Morphological instability induced by the interaction of a particle with a solidifying interface

We show that the interaction of a particle with a directionally solidified interface induces the onset of morphological instability provided that the particle-interface distance falls below a critical value. This instability occurs at pulling velocities that are below the threshold for the onset of the Mullins-Sekerka instability. The expression for the critical distance reveals that this instability is manifested only for certain combinations of the physical and processing parameters. Its occurence is attributed to the reversal of the thermal gradient in the melt ahead of the interface and behind the particle.Comment: 5 pages, 4 figures. To appear in Eur. Phys. J.

A Simple Analytical Model for Estimating the Dissolution-Driven Instability in a Porous Medium

This article deals with the stability problem that arises in the modeling of the geological sequestration of carbon dioxide. It provides a more detailed description of the alternative approach to tackling the stability problem put forth by Vo and Hadji (Physics of Fluids, 2017, 29, 127101) and Wanstall and Hadji (Journal of Engineering Mathematics, 2018, 108, 53–71), and it extends two-dimensional analysis to the three-dimensional case. This new approach, which is based on a step-function base profile, is contrasted with the usual time-evolving base state. While both provide only estimates for the instability threshold values, the step-function base profile approach has one great advantage in the sense that the problem at hand can be viewed as a stationary Rayleigh–Bénard problem, the model of which is physically sound and the stability of which is not only well-defined but can be analyzed by a variety of existing analytical methods using only paper and pencil

Analytical investigations of convective effects on a solid-liquid interface

A thin layer of a single-component Boussinesq fluid, contained between two rigid horizontal plates of low thermal conductivity, is cooled from above and heated from below. In the steady-static state, a heat flux traverses the system so that the temperature attained at the upper boundary of the layer is in the solid phase. An interface is planar in the conductive state, and corrugated in the convective regime. A small amplitude expansion study reveals that the critical Rayleigh number and the critical wavenumber for the onset of the interface deformation increase with the solid layer thickness. A weakly nonlinear stability analysis reveals that there is subcritical instability irrespective of the interface shape. The stable forms of the solidified front are then found. In the case of hexagonal pattern, the fluid motion is shown to be upward at the cells centers. Hexagons are also found to exhibit a higher heat flux than either rolls or squares. The non-planar interface is shown to have a cellular structure identical to the convection patterns which arise in this situation.A long wavelength approximation is used to derive a non-linear evolution equation for the leading order interface pertubation. This evolution equation is found to be ill-posed when the dimensionless thickness of the solid layer A exceeds 0.256. The equation is then solved numerically for A between 0 and 0.256. The curve shifts to the right as A approaches the value 0.256.The linear stability analysis for the binary alloy case is performed. This part complements the work done by Caroli et al. (1985). The critical value for the pulling velocity V at which both the morphological and convective instabilities are excited at the same concentration level is determined numerically.U of I OnlyETDs are only available to UIUC Users without author permissio

Long Wavelength Analysis of a Model for the Geographic Spread of a Disease

We investigate the temporal and spatial evolution of the spread of an infectious disease by performing a long-wavelength analysis of a classical model for the geographic spread of a rabies epidemic in a population of foxes subject to idealized boundary conditions. We consider twodimensional and three-dimensional landscapes consisting of an infinite horizontal strip bounded by two walls a finite distance apart and a horizontal region bounded above and below by horizontal walls, respectively. A nonlinear partial differential evolution Equation for the leading order of infectives is derived. The Equation captures the space and time variations of the spread of the disease in the weakly super critical region

Unconditionally stable time splitting methods for the electrostatic analysis of solvated biomolecules

