### Hangjie Ji

I'm from Hangzhou, China. Currently in Los Angeles, USA. Passionate about mathematics and technology.

e-Mail

# Research

### Finite-time thin film rupture driven by generalized evaporative loss

#### with Thomas P. Witelski

Rupture is a nonlinear instability resulting in a finite-time singularity as a fluid layer approaches zero thickness at a point. We study the dynamics of rupture in a generalized mathematical model of thin films of viscous fluids with evaporative effects. The governing lubrication model is a fourth-order nonlin- ear parabolic partial differential equation with a non-conservative loss term due to evaporation. Several different types of finite-time singularities are observed due to balances between evaporation and surface tension or intermolecular forces. Non-self-similar behavior and two classes of self-similar rupture solutions are analyzed and validated against high resolution PDE simulations.

### Global existence of solutions to a tear film model with locally elevated evaporation rates

#### with Yuan Gao, Jian-Guo Liu and Thomas P. Witelski

Motivated by a model proposed by Peng et al. [Advances in Coll. and Interf. Sci. 206 (2014)] for break-up of tear films on human eyes, we study the dynamics of a generalized thin film model. The governing equations form a fourth-order coupled system of nonlinear parabolic PDEs for the film thickness and salt concentration subject to non-conservative effects representing evaporation. We analytically prove the global existence of solutions to this model with mobility exponents in several different ranges and present numerical simulations that are in agreement with the analytic results. We also numerically capture other interesting dynamics of the model, including finite- time rupture-shock phenomenon due to the instabilities caused by locally elevated evaporation rates, convergence to equilibrium and infinite-time thinning.

### Weak solutions of a continuum model for vicinal surface in the attachment-detachment-limited regime with elastic interactions

#### with Yuan Gao, Jian-Guo Liu and Thomas P. Witelski

We study a continuum model for solid films that arises from the modeling of one-dimensional step flows on a vicinal surface in the attachment-detachment-limited regime. The resulting nonlinear partial differential equation, $u_t = −u^2(u^3 + \alpha u)_{hhhh}$, gives the evolution for the surface slope u as a function of the local height h in a monotone step train. Subject to periodic boundary conditions and positive initial conditions, we prove the existence, uniqueness and positivity of global strong solutions to this PDE using two Lyapunov energy functions. The long time behavior of u converging to a constant that only depends on the initial data is also investigated both analytically and numerically.

### Numerical solution of nonlinear shallow shell problems with thermal stresses and partially clamped boundary conditions

#### with Longfei Li

We investigate the classic problem of deflection of a rectangular elastic shallow shell with precast nonlinear shape subjected to localized thermal stresses. The shell is simply supported or free on two opposite edges and has mixed boundary conditions on the other two edges, clamped on the center part and free or simply supported on the remainder. A finite difference scheme for biharmonic equations with special treatment of singularities due to mixed boundary conditions is derived based on the local Wiener-Hopf solution in order to obtain a second-order accurate solution. Using this scheme as a basis, the nonlinear coupled shallow shell equations are then solved iteratively by a Picard-type method. Numerical results are presented to demonstrate the effects of the precast shell shape, thermal stresses and mixed boundary conditions on the deflection of shallow shell. In particular, critical thermal loads of snap-through buckling with various mixed boundary conditions are examined numerically and compared with previous studies.