Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

Wall laminar nanofluid jet flow and heat transfer

Abstract: Purpose This work aims to investigates exact solutions of the classical Glauert’s laminar wall jet mass and heat transfer under wall suction, wall contraction or dilation, and two thermal transport boundary conditions; prescribed constant surface temperature and prescribed constant surface flux in nanofluidic environment. Design/methodology/approach The flow system arranged in terms of partial dif- ferential equations is non-dimensionalized with suitable dimensionless transformation variables, and this new set of equations is reduced into ordinary differential equations via a set of similarity transformations, where they are treated analytically for closed form solutions. Findings Exact solutions of nanofluid flow for velocity distributions, momentum flux, wall shear stress and heat transfer boundary layers for commonly studied nanoparticles; namely copper, alumina, silver, and titanium oxide are presented. The flow behavior of alumina and titanium oxide is identical, and a similar behavior is seen for copper and silver, making two pairs of identical traits. The mathematical expressions as well as visual analysis of wall shear drag and temperature gradient which are of practical interest are analyzed. It is shown that wall stretching or shrinking, wall transpiration and velocity slip together influences the jet flow mechanism and extends the original Glauert’s jet solutions. The exact solutions for the two temperature boundary layer conditions and temperature gradients are analyzed analytically. It is found that the effect of nanopar- ticles concentration on thermal boundary layer is intense, causing temperature uplift, whereas the wall transpiration causes a decrease in thermal layers. Originality/value The analysis carried out in nanofluid environment is genuinely new and unique, as our work generalizes the Glauert’s classical regular wall jet fluid problem.

On radiating laminar free-mixing: similarity analysis,

Abstract: : Radiating, planar, laminar free-mixing is investigated by a boundary layer type analysis with similarity methods. The radiative process is described by either the optically thick, the optically thin or the plane parallel layer approximations. The jet, symmetric wake, asymptotic symmetric wake and the merging of two semi-infinite streams are considered. Although non-trivial solutions are admitted in each of the flows, a non-restrictive similarity solution is obtained only in the case of the asymmetric wake with constant external flow properties for an optically thick gas. The corresponding analysis for axisymmetric free-mixing is not included here, since it previously has been demonstrated that in non-radiative compressible flows only extremely restrictive similarity solutions exist. (Author)

Similarity solution for the propagation of shock waves with varying energy in non-uniform medium

Abstract: A model of self-similar propagation of shock waves driven by a flare energy release in a non-uniform atmosphere has been considered. The total energy content of the model is assumed to be increased with time within the inner expanding surface and shock front. Finally the variation of velocity, pressure, density, and energy of the model have been discussed. The gas is assumed to be grey and opaque.

Similarity solution for unsteady gravity-driven dry patch in a non-Newtonian fluid flow

Abstract: We consider an unsteady thin-film flow of a non-Newtonian fluid around a dry patch subject to gravitational acceleration on an inclined plane. The general governing partial differential equation is transformed into the second-order ordinary differential equation using a unique travelling-wave similarity transformation. The analysis shows that the dry patch has a parabolic shape and the film thickness was found to increase monotonically away from the dry patch. Numerical solutions of the similarity equation are obtained for the velocity of the dry patch. These numerical solutions are also compared with the asymptotic solutions in the certain limits. The effects of power-law index on the behavior and patterns of the solutions are also discussed.

Bounds, monotonicity, uniqueness, and analytical calculation of a class of similarity solutions for the fluid flow over a nonlinearly stretching sheet

Abstract: Invoking some estimates obtained in [F.T. Akyildiz et al., Mathematical Methods in the Applied Sciences 33 (2010) 601–606] (which presented an alternate method of proof for the present problem), we correct the parameter regime considered in [R.A. Van Gorder, K. Vajravelu, and F. T. Akyildiz, Existence and uniqueness results for a nonlinear differential equation arising in viscous flow over a nonlinearly stretching sheet, Applied Mathematics Letters 24 (2011) 238–242] and add some details, which were omitted in the original proof. After this is done, we formulate a more elegant method of proof, converting the nonlinear ODE into a first nonlinear order system. This gives us a more natural way to view the problem and lends insight into the behavior of the solutions. Finally, we give a new way to approximate the shooting parameter α = f ′ ′ (0) analytically, through minimization of the L2([0, ∞ )) norm of residual errors. This approximation demonstrates the behavior of the parameter α we expect from the proved theorems, as well as from numerical simulations. In this way, we obtain a concise analytical approximation to the similarity solution. In summary, from this analysis, we find that monotonicity of solutions and their derivatives is essential in determining uniqueness, and these monotone solutions can be approximated analytically in a fairly simple way. Copyright © 2014 John Wiley & Sons, Ltd.