Large-time Solution Research Articles

Unsteady mixed convection flow in the stagnation region of a heated vertical plate due to impulsive motion

The unsteady mixed convection in the stagnation flow on a heated vertical plate is studied where the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase in the surface temperature (heat flux). The short time as well as the long time solutions are included in the analysis. Both prescribed surface temperature and prescribed surface heat flux conditions are considered. The partial differential equations governing the flow and the heat transfer have been solved numerically using an implicit finite difference scheme. Also, the asymptotic behaviour of the solution for large value of the independent variable is examined when the flow becomes steady. There is a smooth transition from the small-time solution to the large-time solution. The surface shear stress and the heat transfer increase with time and buoyancy parameter. The heat transfer increases with the Prandtl number, but the surface shear stress decreases.

On nonlinear transient free-surface flows over a bottom obstruction

A fully nonlinear integral-equation model based on the potential theory is used to study transient free-surface flow induced by a bump moving along a flat bottom. Comparison is made between the large-time solution of the transient model and the fully nonlinear steady-state solution in Zhang and Zhu [J. Eng. Math. 30, 487 (1996)] for both subcritical and supercritical flows, and asymptotic agreement between the two is observed. For the transcritical flow, some features associated with the resonant flow motion previously found by approximate theories are confirmed. A diagram of the parameter space is presented showing the regions of different solution categories including wave breaking. Through this diagram, a discrepancy between two previous steady-state models can now be clearly explained. Interestingly, the dividing line separating those solutions with or without wave breaking levels out asymptotically for very large Froude numbers, indicating that there is a limiting dimension of the bump, beyond which physically meaningful steady-state solutions do not exist, although they may well be obtained mathematically.

Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system

The initiation and propagation of reaction-diffusion travelling waves in two regions coupled together by the linear diffusive interchange of the autocatalytic species is considered via an initial-value problem in which amounts of the autocatalyst are introduced locally into otherwise uniform concentrations of the other species. The reaction in one region is given by quadratic autocatalysis, while the reaction in the other is given by quadratic autocatalysis together with the linear decay of the autocatalyst. A priori bounds for the initial-value problem are obtained first. These, together with the solution valid for small inputs of the autocatalyst, enable conditions to be derived under which travelling waves can be initiated giving a wave for all γ with k <2, or if k <(2γ−1)/(γ−1), where k and γ are dimensionless groups corresponding to the rate of chemical decay of the autocatalyst and to the strength of coupling respectively. The global asymptotic stability of the unreacted state is then discussed. A solution valid for strong coupling between the two regions is then derived. The equations governing the permanent-form travelling waves are treated in some detail, general properties of their solution and a solution valid for weak coupling being derived. Finally, the large-time solution of the initial-value problem is considered. This shows that, when travelling waves are initiated, they travel with their minimum possible speed: υ0=2[2−k−2γ+(k2+4γ2)1/2]1/2

Propagating reaction-diffusion waves in a simple isothermal quadratic autocatalytic chemical system

The propagating reaction-diffusion waves that develop in the isothermal autocatalytic system A + B → 2B from a local initial input of reactant B are considered. A solution valid for a small initial input of B is obtained first, and this is augmented by numerical solutions of the general problem. These show that, asymptotically, the reaction-diffusion wave propagates with the minimum, physically acceptable, wave speed. The large-time solution for the general case is then discussed and it is shown that ahead of the reaction-diffusion front is a weak diffusion-controlled region. It is the matching between these two regions that fixes the wave speed, so that the speed of propagation of the reaction-diffusion front is controlled by the rate at which B can spread forward by diffusion.

An asymptotic, large time solution of the convection Stefan problem with surface radiation

An asymptotic, large time solution has been obtained for the convection Stefan problem with surface radiation. The moving boundary problem has been reformulated as a fixed boundary problem where Lagrange-Burmann expansions are used to complete the variable transformation. An asymptotic solution of the problem is obtained by requiring that the asymptotic expansions assumed for the interface position X( t) and wall temperature u w ( t) for large times are consistent with the resulting interfacial Lagrange-Bürmann expansions. It is found that the asymptotic expansions admit Neumann's solution as the leading terms and that logarithmic terms start intervening at the third-order terms of the expansions for nonzero Stefan number.

Influence functions for the unsteady surface element method

The unsteady surface element methods provide analytical and numerical procedures for the efficient solution of certain transient heat conduction problems. The methods are based on Duhamel's theorem which requires influence functions. Because the influence functions implicitly include the effects of boundary conditions and geometries, many different ones are needed. Sometimes the most difficult influence functions to evaluate are those at the heated surface and at the smallest dimensionless times. This paper provides some early time influence functions for some basic one-dimensional geometries as well as for some twoand three-dimensional cases for various heated regions on the surface of a semi-infinite body. A given early time solution has the advantage of being valid for a variety of boundary conditions distant from the surface. Some large-time influence functions are also given. An example is given that illustrates that a combination of a short-time solution and a large-time solution can sometimes adequately cover the complete time domain.

An approximate solution for planar solidification with Newton cooling at the surface

The classical moving boundary problem for the planar freezing of a semi-infinite saturated liquid with Newton cooling at the wall is well known not to admit an exact solution. Existing perturbation solutions are valid only when the Stefan number is large. Further, since the implementation of the Newton cooling condition involves the step size, numerical solutions are only accurate if extremely small sizes are taken which involves large computing times. Here two new approximate analytic solutions are obtained, the first is an initial or starting solution while the second is valid for subsequent times. In the limit of large Stefan numberα the pseudo-steady state and first order corrected motions arise from both approximations. Further, in the limit of no Newton cooling at the wall the large time solution gives rise to precisely the well known Stefan or Neumann solution. The validity of the approximations is illustrated numerically with reference to previous work and known upper and lower bounds.

