Thin liquid films are often studied by reducing the Navier-Stokes equations
using Reynolds lubrication theory, which leverages a small aspect ratio
to yield simplified governing equations. In this dissertation a plate
coating application, in which polydimethylsiloxane coats a silicon substrate,
is studied using this approach. Thermal Marangoni stress
drives fluid motion against the resistance of gravity, with the parameter
regime being chosen such that these stresses lead to a stable advancing front.
Additional localized thermal Marangoni stress is used to control the thin film;
in particular, coating thickness is modulated through the intensity of such
localized forcing. As thermal effects are central to film dynamics, the dissertation
focuses specifically on the effect that incorporating temperature dependence
into viscosity, surface tension, and density has on film dynamics and control.
Incorporating temperature dependence into viscosity, in particular,
leads to qualitative changes in film dynamics.
A mathematical model is developed in which the temperature dependence
of viscosity and surface tension is carefully taken into account.
This model is then
studied through numerical computation of solutions, qualitative analysis,
and asymptotic analysis. A thorough comparison is made between the
behavior of solutions to the temperature-independent and
temperature-dependent models. It is shown that using
localized thermal Marangoni stress as a control mechanism is feasible
in both models. Among constant steady-state solutions
there is a unique such solution in the temperature-dependent model,
but not in the temperature-independent model, a feature that
better reflects the known dynamics of the physical system.
The interaction of boundary conditions with finite domain size is shown
to generate both periodic and finite-time blow-up solutions, with
qualitative differences in solution behavior between models.
This interaction also accounts for the fact that locally perturbed solutions,
which arise when localized thermal Marangoni forcing is too weak
to effectively control thin film thickness, exist only for a discrete
set of boundary heights.
Modulating the intensity of localized thermal Marangoni forcing is
an effective means of modulating the thickness of a thin film
for a plate coating application; however, such control must be initiated before
the film reaches the full thickness it would reach in the absence of
such localized forcing. This conclusion holds for both the temperature-independent
and temperature-dependent mathematical models; furthermore, incorporating
temperature dependence into viscosity causes qualitative changes in solution
behavior that better align with known features of the underlying physical system.
