Perturbation and operator methods for solving Stokes flow and heat flow problems

Abstract

This research investigates a number of problems, related to Stokes flow and to heat flow. The Stokes flow is inspired by glass flow in the process of making bottles or jars. The heat flow is related to a heat conduction model problem, and a problem about hot-spots formation in the microwave heating. We will discuss the first problem. There are two phases during the industrial process of making glass, viz. the pressing phase and the blowing phase. We consider some mathematical aspects of the pressing phase. The motion of glass at temperatures above 6000C can be described by the Navier-Stokes equations. Since glass is a highly viscous fluid, those equations can be simplified to the Stokes equations. We use two different methods to solve these equations, viz. perturbation and operator methods. The perturbation method is based on the geometry being slowly varying. As a result, we obtain the velocity analytically. This result has a good agreement with numerical results based on finite element modelling. Using the velocity obtained we derive the formula for the force on the plunger. Next, we consider the operator method. Using this method, the Stokes equations can be transformed into an operator equation on the boundary ??? with a tangent vector field a on the boundary ??? as unknown. Solving this operator equation shows, that the solutions of the Stokes equations can be parameterized by aH, the harmonic extension of a to the interior of the domain ??. As an application, we present some full explicit solutions of the Stokes equations for several domains such as the interior and exterior of the unit ball and of the unit disk, an infinite strip, a half space, and a wedge. In the second problem, we consider the heat conduction problem inside two types of geometry, viz. slowly and slightly varying geometry. Using this problem, we show the difference between those geometries. An example that involves the boundary layers at the ends is presented. Finally, we consider a simplified model of the microwave heating of a one-dimensional unit slab. This slab consists of three layers that have different thermal conductivities. We consider only the steady state problem with Dirichlet boundary conditions and continuity of heat flux across the layers. Using a fundamental-mode approximation of eigenfunction expansion, we investigate the effect of thermal conductivity on the formation of hot-spots where the temperature increases catastrophically as a function of d, the amplitude of the applied electric field. First, we consider a unit slab geometry. In this geometry, we find the critical value dcr, for which slight changes in d yields a sudden jump to another stable solution, now with a much higher temperature. Next, we consider a unit slab consisting of three layers of material with different thermal conductivity (µ). We assume the inner layer has the smallest value of µ. We find the temperature in this layer is much higher than that in other layers. Then, we consider only the inner layer. For a given value of d and changing values of µ, we get a temperature jump near some values of µ. This jump shows that there is a critical value of µ and signifies the formation of a hot-spot.