Topic
Similarity solution
About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.
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TL;DR: In this paper, the authors report on measurements of pinch-off dynamics for a lyotropic surfactant/water solution in the lamellar phase and a thermotropic liquid crystal in the smectic phase.
Abstract: Droplet pinch-off of fluids with liquid crystalline order is a common yet poorly understood process. We report on measurements of pinch-off dynamics for a lyotropic surfactant/water solution in the lamellar phase and a thermotropic liquid crystal in the smectic phase. We find pinch-off is universal and well described by a similarity solution for a strain thinning power-law fluid. This finding is consistent with bulk rheology measurements which show these materials shear thin with the appropriate power-law dependence. Remarkably, we find depending on material processing, this universal pinch-off cuts off at different length scales. Collectively, these phenomena lead to an exceptional form of singularity where pinch-off is both universal and dependent on initial conditions.
31 citations
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TL;DR: In this paper, a new model for unsaturated flow in porous media, including capillary hysteresis and dynamic capillary effects, is analyzed, and the existence and uniqueness of solutions are established and qualitative and quantitative properties of (particular) solutions are analyzed.
Abstract: A new model for unsaturated flow in porous media, including capillary hysteresis and dynamic capillary effects, is analyzed. Existence and uniqueness of solutions are established and qualitative and quantitative properties of (particular) solutions are analyzed. Some results of numerical computations are given. The model under consideration incorporates simple ‘play’-type hysteresis and a dynamic term (time-derivative with respect to water content) in the capillary relation. Given an initial water content distribution, the model determines which parts of the flow domain are in drainage and which parts are in imbibition. The governing equations can be recast into an elliptic problem for fluid pressure and an evolution equation for water content. Standard methods are used to obtain numerical results. A comparison is given between J.R. Philip's semi-explicit similarity solution for horizontal redistribution in an infinite one-dimensional domain and solutions of the new model.
31 citations
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TL;DR: In this article, it is shown that the solutions of a class of initial boundary value problems converge to a similarity solution as $t \to \infty $. And the optimal rate of convergence is established.
Abstract: A nonlinear diffusion equation which arises in the theory of plasma physics is discussed. It is shown that the solutions of a class of initial boundary value problems converge to a similarity solution as $t \to \infty $. The optimal rate of convergence is established.
31 citations
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TL;DR: In this article, an analysis of the similarity solution of the linear, homogeneous, fragmentation equation for a general volume-conserving daughter-fragment distribution is given for the special case of a polynomial daughter-distribution of degree p, and an exact, basic similarity solution for the time evolution of the particle-volume distribution is derived.
Abstract: An analysis of the similarity solution of the linear, homogeneous, fragmentation equation is given for a general volume-conserving daughter-fragment distribution. For the special case of a polynomial daughter-distribution of degree p, an exact, basic similarity solution for the time evolution of the particle-volume distribution is derived. The solution is proportional to the Meijer G-function which may be represented as a linear combination of generalized hypergeometric functions. The properties of generalized hypergeometric functions and G-functions are given in the special function literature and include the continuation of the solution to large values of the similarity variable which is given here for special cases.
30 citations
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TL;DR: Some similarity reductions are found for the regularized Ostrovsky-Grimshaw model with symbolic computation, to a coupled set of nonlinear ordinary differential equations.
Abstract: The Ostrovsky model is widely used to describe mechanical and physical problems such as internal or surface waves in the oceans and magnetic sounds in plasmas. This model has recently been Grimshaw-regularized for certain continuity in the mass field, while computerized symbolic computation becomes a branch of artificial intelligence. In this paper, some similarity reductions are found for the regularized Ostrovsky-Grimshaw model with symbolic computation, to a coupled set of nonlinear ordinary differential equations. The micropterons and macropterons are analytically presented and discussed, and have been found to contain certain solitonic cores plus a number of sinusoidal ``wings''. Examples are the micropterons and macropterons for fluid velocities in the wave propagation direction and transverse direction, respectively.
30 citations