Showing papers on "Similarity solution published in 1984"

The collapse of the cores of slowly rotating isothermal clouds

Abstract: A generalized model which accounts for the effects of initially uniform and slow rotation is defined for the spherical collapse of a singular isothermal sphere such as protosolar and binary nebulae. An initial unstable equilibrium state is described for a sound speed of 0.35 km/sec and a rotation rate of 10 to the -14th/sec for the molecular cloud surrounding the accreting core. The total angular momentum and mass of the inner cloud is set equal to solar system values. The evolution of the collapse is traced by applying a perturbation analysis to the similarity solution for a nonrotating condition, and matched asymptotic expansions solve the hydrodynamic equations. The model is concluded a valid tool for studying star and nebular disk formation.

The three‐dimensional flow due to a stretching flat surface

Abstract: An exact similarity solution of the Navier–Stokes equations is found. The solution represents the three‐dimensional fluid motion caused by the stretching of a flat boundary.

Flow of a viscoelastic fluid over a stretching sheet

Abstract: This paper presents a study of the flow of an incompressible second-order fluid past a stretching sheet. The problem has a bearing on some polymer processing application such as the continuous extrusion of a polymer sheet from a die.

Flow development in a porous channel and tube

Abstract: The spatial development of the inlet velocity profile in a uniformly porous channel and tube is investigated. For tubes which are very long compared with their radius, it is shown that above a critical Reynolds number of 2.3 the inlet velocity profile does not necessarily decay into the fully developed, similarity profile for an infinite tube. Rather, the structure of the flow throughout the entire tube is influenced by the inlet profile. This loss of validity of the similarity solution is due to the fact that the tube is of finite length and the inlet profile is not of the similarity form. The actual length of the tube is, however, unimportant. Analogous results hold for the flow in a porous channel, with a critical Reynolds number of approximately 6.

Transient Heat Transfer in Superfluid Helium — Part II

Abstract: Three classical problems associated with the ordinary diffusion equation concern the temperature in (1) a half-space with clamped heat flux at the free face, (2) a half-space with clamped temperature at the free face, and (3) an infinite medium with a pulsed plane heat source. These problems are also important for the nonlinear diffusion equation based on the Gorter-Mellink relation, which describes heat transport in superfluid helium. A similarity solution to problem (1) , the champed-flux problem, has already been found1 and compared, with goou agreement, with experimental data of van Sciver.2 [A similarity solution is one in which the profiles of temperature rise AT versus distance Z at different times t can be obtained from one another by suitable (different) stretching of the temperature and distance axes.] In this paper, I give similarity solutions in analytic form to problems (2) and (3) , the clamped-temperature and pulsed-source problems.

Similarity solutions for buoyancy induced flows in a saturated porous medium adjacent to impermeable horizontal surfaces

Abstract: In the present paper similarity solutions for the convective flow induced by buoyancy in a saturated porous medium adjacent to horizontal impermeable surfaces are obtained. The analysis incorporates the variation of permeability from the wall and expressions for boundary layer thickness, local and overall surface heat-flux are obtained. Applications of the results to convective flows in a geothermal reservoir are discussed.

Mixed Convective Burning of a Fuel Surface With Arbitrary Inclination

Abstract: An analysis is developed for mixed, forced, and free convective combustion on a flat fuel surface of arbitrary inclination that makes use of the laminar boundary layer approximation to describe the gas flow and of the flame-sheet approximation to describe the gas-phase reactions. A mixed-convection parameter (Re/sub x//sup n/+Gr/sub x//sup m/)/sup 1/2n/ that properly scales the dependent and independent variable fields and a mixed convection ratio (Gr/sub x//sup m//Re/sub x//sup n/)/sup 1/2/ that plays the role of the downstream local similarity coordinate are introduced in the nondimensionalization of the equations. It is shown that these two parameters, rather than the standard Reynolds, Re/sub x/, and Grashof, Gr/sub x/, numbers are the optimum choice of governing nondimensional groups for this problem. The values of m and n are selected to obtain a similarity solution of the governing equations in the pure free convection limit for a vertical (m = 2, n = 4) and a horizontal (m = 2, n = 5) surface, which are the cases solved in this work. With this formulation, the solution of the problem provides for both cases smooth transition of all physical variables from one convective limit to the other. Results are obtained from numericalmore » integration of the governing equations and from application of the local similarity approximation. It is shown that the range of validity of local similarity is extended beyond that obtained with alternate formulations and that the proper limits are approached. For use in practical applications, the results suggest that explicit expressions for the mass burning rate and for the fraction of unburnt pyrolyzate can be found that will suffice over the whole range of mixed-flow intensity.« less

Application of the similarity method to the nonlinear Boltzmann equation

Abstract: A consequent application of the similarity method to the nonlinear Boltzmann equation leads to the general form of exact similarity solutions and allows a group-theoretic classification. Classes of similarity solutions depending very strongly on the source term but different from the Bobylev-Krook-Wu solution will be discovered and discussed.

Nonlinear Moving Boundary Problems and a Keller Box Scheme

Abstract: A modification of the original Kelley box scheme is used to approximate nonlinear parabolic partial differential equations with moving boundaries. The proposed scheme solves two systems of linear algebraic equations at each time-step to produce second-order approximations to the solution, its derivative (the flux) and the position of the moving boundary. The scheme is applied to a physical problem which has a known similarity solution. A comparison of numerical results with the similarity solution gives evidence of stability and convergence.

