Hydrodynamic Interactions Between Solid Particles at a Fluid–Gas Interface - Drops and Bubbles in Contact with Solid Surfaces - page 99

subspace of the symmetric multipole functions, stands for the multiple scattering
series.
Our goal is to evaluate the quasi-two-dimensional mobility

¯ ×

μ ,whichisa3 N

=

3 N matrix depending on the configuration of the sphere centers in the plane z a ,

= ζ − 1 . μ ¯ (12)

Finally, after some additional algebraic steps (cf. [41] for details), the Q2D mo-
bility is obtained in form of a multiple scattering series,

1

¯ =

μ F +

μ 2 μ F Z F G 0 Z F μ F , (13)

2 G 0 Z F 1 +

Z F is defined as

where the quasi-two-dimensional operator

Z F =

Z F μ F Z F . (14)
Here, μ F is the Q2D mobility operator of a single sphere touching the free surface,
i.e.,

Z F −

μ F =[ P F Z F P F ] − 1 . (15)

C. Single-Sphere Mobilities

From evaluating the multipole matrix elements of Z F and transforming them to
Cartesian coordinates, we obtain the Q2D single-sphere mobility tensor μ F for a
sphere touching the planar free surface. By definition, its translation-translational,
μ t F , and rotation-rotational, μ r F , components relate the sphere translational, U ,and
angular, , velocities to the external force, F , and torque, T ,

μ t F · F , U =

(16)

μ r F · T = . (17)

The single sphere mobility has the form,

μ t F = μ t F [ R R + R ⊥ R ⊥ ] , (18)

μ t F = μ r F ˆ z z . ˆ (19)

Here μ t F and μ r F

× ×

1 tensors, the components
of the reduced mobility, which describes only the relevent degrees of freedom, or,
alternatively, as the standard 3

can be interpreted as 2 2, and 1

3 tensors with a number of zeros, which eliminate
the unphysical motions. The scalar coefficients are,
μ t F /μ t 0

×

= 1 . 3799554 , (20)

μ r F /μ r 0 = 1 . 10920983 . (21)

Here, μ t 0 ] − 1 are the translation-translational and

rotation-rotational mobilities, respectively, of a single sphere in an unbounded fluid.

] − 1 and μ r 0 ηa 3 =[ 6 π ηa =[ 8 π