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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: The similarity solution using the composite similarity variable appears to be applicable to a broad class of reactive transport problems involving mineral reactions in fracture–matrix systems and reproduces the solutions for non-reactive solute and heat transport when diffusion/dispersion/conduction are neglected in the fracture.
Abstract: We propose a new composite similarity variable, based on which a similarity solution is derived for reaction front propagation in fracture-matrix systems. The similarity solution neglects diffusion/dispersion within the fracture and assumes the existence of a sharp reaction front in the rock matrix. The reaction front location in the rock matrix is shown to follow a linear decrease with distance along the fracture. The reaction front propagation along the fracture is shown to scale like diffusion (i.e. as the square root of time). The similarity solution using the composite similarity variable appears to be applicable to a broad class of reactive transport problems involving mineral reactions in fracture-matrix systems. It also reproduces the solutions for non-reactive solute and heat transport when diffusion/dispersion/conduction are neglected in the fracture. We compared our similarity solution against numerical simulations for nonlinear reactive transport of an aqueous species with a mineral in the rock matrix. The similarity solutions agree very well with the numerical solutions, especially at later times when diffusion limitations are more pronounced.This article is part of the themed issue 'Energy and the subsurface'.
9 citations
TL;DR: In this article, a Stefan problem is formulated for a diffusion equation where the Fickian flux is supplemented by a stress gradient term, and the stress in turn obeys a concentration-dependent evolution equation that includes impulsive and relaxation effects.
Abstract: A Stefan problem is formulated for a diffusion equation where the Fickian flux is supplemented by a stress gradient term. The stress in turn obeys a concentration-dependent evolution equation that includes impulsive and relaxation effects. The speed of the free boundary is taken to be proportional to the flux at the front. For short times, the front position is found to be proportional to $t^{1/2} $. Using singular perturbation and numerical methods, three types of “long-time” behavior are found, depending on relationships between the coefficient functions and the parameters in the problem: (i) the front position is proportional to $t^{1/2} $; (ii) the front position is proportional to t; or (iii) the front stops in a finite time and the solution becomes singular (infinite gradients) at that point.
9 citations
TL;DR: In this paper, it is shown that a perturbation scheme for the stream function consisting of a series of inverse powers of ξ and which uses the asymptotic similarity solution as the basic solution is inadequate, and a modification to the series so as to include terms like ξn(ln ζ)m must be adopted in order to satisfy the boundary conditions.
Abstract: A general similarity solution suggested by Watson for the problem of the laminar, radial, free-jet with swirl has been previously discussed by Riley who also calculated the order to which the solution was valid. That problem is considered in more detail here and higher order terms are given. It is shown that a perturbation scheme for the stream function consisting of a series of inverse powers of ξ and which uses the asymptotic similarity solution as the basic solution is inadequate, and a modification to the series so as to include terms like ξn(ln ζ)m must be adopted in order to satisfy the boundary conditions. It is also shown that the general similarity solution may be obtained from the asymptotic series representing the general case with swirl for certain special values of the free constants and also for the no-swirl or free-jet problem. The asymptotic series is given to order ξ−13 for the case of swirl and to order ζ−29 when there is no swirl.
9 citations
TL;DR: In this paper, it was shown that the generalized Riemann problem admits a unique global piecewise C1 solution u = u(t, x) containing only n shock waves with small amplitude on t ≥ 0.
Abstract: It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C1 solution u = u(t, x) containing only n shock waves with small amplitude on t ≥ 0 and this solution possesses a global structure similar to that of the similarity solution u = U (x/t) of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data.
9 citations
09 Apr 2020
TL;DR: In this paper, the authors revisited the problem of Aurangzaib et al. and reproduced the results for triple solutions, which revealed that there are ranges of multiple solutions, triple solutions and a single solution.
Abstract: In this article, we reconsidered the problem of Aurangzaib et al., and reproduced the results for triple solutions. The system of governing equations has been transformed into the system of non-linear ordinary differential equations (ODEs) by using exponential similarity transformation. The system of ODEs is reduced to initial value problems (IVPs) by employing the shooting method before solving IVPs by the Runge Kutta method. The results reveal that there are ranges of multiple solutions, triple solutions, and a single solution. However, Aurangzaib et al., only found dual solutions. The effect of the micropolar parameter, suction parameter, and Prandtl number on velocity, angular velocity, and temperature profiles have been taken into account. Stability analysis of triple solutions is performed and found that a physically possible stable solution is the first one, while all leftover solutions are not stable and cannot be experimentally seen.
9 citations