Effective surface viscosities of a particle-laden fluid interface

The Einstein formula for the effective shear viscosity of low Reynolds number suspension flows is generalized to the case of flat, low-concentration, particle-laden interfaces separating two immiscible fluids. The effective surface shear and dilational viscosities of this system is found to be eta_s=5/3(eta_1+eta_2)Rphi and zeta_s=5(eta_1+eta_2)Rphi, correspondingly, where eta1 and eta2 are the shear viscosities of two bulk fluids and phi is the surface concentration of spherical particles of radius R. The formula is found to be in excellent agreement with data obtained using multicomponent lattice Boltzmann equation simulation.