then there will be a range of Re, namely Hm<Re<1, in which significant departure from sphericity might occur.

For high Reynolds numbers (Re » 1) the terminal velocity is given by Fr≈O(1) and distortion will occur if We>1. Using Fr=1 and Hm=We3Fr-2Re-4 it follows that departure from sphericity will occur when
 ......(5.105)
Consequently, in the common circumstances in which Hm<1, there exists a range of Reynolds numbers, Re<Hm, in which sphericity is maintained; nonspherical shapes occur when Re>Hm. For Hm>1 departure from sphericity has already occurred at Re<1 as discussed above.

Figure 5.10 Photograph of a spherical cap bubble rising in water (from Davenport, Bradshaw, and Richardson 1967).

Figure 5.11 Notation used to describe the geometry of spherical cap bubbles.

Experimentally, it is observed that the initial departure from sphericity causes ellipsoidal bubbles that may oscillate in shape and have oscillatory trajectories (Hartunian and Sears 1957). As the bubble size is further increased to the point at which We≈20, the bubble acquires a new asymptotic shape, known as a ``spherical-cap bubble.'' A photograph of a typical spherical-cap bubble is shown in Figure 5.10; the notation used to describe the approximate geometry of these bubbles is sketched in figure 5.11. Spherical-cap bubbles were first investigated by Davies and Taylor (1950), who observed that the terminal velocity is simply related to the radius of curvature of the cap, Rc, or to the equivalent volumetric radius, RB, by
 ......(5.106)
Assuming a typical laminar drag coefficient of CD=0.5, a spherical solid particle with the same volume would have a terminal velocity,
 ......(5.107)
which is substantially higher than the spherical-cap bubble. From Equation 5.106 it follows that the effective CD for spherical cap bubbles is 2.67 based on the area πR2B.

Wegener and Parlange (1973) have reviewed the literature on spherical cap bubbles. Figure 5.12 is taken from from their review and shows that the value of W/(gRB)½ reaches a value of about 1 at a Reynolds number, Re=2WRB, of about 200 and, thereafter, remains fairly constant. Visualization of the flow reveals that, for Reynolds numbers less than about 360, the wake behind the bubble is laminar and takes the form of a toroidal vortex (similar to a Hill (1894) spherical vortex) shown in the left-hand photograph of Figure 5.13. The wake undergoes transition to turbulence about Re=360, and bubbles at higher Re have turbulent wakes as illustrated in the right side of Figure 5.13. We should add that scuba divers have long observed that spherical cap bubbles rising in the ocean seem to have a maximum size of the order of 30cm in diameter. When they grow larger than this, they fission into two (or more) bubbles. However, the author has found no quantitative study of this fission process.

Figure 5.12 Data on the terminal velocity, W/(gRB)½, and the conical angle, θM , for spherical-cap bubbles studied by a number of different investigators (adapted from Wegener and Parlange 1973).

Figure 5.13 Flow visualizations of spherical-cap bubbles. On the left is a bubble with a laminar wake at Re≈180 (from Wegener and Parlange 1973) and, on the right, a bubble with a turbulent wake at Re≈17000 (from Wegener, Sundell and Parlange 1971, reproduced with permission of the authors).

In closing, we note that the terminal velocities of the bubbles discussed here may be represented according to the functional relation of Equations 5.103 as a family of CD(Re) curves for various Hm. Figure 5.14 has been extracted from the experimental data of Haberman and Morton (1953) and shows the dependence of CD(Re) on Hm at intermediate Re. The curves cover the spectrum from the low Re spherical bubbles to the high Re spherical cap bubbles. The data demonstrate that, at higher values of Hm, the drag coefficient makes a relatively smooth transition from the low Reynolds number result to the spherical cap value of about 2.7. Lower values of Hm result in a deep minimum in the drag coefficient around a Reynolds number of about 200.

Figure 5.14 Drag coefficients, CD, for bubbles as a function of the Reynolds number, Re, for a range of Haberman-Morton numbers, Hm, as shown. Data from Haberman and Morton (1953).

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Last updated 12/1/00.
Christopher E. Brennen