Quarterly of Applied Mathematics

Complex variable techniques are employed to characterize two-dimensional solutions u{x, y) of Laplace's equation which satisfy the boundary condition [fi(d2u/ dy2) + (du/dx)]x„0 = 0, where /? is referred to as the surface-stiffness parameter. Simple closed-form singular solutions are derived which satisfy this boundary condition and represent source and dislocation singularities. The former is used to synthesize the field generated by a small inclusion of arbitrary shape on which u = 1, in the presence of a boundary at y — 0 on which u = 0. At points not near the inclusion the field has the form of a function of position and surface stiffness multiplied by a strength factor which depends on the size and shape of the inclusion and the surface stiffness. Detailed calculations are presented for two extreme shapes of inclusions—a shallow, wide inclusion on the surface and a deep, narrow inclusion penetrating below the surface—which exhibit the relation between the field near the inclusion and the distant field, and show explicitly the dependence of the strength factor on surface stiffness and inclusion size and shape. The nature and strength of the singularities at the tips of the inclusions are also examined and it is found that a tip singularity at the surface changes character as the surface stiffness varies. Introduction. There exists an important class of physical problems which requires the determination of a function u(x, y) satisfying Laplace's equation in a plane region and also satisfying a boundary condition of the form fi(d2u/ds1) + (du/dn) = 0 (1) on one or more rectilinear boundaries of that region, where s and n denote respectively the directions parallel and normal to the boundary and /? is a real number which can assume any value between zero and infinity. For example, this problem is of central importance in that branch of surface chemistry which deals with the measurement of the "surface viscosity" of molecular monolayers and bilayers [1], Fig. 1 illustrates the typical physical situation. A cylindrical viscometer is filled with a substrate liquid, usually water, up to a level x = 0 and a thin layer of the substance under investigation is spread on this surface. A circular disk or knife-edge is placed in contact with the surface and caused to rotate at constant angular velocity: the cross-section of this perturbing body appears as the shaded region in Fig. 1, and the rigid wall of the viscometer appears as the boundary at y = 0. If the radius of the viscometer is large compared to the distance between the wall * Received December 1, 1978. The author is grateful to Dr. Hiroshi Tada for his helpful comments on this paper.