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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01 Jan 1970
TL;DR: In this article, a theoretical study of laminar natural convection over a semi-infinite vertical plate is presented, where the plate is maintained at a given concentration in some chemical species and convection is induced by diffusion into and chemical reaction with the ambient fluid.
Abstract: This paper deals with a theoretical study of laminar natural convection over a semi-infinite vertical plate. It is assumed that the plate is maintained at a given concentration in some chemical species and convection is induced by diffusion into and chemical reaction with the ambient fluid. The fluid is assumed to be viscous and the induced fluid flow, steady. A similarly transform to one variable is possible in the absence of chemical reaction. However, when the chemical reaction takes place between the plate and the ambient fluid, a similarity solution is not available. Thus to obtain an analytical solution of the problem, perturbation expansions about an additional similarity variable which is dependent on the reaction rate must be employed. It has been found that the Schmidt number Sc and the reaction order n are the two fundamental parameters of the problem.
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TL;DR: In this article, a new similarity transformation applies to the boundary layer equations, which govern laminar, steady, and incompressible flows, and conclusions are drawn regarding the tolerance of these boundary layers to flow separation under an adverse pressure gradient.
Abstract: A new similarity transformation applies to the boundary layer equations, which govern laminar, steady, and incompressible flows. This transformation is proved to be more consistent and more complete than the well known Falkner—Skan transformation. It applies to laminar, incompressible, and steady boundary layer flows with a power-law u e (x)=cx m or exponential profile u e (x)=ce mx of the outer velocity. This family of "similar solutions" is resolved for various values of the exponent m. A physical interpretation of these velocity profiles is presented, and conclusions are drawn regarding the tolerance of these boundary layers to flow separation under an adverse pressure gradient.
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TL;DR: In this article, a diffusion-controlled moving-boundary problem in drug dissolution is considered, where the moving front passes from one medium to another for which the diffusion coefficient is many orders of magnitude smaller.
Abstract: This paper is concerned with a diffusion-controlled moving-boundary problem in drug dissolution, in which the moving front passes from one medium to another for which the diffusion coefficient is many orders of magnitude smaller. It has been shown in an earlier paper that a similarity solution exists while the front is passing through the first layer, but that this breaks down in the second layer. Asymptotic methods are used to understand what is happening in the second layer. Although this necessitates numerical computation, one interesting outcome is that only one calculation is required, no matter what the diffusion coefficient is for the second layer.
01 Jan 2014
TL;DR: In this paper, the longitudinal dispersion phenomenon of miscible fluids in porous media is discussed by regarding the cross-sectional flow velocity as time dependent in a specific form, and the mathematical formulation of the phenomenon yields a non-linear partial differential equation.
Abstract: The longitudinal dispersion phenomenon of miscible fluids in porous media is discussed by regarding the cross- sectional flow velocity as time dependent in a specific form. The mathematical formulation of the phenomenon yields a non-linear partial differential equation. This partial differential equation is transformed into ordinary differential equation by using infinitesimal transformations group technique of similarity analysis. An analytical solution of the later is derived in terms of error function.The solution obtained is physically consistent with the results of earlier re- searchers and which is also more classical than other results obtained by various researchers.This type of phenomenon has been of great concern to hydrologists who have been studying the problem of displacement of fresh water by sea water in coastal areas. The oil industry has also become involved in miscible displacement studies in connection with the possibility of flushing oil by solvents from reservoirs.
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TL;DR: In this article, the solutions of the Cauchy problem for the equations of gas dynamics are shown to be unbounded in the non-isentropic case, and the solution of the problem is shown to work in the case of nonisentropy.
Abstract: THE solutions of the Cauchy problem for the equations of gas dynamics are shown to be unbounded in the non-isentropic case.