A Numerical and Asymptotic Study of Some Third-Order Ordinary Differential Equations Relevant to Draining and Coating Flows

Some draining or coating fluid-flow problems, in which surface tension forces are important, can be described by third-order ordinary differential equations. Accurate computations are provided here for examples such as $y'''(x) = - 1 + {1 / {y^2 }}$ that permit the boundary condition $y \to 1$ as $x \to - \infty $, so modelling a layer of fluid that is asymptotically uniform behind the draining front. The ultimate fate of the solution as x increases is studied for the above example, and for a generalisation involving a small parameter $\delta $ such that this example is recovered in the limit as $\delta \to 0$, but which is such that $y \to \delta $ as $x \to + \infty $, so modelling draining over an already-wet surface. Matched asymptotic expansions are then used to derive limiting results for small $\delta $, this being a singular perturbation since the problem with $\delta = 0$ does not permit $y = 0$. The physical basis for this singularity is the well-known impossibility of moving a contact line over...