Spreading of a Thin Liquid Drop Under the Influence of Gravity, Rotation and Non-Uniform Surface Tension

Abstract

Figure 1 shows an axisymmetric thin liquid drop that is spreading under the influence of gravity, rotation and non-uniform surface tension. This process is modeled using the Lie group method. Functional forms of the non-uniform surface tension ∑(x) are obtained. A fourth-order nonlinear partial differential equation describing the evolution of the free surface, u(t,x), of the liquid drop is derived by imposing the thin-film approximation on the Navier-Stokes equations and solving the resulting system of equations subject to boundary conditions on the surface of the rotating disk and on the free surface of the liquid drop. Exact and approximate Lie point symmetry generators admitted by the free surface equation are determined. These symmetry generators are imposed on the moving boundary, x=R(t), where R(t) is the radius of the foot of the liquid drop at time t and x is the radial coordinate fixed on the rotating disk. Functional forms of R(t) are determined. The rate of spreading dR(t)/dt can then be easily calculated.