Similarity solution

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

The motion induced between a rotating plate and a radially stretching or shrinking plate

Abstract: The flow induced between parallel plates separated by a distance h executing different types of in-plane motion is investigated. The upper plate radially stretches at strain rate a and the lower plate rotates at angular velocity Ω about a common axis. A similarity solution form reduces the Navier-Stokes equations to a coupled pair of ordinary differential equations in two parameters: R = |a| h2/ν and σ = Ω/|a|, where ν is the kinematic viscosity of the fluid. Solutions are obtained for both stretching and shrinking upper plates and numerical results for pressure gradient and wall shear stress parameters are found and compared with low-R series solutions and large-R asymptotic behaviors. Sample radial and azimuthal velocity profiles reveal regions of zero radial and azimuthal wall shear stress which are studied in detail. This work represents the first study of the flow induced between parallel plates for which each plate executes a different type of in-plane motion.

Approximate solution of laminar thermal boundary layer over a thin plate heated from below by convection

Abstract: In this paper, an integration of a symbolic power series method - Pade approximation technique (PS - Pade), was utilized to solve a system of nonlinear differential equations arising from the similarity solution of laminar thermal boundary layer over a flat plate subjected to a convective surface boundary condition. As both boundary conditions tended to infinity, the combination of series solutions with the Pade approximants was used for handling boundary conditions on the semi-infinite domain of solution. The combination of power series and Pade proposed an alternative approach of solution which did not require small parameters and avoided linearization and physically unrealistic assumptions. The results of the present approach were compared with numerical results as well as those of previous works reported in the literature. The obtained results represented remarkable accuracy in comparison with the numerical ones. Finally, reduced Nusselt number, as an important parameter in heat transfer, was calculated by the obtained analytical solution. The present power series-Pade technique was very simple and effective, which could develop a simple analytic solution for flow and heat transfer over the flat plate. The results of the present study could be easily used in practical applications.

One-Dimensional Nonlinear Stefan Problems in Storm's Materials

Abstract: We consider two one-phase nonlinear one-dimensional Stefan problems for a semi-infinite material x > 0; with phase change temperature Tf: We assume that the heat capacity and the thermal conductivity satisfy a Storm's condition. In the first case, we assume a heat flux boundary condition of the type q(t) = q0 p t , and in the second case, we assume a temperature boundary conditionT = Ts < Tf at the fixed face. Solutions of similarity type are obtained in both cases, and the equivalence of the two problems is demonstrated. We also give procedures in order to compute the explicit solution.

Turbulent jet in confined counterflow

Abstract: The mean flowfield of a turbulent jet issuing into a confined, uniform counterflow was investigated computationally. Based on dimensional analysis, the jet penetration length was shown to scale with jet-to-counterflow momentum flux ratio. This scaling and the computational results reproduce the well-known correct limit of linear growth of the jet penetration length for the unconfined case when the momentum flux ratio is small. However, for the high momentum flux ratio case corresponding to the confinement, the jet penetration length is shown to reach an asymptotic limit of about 3.57 times the confining duct diameter. This conclusion is contrary to the existing results which predict indefinite growth. A simple modification of an existing similarity solution for the jet in an unconfined counterflow provides a convenient framework for presenting the results of the flowfield and jet penetration length.

Developing flow of a power-law liquid film on an inclined plane

Abstract: Developing flow of a liquid film along a stationary inclined wall is analyzed for a power-law constitutive equation. For films with appreciable inertia and therefore small interfacial slopes, the boundary-layer approximation may be used. The boundary-layer equations are solved numerically through the von Mises transformation that gives a partial differential equation over a semi-infinite strip and approximately by the method of von Karman and Polhausen that gives an ordinary differential equation for the film thickness, called a film equation. Film equations derived from self-similar velocity profiles fail when the film thickens and the flow undergoes a supercritical to subcritical transition; a nonremovable singularity arises at the critical point, the location of the flow transition. A film equation is developed that accommodates this transition. Predictions exhibit a standing wave where hydrostatic pressure becomes important and opposes inertia. This thickening effect is accentuated for small angles of inclination at moderate Reynolds numbers. In the limit of small film thickness in which gravitational effects are negligible, the thickness profile is nonlinear in agreement with an independent and new similarity solution. This result contrasts with the established linear thickness profile for a Newtonian liquid. The circumstances in which the film equation gives results close to the full boundary layer equation are identified.