Browsing by Author "Lowengrub, J.S."

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    Boundary integral methods for multicomponent fluids and multiphase materials
    (2001-06) Hou, T.Y.; Lowengrub, J.S.; Shelley, M.J.
    In this paper, we present an overview of the application of boundary integral methods in two dimensions to multicomponent fluid flows and multiphase problems in materials science. We focus on the recent development and outcome of methods which accurately and efficiently include surface tension. In fluid flows, we examine the effects of surface tension on the Kelvin-Helmholtz and Rayleigh-Taylor instabilities in inviscid fluids, the generation of capillary waves on the free surface and problems in Hele-Shaw flows involving pattern formation through the Saffman-Taylor instability, pattern selection and singularity formation. In materials science, we discuss microstructure evolution in diffusional phase transformations and the effects of the competition between surface and elastic energies on the microstructure morphology. A common link between these different physical phenomena is the utility of an analysis of the appropriate equations of motion at small spatial scales to develop accurate and efficient time stepping methods.
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    Focusing of an elongated hole in porous medium flow
    (2001-06) Angenent, Sigurd; Aronson, D.G.; Betelu, Santiago; Lowengrub, J.S.
    In the focusing problem we study solutions to the porous medium equation $u_t=\Delta u^m$ whose initial distributions are positive in the exterior of a compact two-dimensional region and zero inside. We assume that the initial interface is elongated and possesses reflectional symmetry with respect to both the x- and y- axes. We implement a numerical scheme that adapts the numerical grid around the interface so as to maintain a high resolution as the interface shrinks to a point. We find that as t tends to the focusing time T, the interface becomes oval-like with the lengths of the major and minor axes $O(\sqrt{T-t})$ and $O(T-t)$ respectively. Thus, the aspect ratio is $O(1/\sqrt{T-t})$. By scaling and formal asymptotic arguments, we derive an approximate solution which is valid for all m. This approximation indicates that the numerically observed power behavior for the major and minor axes is universal for all m>1.
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    Modelling pinchoff and reconnection in a Hele-Shaw cell. Part I: The models and their calibration
    (2001-06) Lee, Hyeong-Gi; Lowengrub, J.S.; Goodman, J.
    This is the first paper in a two-part series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [1]) to flow in a Hele-Shaw cell. The system takes into account the chemical diffusivity between different components of a fluid mixture and the reactive stresses induced by inhomogeneity. In one of the systems we consider (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we motivate, present and calibrate the HSCH/BHSCH equations so as to yield the classical sharp interface model as a limiting case. We then analyze their equilibria, one dimensional evolution and linear stability. In the second paper (Part II [2]), we analyze the behavior of the models in the fully nonlinear regime. In the BHSCH system, the equilibrium concentration profile is obtained using the classical Maxwell construction [3] and does not depend on the orientation of the gravitational field. We find that the equilibria in the HSCH model are somewhat surprising as the gravitational field actually affects the internal structure of an isolated interface by driving additional stratification of light and heavy fluids over that predicted in the Boussinesq case. A comparison of the linear growth rates indicates that the HSCH system is slightly more diffusive than the BHSCH system. In both, linear convergence to the sharp interface growth rates is observed in a parameter controlling the interface thickness. In addition, we identify the effect that each of the parameters, in the HSCH/BHSCH models, has on the linear growth rates. We then show how this analysis may be used to suggest a set of modified parameters which, when used in the HSCH/BHSCH systems, yield improved agreement with the sharp interface model at a finite interface thickness. Evidence of this improved agreement may be found in Part II [2].
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    Modelling pinchoff and reconnection in a Hele-Shaw cell. Part II: Analysis and simulation in the nonlinear regime
    (2001-06) Lee, Hyeong-Gi; Lowengrub, J.S.; Goodman, J.
    This is the second paper in a two part series in which we analyze two diffuse interface models to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. Diffusion between the components is limited if the components are macroscopically immiscible. In one of the systems (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we focus on buoyancy driven flow and the Rayleigh-Taylor instability. In the fully nonlinear regime before pinchoff, results from the HSCH and BHSCH models are compared to highly accurate boundary-integral simulations of the classical sharp interface system. In this case, we find that the diffuse interface models yield nearly identical results and we demonstrate convergence to the boundary-integral solutions as the interface thickness vanishes. We find that the break-up of an unstably stratified fluid layer is smoothly captured by both models. The HSCH model seems to be more diffusive than the BHSCH model and predicts an earlier pinchoff time which causes subtle differences between the two in the pinchoff region. Further, in the limit of zero interface thickness, we find that the effect of compressibility does not vanish at pinchoff. This distinguishes the HSCH model from all others in which compressibility effects are neglected. It may turn out, for example, that characterizing the limiting effect of compressibility at pinchoff may suggest a physically-based selection mechanism for cutting and reconnecting sharp interfaces. Varying the gravitational force and viscosities of the fluids yields different pinchoff times and numbers of satellite drops. Moreover, using the analysis of the linear growth rates from our first paper (Part I [1]), we confirm that the modified HSCH/BHSCH parameters suggested in that work lead to improved agreement with sharp interface results at finite interface thicknesses. Lastly, we also consider a case in which the fluid components are miscible. We find competition between buoyancy, viscous, diffusional and, at very early times, surface tension-like forces.
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    On an elastically induced splitting instability
    (2001-06) Leo, P.H.; Lowengrub, J.S.; Nie, Qing
    We show that a morphological instability driven by deviatoric applied stresses can generate elastically induced particle splitting during phase transformations. The splitting instability occurs when the elastic fields are above some critical value. For elastic fields below critical, one observes a small perturbation of the particle shape consistent with splitting, but this perturbation is stabilized by surface tension. Both the onset of the splitting instability and the nonlinear evolution of the particle towards splitting depend on the precise form of the applied stress, the elastic constants of the precipitate and matrix, and the initial shape of the precipitate. We also investigate whether non-dilatational mistif strains can generate splitting instabilities in the absence of applied stress; however the results are inconclusive.