Stretching of a liquid layer underneath a semi-infinite fluid

Exact similarity solutions of the Navier–Stokes equation are derived describing the flow of a liquid layer coated on a stretching surface underneath another semi-infinite fluid. In the absence of hydrodynamic instability, the interface remains flat as the layer thickness decreases in time. When the physical properties of the fluids are matched, we obtain Crane’s analytical solution for two-dimensional (2D) flow and a corresponding numerical solution for axisymmetric flow. When the rate of stretching of the surface is constant in time, the temporal evolution of the interface between the layer and the overlying fluid is computed by integrating in time a system of coupled partial differential equations for the velocity in each fluid together with an ordinary differential equation expressing kinematic compatibility at the interface, subject to appropriate boundary, interfacial, and far-field conditions. Multiple solutions are found in certain ranges of the density and viscosity ratios. Additional similarity solutions are presented for accelerated 2D and axisymmetric stretching. The numerical prefactors that appear in the analytical expressions for the interface location and wall shear stress are presented for different ratios of the densities and viscosities of the two fluids.