Similarity solution

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

Similarity solution of air and nanofluid impingement cooling of a cylindrical porous heat sink

Abstract: In the present paper, a numerical study on flow and heat transfer of air/nanofluid impinging jet flow through a cylindrical porous heat sink has been done The cooling fluid flows uniformly through the porous foam over the hot disk Conservation equations of mass, momentum and energy in form of partial differential equation, by using similarity variables, converted to nonlinear ordinary differential equations and finally these equations have been solved numerically Also, results are validated through comparison with those of a commercial CFD code with an acceptable agreement In this study, effects of dimensionless parameters in the form of geometrical, porous media characteristics and cooling fluid properties on the velocity profile, temperature profile and average Nusselt number have been scrutinized Results indicate that addition of nanoparticles, thermal conductivity ratio increment and aspect ratio reduction will enhance the heat sink thermal performance

Asymptotic Behaviour of Velocity Profiles in the Prandtl Boundary Layer Theory

Abstract: Consider the Prandtl boundary layer equation for the steady two-dimensional laminar flow of an incompressible viscous fluid past a rigid wall. On the basis of an arbitrary velocity profile at some initial position on the wall, the analysis presented shows that, for a power law streaming speed U(x) = C(x + d)$^m$ (m $\geqslant$ 0), the velocity profile which develops downstream is asymptotically given by the well known Falkner-Skan similarity solution. Moreover, for a streaming speed satisfying (6), the velocity profile which develops downstream is asymptotically unique, though of course the particular form of the resulting profile depends on the precise nature of the exterior stream. The rate of convergence for this asymptotic behaviour is estimated, as well as corresponding rates for the convergence of the skin friction coefficient. This result verifies the tacit assumption of a number of writers that the downstream velocity profile is essentially independent of the initial profile, and also supplies a theoretical justification for the role of similar solutions in boundary layer theory. We also prove the existence of concave velocity profiles whenever the pressure gradient is favourable. It follows that, for streaming speeds which correspond to a favourable pressure gradient, concave velocity profiles play somewhat the same role as similarity profiles do for a power law streaming speed.

Similarity Analysis of Creep Indentation Tests

Abstract: The general theory of axisymmetric hardness tests on nonlinear media is approached from the standpoint of similarity transformations. It is shown how an entire process of indentation can be made to depend on the solution of just one boundary-value problem in scaled variables and with a fixed geometry. Once this single auxiliary solution has been obtained, the values of all physical quantities in the original problem can be generated readily at any stage without further numerical error. Even by themselves the similarity relations provide valuable information about (for example) an invariant connection between the depth of penetration and the radius of contact, or about the variation of penetration with time in a creep test under dead load. Two kinds of material behaviour are considered: (a) nonlinear elastic (modelling strain-hardening plasticity) and (b) nonlinear viscous (modelling secondary creep). In either category the constitutive specification is sufficiently flexible to represent a wide range of actual responses in the context of hardness testing. The analysis for case (a) extends a theory of ball indentation by Hill et al. to a class of indenters with shapes varying from flat to conical. It also prepares the ground for case (b) which is more difficult and calls for a quite different auxiliary problem.

Melting from a flat plate embedded in a porous medium in the presence of steady natural convection

Abstract: The problem of melting from a flat plate embedded in a porous medium is studied. The main focus is to determine the effect of natural convection flow in the liquid phase on the melting phenomenon. Two configurations an considered and modeled mathematically: a vertical plate and a horizontal plate. The final similarity equations art integrated numerically by use of the fourth-order Runge-Kutta method. Systematic “shooting” is required to satisfy the boundary conditions at infinity. Results are reported for the temperature and flow fields in the melt region. The melting phenomenon decreases the local Nusselt number at the solid-liquid interface.