Date of Award

Spring 5-31-2018

Document Type


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Shahriar Afkhami

Second Advisor

Lou Kondic

Third Advisor

Shawn Alexander Chester

Fourth Advisor

Denis L. Blackmore

Fifth Advisor

Christopher Batty


This dissertation is developed in the field of Computational Fluid Dynamics (CFD) and it focuses on numerical simulations of the dynamics of thin viscoelastic films in different settings. The first part of this dissertation presents a novel computational investigation of thin viscoelastic films and drops, that are subject to the van der Waals interaction force, in two spatial dimensions. The liquid films are deposited on a flat solid substrate, that can have a zero or nonzero inclination with respect to the base. The equation that governs the interfacial dynamics of the thin films and drops is obtained within the long-wave approximation of the Navier-Stokes equations, with the Jeffreys model for viscoelastic stresses. The effects of viscoelasticity and the substrate slippage on the dynamics of thin viscoelastic films are investigated. Moreover, the effects of viscoelasticity on drops that spread or recede on a prewetted flat substrate are analyzed. For dewetting films, the numerical results show the presence of multiple secondary droplets for higher values of the relaxation time, consistently with experimental findings. These secondary length scales are found to be suppressed by gravitational effects when the case of dewetting films on inverted planes is analyzed. For spreading and receding drops on flat, prewetted substrates, viscoelastic effects are found to lead to deviations from the Cox-Voinov law for partially wetting fluids. In general, viscoelasticity enhances the spreading and suppresses the retraction of viscoelastic drops, compared to Newtonian ones.

The second part of this dissertation presents a novel numerical investigation of the dynamics of free-boundary flows of viscoelastic liquid membranes, not necessarily deposited on solid substrates. The governing equation describes the balance of linear momentum, in which the stresses include the viscoelastic response to deformations of Maxwell type. A penalty method is utilized to enforce near incompressibility of the viscoelastic media, in which the penalty constant is proportional to the viscosity of the fluid. A finite element method is used, in which the slender geometry representing the liquid membrane is discretized by linear three-node triangular elements under plane stress conditions. Two applications of interest are considered for the numerical framework provided: shear flow, and extensional flow in drawing processes. Finally, the last part of this dissertation considers the expansion of the study of the dynamics of viscoelastic membranes by applying the general theory of shells, in which any application of loading or external forces causes both bending and stretching, so that buckling or wrinkling phenomena can be investigated as future work.

Included in

Mathematics Commons