Effective surface-shear viscosity of an incompressible particle-laden fluid interface

The presence of even a small amount of surfactant at the particle-laden fluid interface subjected to shear makes surface flow incompressible if the shear rate is small enough [T. M. Fischer et al., J. Fluid Mech. 558, 451 (2006)]. In the present paper the effective surface shear viscosity of a flat, low-concentration, particle-laden incompressible interface separating two immiscible fluids is calculated. The resulting value is found to be 7.6% larger than the value obtained without account for surface incompressibility.

Figure 1

Streamlines of the interfacial flow ($z=0$) near the particle.

Figure 2

Perturbation velocity component $v_x^{\left(1\right)}$ at $y=0$, $z=0$ as a function of the distance $x$ from the center of the particle with (solid line) and without (dashed line) account for incompressibility of interfacial flow.

Figure 3

The coefficient $K$, defined by Eq. (35), as a function of the truncation parameter $l_{\max }$. Dashed line corresponds to quadratic extrapolation, cross indicates the value of $K$ at $l_{\max }\to \infty$.