Samenvatting
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 hotspots 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 NavierStokes 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 onedimensional
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 fundamentalmode approximation of eigenfunction
expansion, we investigate the effect of thermal conductivity on the formation of hotspots
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 hotspot.
Originele taal2  Engels 

Kwalificatie  Doctor in de Filosofie 
Toekennende instantie 

Begeleider(s)/adviseur 

Datum van toekenning  22 mei 2002 
Plaats van publicatie  Eindhoven 
Uitgever  
Gedrukte ISBN's  9038605420 
DOI's  
Status  Gepubliceerd  2002 
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Citeer dit
Chandra, T. D. (2002). Perturbation and operator methods for solving Stokes flow and heat flow problems. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR554634