This work introduces novel unconditionally stable operator splitting methods for solving the time dependent nonlinear Poisson-Boltzmann (NPB) equation for the electrostatic analysis of solvated biomolecules. In a pseudo-transient continuation solution of the NPB equation, a long time integration is needed to reach the steady state. This calls for time stepping schemes that are stable and accurate for large time increments. The existing alternating direction implicit (ADI) methods for the NPB equation are known to be conditionally stable, although fully implicit. To overcome this difficulty, we propose several new operator splitting schemes, including the multiplicative locally one-dimensional (LOD) schemes and additive operator splitting (AOS) schemes. The nonlinear term is integrated analytically in these schemes, while standard discretizations with finite differences in space and implicit time integrations are used. The proposed schemes become much more stable than the ADI methods, and some of them are indeed unconditionally stable in dealing with solvated proteins with source singularities and non-smooth solutions. Numerically, the orders of convergence in both space and time are found to be one. Nevertheless, the precision in calculating the electrostatic free energy is low, unless a small time increment is used. Further accuracy improvements are thus considered, through constructing a Richardson extrapolation procedure and a tailored recovery scheme that replaces the fast Fourier transform method by the operator splitting method in the vacuum case. After acceleration, the optimized LOD method can produce a reliable energy estimate by integrating for a small and fixed number of time steps. Since one only needs to solve a tridiagonal linear system in each independent one dimensional process, the overall computation is very efficient. The unconditionally stable LOD method scales linearly with respect to the number of atoms in the protein studies, and is over 20 times faster than the conditionally stable ADI methods. In addition, some preliminary results on increased stability for ADI methods using a regularization scheme are presented. (Published By University of Alabama Libraries

Augmented Lagrangian method for Euler's elastica based variational models

Euler's elastica is widely applied in digital image processing. It is very challenging to minimize the Euler's elastica energy functional due to the high-order derivative of the curvature term. The computational cost is high when using traditional time-marching methods. Hence developments of fast methods are necessary. In the literature, the augmented Lagrangian method (ALM) is used to solve the minimization problem of the Euler's elastica functional by Tai, Hahn and Chung and is proven to be more efficient than the gradient descent method. However, several auxiliary variables are introduced as relaxations, which means people need to deal with more penalty parameters and much effort should be made to choose optimal parameters. In this dissertation, we employ a novel technique by Bae, Tai, and Zhu, which treats curvature dependent functionals using ALM with fewer Lagrange multipliers, and apply it for a wide range of imaging tasks, including image denoising, image inpainting, image zooming, and image deblurring. Numerical experiments demonstrate the efficiency of the proposed algorithm. Besides this, numerical experiments also show that our algorithm gives better results with higher SNR/PSNR, and is more convenient for people to choose optimal parameters. (Published By University of Alabama Libraries

Linear and nonlinear Rayleigh-Bénard convection in the absence of horizontal boundaries

In the first part of the thesis, we will investigate the linear and weakly non- linear solutions to a convection problem that was first studied by Ostroumov in 1947. The problem pertains to the stability of the equations governing convective motion in an infinite vertical fluid layer that is heated from below. Ostroumov's linear stability analysis yields instability threshold conditions that are characterized by zero wave number for the Fourier mode in the vertical direction and by eigenfunctions that are independent of the vertical coordinate. Thus, any undertaking at determining the super critical nonlinear solutions and their stability through a small amplitude expansion fails. This failure is due to the fact that the terms induced by the nonlinear interaction of the linear modes vanish identically. Here, we put forth exact and stable solutions to the Ostroumov problem. These solutions are characterized by the same critical conditions for linear instability as the Ostroumov solutions. Moreover, we are able to use a small amplitude analysis to extend the analysis to the super critical regime and obtain the nonlinear steady stable solutions. Furthermore, when the analysis is extended to the case where the fluid layer thickness is also allowed to be infinite, we found that the infinite fluid region becomes linearly unstable through a Batchelor-Nitsche instability mechanism. The nonlinear solutions as well as similarity type solutions are then provided. Finally, numerical solutions of the full nonlinear problem are also presented which shed light on the flow patterns and temperature distribution induced by these new solutions. In the second part we consider Rayleigh-Bénard convection with a static density distribution whose unstably stratifed part occupies a very thin central layer. We get asymptotic relations for the critical Rayleigh number for small and large values of the thickness control parameter. Some limiting cases corresponding to the linear eigenvalue problem are treated analytically and the results confirmed by a detailed numerical investigation. For the moderate values of the thickness control parameter an analytical nonlinear stability three-dimensional study is under- taken in the case of poorly conducting boundaries. A weakly nonlinear evolution equation for the leading order temperature perturbation is also derived and solved numerically as function of thickness control parameter " and Prandtl number. (Published By University of Alabama Libraries