Modeling Temperature Dependence in Marangoni-driven Thin Films

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.