Similarity solution

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

Flow of an eyring-powell non-newtonian fluid over a stretching sheet

Abstract: This article is devoted to the study of the boundary layer flow of a non-Newtonian fluid over a stretching sheet. The non-Newtonian behavior of the fluid is characterized by the constitutive equation due to Powell and Eyring (1944). A second-order approximation of the Eyring-Powell model is used to obtain the flow equations. A local similarity solution of the governing problem is obtained numerically using an implicit finite difference scheme known as the Keller box method. The influence of pertinent non-Newtonian fluid parameters M and λ on the velocity and skin-friction coefficient is analyzed through graphical and tabular results.

Stagnation-point flow of upper-convected Maxwell fluids

Abstract: Two-dimensional stagnation-point flow of viscoelastic fluids is studied theoretically assuming that the fluid obeys the upper-convected Maxwell (UCM) model. Boundary-layer theory is used to simplify the equations of motion which are further reduced to a single non-linear third-order ODE using the concept of stream function coupled with the technique of the similarity solution. The equation so obtained was solved using Chebyshev pseudo-spectral collocation-point method. Based on the results obtained in the present work, it is concluded that the well-established but controversial prediction that in stagnation-point flows of viscoelastic fluids the velocity inside the boundary layer may exceed that outside the layer may just be an artifact of the rheological model used in previous studies (namely, the second-grade model). No such peculiarity is predicted to exist for the Maxwell model. For a UCM fluid, a thickening of the boundary layer and a drop in wall skin friction coefficient is predicted to occur the higher the elasticity number. These predictions are in direct contradiction with those reported in the literature for a second-grade fluid.

Convergence rate for compressible Euler equations with damping and vacuum

Abstract: We study the asymptotic behavior of L∞ weak-entropy solutions to the compressible Euler equations with damping and vacuum. Previous works on this topic are mainly concerned with the case away from the vacuum and small initial data. In the present paper, we prove that the entropy-weak solution strongly converges to the similarity solution of the porous media equations in Lp(R) (2≤p<∞) with decay rates. The initial data can contain vacuum and can be arbitrary large. A new approach is introduced to control the singularity near vacuum for the desired estimates.

An exact solution of the Navier-Stokes equations for magnetohydrodynamic flow

Abstract: Viscous flow past a stretching sheet in the presence of a uniform magnetic field is considered. An exact similarity solution for velocity and pressure of the two-dimensional Navier-Stokes equations is presented, which is formally valid for all Reynolds numbers. The solution for the velocity field turns out to be the identical solution derived earlier by Pavlov [1] within the framework of high-Reynolds-number boundary layer theory, in which the pressure distribution cannot be determined.

Long-Time Asymptotics of Kinetic Models of Granular Flows

Abstract: We analyze the long-time asymptotics of certain one-dimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi-elastic limit of a model Boltzmann equation with dissipative collisions and variable coefficient of restitution. These nonlinear equations, classified as nonlinear friction equations, split naturally into two classes, depending on whether or not the temperature of their similarity solutions (homogeneous cooling states) reduce to zero in finite time. For both classes, we show uniqueness of the solution by proving decay to zero in the Wasserstein metric of any two solutions with the same mass and mean velocity. Furthermore, if the temperature of the similarity solution decays to zero in finite time, we prove, by computing explicitly upper bounds for the lifetime of the solution in terms of the length of the support, that the temperature of any other solution with initially bounded support must also decay to zero in finite time.