Pergamon Chemical Enftineering Science, Vol, 50, No, 2, pp. 263 277, 1995 Copyright (C3 1995 Elsevier Science Ltd Printed in Great Britain, All rights reserved 0009 2509/95 $9.50 + 0.00 ooog-2sog(94)oo23a-x INFLUENCE OF THE SURFACE VISCOSITY ON THE DRAG AND TORQUE COEFFICIENTS OF A SOLID PARTICLE IN A THIN LIQUID LAYER K. D. DANOV Laboratory of Thermodynamics and Physico-ChemicalHydrodynamics, Faculty of Chemistry, University of Sofia, J. Bourchier Ave. l, Sofia 1126, Bulgaria and R. AUST,* F. DURST and U. LANGE Lehrstuhl f/ir Str6mungsmechanik, Universit/it Erlangen-Nfirnberg,Cauerstr. 4, D-91058 Erlangen, Germany (Received 4 May 1994; accepted in revised form 25 July 1994) A~tract--The present paper provides a short literature survey of the treatment of particle motion in fluids as a basis of the author's own work in this field.The work relates to the motion of a small solid particle inside a thin liquid film. The particle motion is treated numerically to provide information of interest to thin film liquid coating. The numerical computations yield information on the local velocity and pressure distribu- tions, but also integral information expressed in drag and torque coefficients.The computations are carried out for stationary flows of low Reynolds and capillary numbers and results are presented for differentva!ues of surface dilatation and shear viscosities. Translational and rotational motions of the particle are considered by integrating the resultant second-order partial differentialequation with an alternating direct implicit method. The numerical results reveal in all cases the strong influenceof the surface viscosityon the motion of the solid particle in the viscous liquid layer when the radius of the particle is of the same order of magnitude as the thickness of the liquid film or when the particle is close to the liquid-gas interface. 1. INTRODUCTION The motion of a single particle in a fluid has fasci- nated fluid dynamists for more than a century and analytical treatments have been put forward by many researchers. Stokes (1851) derived the formula for the friction force exerted on a spherical particle moving with constant velocity in an unlimited incompressible viscous fluid. Kirchhoff (1876) gave the solution for a spherical particle slowly rotating in an unbounded fluid with the same fluid properties as employed by Stokes (1851). Rybczynski (1911) and Hadamard (1911) generalised Stokes' (1851) (low-Reynolds num- ber) solution for the settling velocity of a solid sphere and derived the solution for the case of a spherical fluid droplet. However, experiments [cf. Lebedev (1916) and Silvey (1916)] performed shortly after- wards revealed that fluid droplets of sufficiently small radius settled as if they were solid spheres, obeying Stoke's original formula. Boussinesq (1913), in an at- tempt to resolve this discrepancy between theory and experiment, postulated the existence of a surface vis- cosity, conceived as the two-dimensional equivalent of the conventional three-dimensional viscosity possessed by bulk-fluid phases. The solution for a neutrally buoyant rigid sphere suspended in a *Author to whom correspondence should be addressed. homogenous shear flow that extends to infinity was probably first obtained by Einstein (1956) in his theory of the viscosity of particle suspensions. The theory of Einstein (1956) was extended by Jeffery (1922) to nonspherical particles, where a new feature arises due to the orientation of the particle relative to the principal axes of shear. Jeffery's (1922) classic analysis of the periodic rotation of an ellipsoid of revolution suspended in a simple shearing motion of a viscous fluid. The status of knowledge in this field was summarised and theory and experiment were compared by Goldsmith and Mason (1967). Up to the mid-1960s a great deal of work was done in obtaining first- and higher-order wall corrections for particles in flows that are bounded by plane or cylindrical walls [cf. Happel and Brenner (1965)]. The force and torque exerted on the particle were ex- pressed in a power series according to a/l, where a is the particle radius and l is the distance between the centre of the particle and the wall. Following this idea, Lorentz (1906) derived the motion of a sphere perpen- dicular to a flat wall up to all. Faxen (1921) developed the method of reflection for a sphere moving between two parallel planes in a viscous fluid. Using his method Wakiya (1956) considered the cases of motion in Poiseuille- and Couette-flow. As was pointed out by Hetsroni (1982), the method employed by Wakiya (1956) does not yield physically correct results for 263