A large time solution of a crystal growth stefan problem

An asymptotic large time solution has been obtained for a Stefan problem of crystal growth. The analysis presented is valid at large times when the rate of crystal growth is limited by the diffusion process in the environmental melt system so that a thermodynamic equilibrium state prevails at the interface. The first three terms of the asymptotic expansions obtained show that the leading term reduces to a similarity solution of Ghez, a counterpart of Neumann's solution in the thermal Stefan problem and that logarithmic terms start intervening at the third power of the expressions.

Large time solution of an inhomogeneous nonlinear diffusion equation

The paper considers the large time solution of the nonlinear diffusion equation ρ ( x ) ∂ u / ∂ t = ∂ / ∂ x { C ( x ) u β ∂ u / ∂ x }, β &gt; 0, for initial data with compact support. The asymptotic behaviour of ρ ( x ) and C ( x ) as | x | → ∞ defines a parameter space, and the nature of the solution at large time depends crucially on the location therein. The paper examines the nature and uniformity of the asymptotic solutions for x ∈(-∞, ∞) in a restricted region of the entire parameter space, and in certain cases gives the leading error terms. A new integral invariant of the solution is also given.

Large-time solution of a nonlinear diffusion equation

In this paper we consider the way in which the solution to a class of initial - boundary-value problems for the plane nonlinear diffusion equation approaches the similarity form as t-&gt; oo. The problem we solve is chosen for two main reasons: firstly the equation above is of widespread use in modelling physical situations and secondly it provides a tractable but significant example of a free boundary problem.

The unsteady expansion of a gas into a non-uniform near vacuum

The paper generalizes an earlier problem of Grundy (1972) by considering the expansion of a (uniform) initially contained gas into a low-density non-uniform ambient atmosphere of density ρ0r−k, where k &gt; 0 and r is a non-dimensional radial co-ordinate. Regarding the flow as a perturbation of the perfect-vacuum expansion, we set up a boundary-value problem with boundary conditions on the contact front separating the two gases and on the strong shock which propagates into the ambient atmosphere. A large time solution to the problem can be developed by constructing an outer expansion valid near the contact front and an inner expansion valid near the shock. The matching process encounters two kinds of difficulty both of which imply that the large time solution is indeterminate from an asymptotic analysis alone.The asymptotic analysis does show however that the shock velocity tends to a constant only for restricted values of k. For the remaining values the shock has a k-dependent power-law behaviour. The paper examines the location of the transition and determines the asymptotic power-law dependence of the shock velocity.

Flow Near the Stagnation Point of a Body Which Undergoes a Sudden Change in a Steady Stream

Unsteady laminar boundary layers near the stagnation point of a body which undergoes a sudden change in a steady stream are analyzed by the method of successive approximations. It is shown that the second approximation which includes the effect of nonlinear convective terms of the equations of motion improves remarkably the first-order theory by the earlier investigators. Also, it seems that when the body is started with velocity increasing gradually with increasing time, the small-time solution obtained thus connects smoothly with the existing large-time solution.

A Study of Relaxing Waves

The equation ut+(1+u)ux−∫0t(1+u)uxf(t−τ)dτ=0 is studied when an initial finite pulse u(x, 0) is given. For the linearized equation, general solutions in terms of Laplace transforms are obtained and more explicit expression for exponential kernels is given. An iteration expansion scheme is established for general kernels. For positive kernels, it is found that the stability condition for the solution is ∫0∞f(t)dt&amp;lt;1. Then the large time solution as well as the solution representing the main disturbance is obtained. For the nonlinear equation, the condition for the shock formation is obtained for the special case f(t) = μe−μt, or when the nonlinearity is weak.

On the formation of weak plane shock waves by impulsive motion of a piston

The piston problem for a viscous heat-conducting gas is studied under the assumption that the piston Mach number ε is small. The linearized Navier–Stokes equations are found to be valid up to times of the order of ε−2mean free times after the piston is set in motion, while at large times the solution is governed by Burgers's equation. Boundary conditions for the large-time solution are supplied by the matching principle of the method of inner and outer expansions, which is also used to construct a composite solution valid both for small and for large times.

Unsteady Stagnation Point Heat Transfer

An analysis is presented for the unsteady laminar, forced-convection heat transfer at a two-dimensional and axisymmetrical front stagnation due to an arbitrarily prescribed wall temperature or heat flux variation. The flow is incompressible and steady. The procedure begins with a consideration of the thermal boundary-layer response caused by either a step change in surface temperature or heat flux. Two appropriate asymptotic solutions, valid for small and large times, respectively, are found and satisfactorily joined for Prandtl numbers ranging from 0.01 to 100. The key to the small time solution is the transformation of the energy equation in the Laplace transform plane to an ordinary differential equation with a large parameter. An essential feature of the large time solution is the use of Meksyn’s transformation variable and the method of steepest descent in the evaluation of integrals. It is found that, for both two-dimensional and axisymmetrical stagnation, the time required for the thermal boundary layer to attain steady condition, for either a step change in surface temperature or heat flux, varies inversely with the free stream velocity and directly with 1/4 power of the Prandtl number of the fluid.

Grain Boundary and Diffusion Metasomatism

Grain boundary and surface diffusion are considered as possible means for extending the range of solid diffusion in metasomatic processes. A mathematical model of a granular solid is proposed, and an asymptotic (large-time) solution for diffusion in this system is obtained, giving the distribution of solute as a function of distance and time. It is assumed that both lattice and grain boundary diffusion are operative, and that unit concentration is maintained at the free surface of the solid for all time. The solution strongly suggests that surface and grain boundary diffusion cannot greatly extend the range of solid diffusion over that obtained by strictly lattice mobility.