Axisymmetric Vortex Solution of Navier-Stokes Equation

Abstract: An exact solution of a viscous incompressible flow is presented for a general initial condition. This flow represents an axisymmetric shear layer superimposed on an irrotational straining flow. The solution incorporates the three representative features of vortex motion: stretching, convection and viscous diffusion. In a particular case of constant straining, the flow approaches to a steady state in which the above three effects are in equilibrium. However if the initial state is composed of the same amount of opposite vorticities, the shear layer disappears exponentially in time. Spectral analysis of the solution shows the cascade of vorticity fluctuations to smaller scales. The general solution includes the vortex solutions given by Oseen (1911) and by Burgers (1948), and partly overlaps the similarity solution found by Bellamy-Knights (1970).

A new similarity solution for reverse shocks in supernova remnants.

Abstract: A new self-similar solution is presented for the early time development of a reverse shock propagating into supernova ejecta with an initially uniform density. The ejecta are taken to be expanding adiabatically into an ambient medium with a power-law density profile rhoproportionalr/sup -//sup s/. The solution is obtained by treating the reverse shock as an infinitely thin freely expanding shell. The solution is compared to hydrodynamic simulations, with satisfactory agreement. The usefulness of this analytic approximate solution lies in the fact that the x-ray emission from reverse shocks in uniform supernova ejecta is dominated by a thin dense shell formed at the contact interface with the ambient medium. The structure of this thin shell, which is crucial to determining the x-ray emission, will be better represented at early times by this analytic approximation than by hydrodynamic simulations, because of the large density and temperature gradients in the shell and the finite grinding of the hydro simuations. The solution may also have some application to the reverse shock structure in freely expanding clumps of ejecta. Various generalizations of the similarity solution are given in an Appendix.

Comparison theorems and similarity solution approximations for a nonlinear diffusion equation arising in the study of soft tissue

Abstract: A nonlinear diffusion equation arising from a biphasic continuum model of soft tissue, such as for articular cartilage, is studied. Through the use of maximum principles and comparison theorems, the quantitative characteristics of the solution are established, independent of the particular nonlinear constitutive equation used for the solid-fluid interaction in the tissue. In conjunction with this, similarity and asymptotic methods are employed to obtain approximate solutions to the problem.

Squeeze-film flow of an Oldroyd-B fluid: Similarity solution and limiting Weissenberg number

Abstract: In this paper we consider the motion of an Oldroyd-B fluid being squeezed between two parallel disks of infinite extent. We show that a similarity solution exists provided that the fluid inertia is neglected and that the squeezing velocity varies exponentially with time. We prove that there is a critical Weissenberg number above which at least one component of the stresses grows unboundedly with time. This critical Weissenberg number is of the order of 0.67. This latter conclusion is also valid in the continuous squeeze-film mode.

Electrical discharges and intense vortices

Abstract: In this paper we consider the axisymmetric flow field generated in a semi­-infinite viscous incompressible fluid by an electric current discharge in the presence of a line vortex which coincides with the axis of the discharge. The system is supposed to be bounded by a fixed plate which is perpen­dicular to the line vortex. A similarity solution is constructed for the non­-linear problem. The idealized solution is, in effect, the superposition of two existing solutions both of which are associated with up-draught velocities and lends qualitative support to theories that associate tornado vortices with electrical activity, though here the agency generating the velocity field is the Lorentz force and not, the usually assumed, electric heating.

Similarity solution for the interaction of the stellar wind with surrounding interstellar medium

Abstract: Similarity solution for the interaction of stellar wind with surrounding interstellar medium is obtained on the base of momentum conservation. The conversion efficiency of kinetic energy of stellar wind into the kinetic energy of expanding shell is derived. The results are compared with the observations on the ring nebulae associated with the Wolf-Rayet star and it is shown that the observed values of energy conversion are in good agreement with the momentum conserving model.

Rapid expansion of polytropes

Abstract: An extremely massive unevolved star may be idealized as an n = 3 polytropic sphere supported by radiation pressure. Such a polytrope is subject to explosive, supersonic expansion, whose later stages can be described by similarity solution of the dynamical equations. This picture is confirmed by numerical hydrodynamic simulation, which shows that an n = 3 polytrope, initially at rest near equilibrium, undergoes an explosive expansion which approaches the similarity solution at large times.

Laminar boundary layer heat transfer along a moving plate with suction or blowing

Abstract: The effects of suction or blowing on the laminar boundary layer flow and heat transfer along a horizontal continuously moving plate are analyzed by a local similarity method and a local nonsimilarity method. The cases of prescribed surface temperature and prescribed wall heat flux are considered. The effects of surface suction or blowing on the boundary layer friction factor and heat transfer rate are studied in detail. It is shown that the local nonsimilarity solution gives more accurate results than the local similarity solution.

A moving boundary problem which arises in several physical situations is described

Abstract: A modification of the original Keller box scheme is used to approximate nonlinear parabolic partial differential equations with moving boundaries. The proposed scheme solves two systems of linear algebraic equations at each time-step to produce second-order approximations to the solution, its derivative (the flux) and the position of the moving boundary. The scheme is applied to a physical problem which has a known similarity solution. A comparison of numerical results with the similarity solution gives evidence of stability and convergence. (1.3) ~~~ut = (K(U) UX)X in 0 O, s - g(s, u)ux on x s(t), t> 0.