Stokes' Law is the name given to the formula describing the force F on a stationary sphere of radius a held in a fluid of viscosity η moving with steady velocity V. This is usually expressed in the form

By translation, this result also applied to a sphere moving with steady velocity V in an otherwise stagnant fluid. Equation (1) can more conveniently be expressed in terms of a drag coefficient and a Reynolds number defined as follows

In terms of these variables, Equation (1) takes the form

Equations (1) and (4) are however applicable for slow flows and should only be used for Re < 1. The reasons for this and the results for higher Reynolds numbers are discussed below.

For a spherical gas bubble the corresponding results are

and

These results can be derived as follows.

It is more convenient to analyze the situation in which the sphere is held stationary in a fluid that moves with velocity V at great distances from the sphere. Since Stokes Law is restricted to slow steady flow we can begin with the Navier Stokes equation and omit both the time-dependent and inertial terms giving

The situation is axially symmetrical and therefore there are two components of Eq. (7). Eliminating the pressure between these two components and expressing the resulting equation in terms of the *Stokes stream function* ψ, gives

where E^{2} is the operator,

For a solid sphere the boundary conditions are that the radial and tangential velocities on the surface are zero by the no-slip condition, i.e.,

and that the velocity tends to V at great distances, i.e.,

The form of these boundary conditions suggests a solution of the form ψ = f(r) sin^{2} θ and the only possible result is

Putting in boundary conditions(10)-(12) gives A = 0, B = V/2, C = −3Va/4 and D = Va^{3}/4. Thus,

Hence we can obtain the velocity components,

For a spherical gas bubble, boundary conditions (10) and (12) still apply, but boundary condition (11) has to be replaced by the condition that there is no shear stress on the surface. Hence the shear strain rate
is zero where
is defined by

Under these circumstances A = 0, B = V/2, C = - Va/2 and D = 0, giving

The shear stress on the surface can be obtained from

For a solid sphere,

and for a bubble, τ_{rθ} is obviously zero.

The skin friction drag F_{s} is given by

from which we find that

for a solid sphere and zero for a bubble.

The pressure distribution can be found by substituting the expression for the velocity components into the Navier Stokes equation, giving

where p_{0} is an arbitrary constant.

For a solid sphere,

For a buble,

The normal stress on the surface is given by

where
is the normal strain rate ∂v_{r}/∂r.

For a solid sphere,

and for a bubble,

The form drag F_{F} is given by

giving

for a solid sphere, and

for a bubble.

These results can be tabulated thus,

giving the results presented above.

It must be emphasized that these results are applicable only for low Reynolds numbers. This is because the inertial terms (v.
)v have been omitted from the analysis. An extension to the theory, known as Ossen's approximation, can be obtained by replacing these inertial terms by (V.
)v, which gives rise to the result

In fact, this provides an over-correction and the empirical result,

gives a good correlation of the experimental results up to a Reynolds numbers of about 1000. The reader is referred to Clift et al. (1972) for alternative correlations and extensions to even higher Reynolds numbers.

For bubbles in nonpolar liquids, Equation (6) may be used up to a Reynolds number of about 1.5. However, polar liquids, such as water, are prone to contamination by surface active agents, which collect on the surface of the bubble, effectively immobilizing the surface. Such a surface can support a shear stress and bubbles in polar liquids behave as solid spheres. Indeed circumstances can arise in which bubbles obey the result for solid spheres over a very much larger range of Reynolds numbers than solid spheres themselves. Details of the behavior of bubbles are given by both Clift et al. (1972) and Wallis (1974). Wallis' correlation is probably the most reliable and convenient currently available.

These results can be used for accelerating spheres and bubbles without much loss of accuracy, but care must be exercised if the accelerations are large as then the *Basset history term* must be included, see Clift et al. (1972).

#### REFERENCES

Clift, R., Grace, J. R., and Weber, M. E. (1972) *Bubbles, Drops and Particles*,Academic Press, New York.

Wallis, G. B. (1974) *Int. J. Multiphase Flow*, 1, 491-511.