Showing papers on "Similarity solution published in 1992"

Magnetohydrodynamic flow of a power-law fluid over a stretching sheet

Abstract: Magnetohydrodynamic flow of an electrically conducting power-law fluid over a stretching sheet in the presence of a uniform transverse magnetic field is investigated by using an exact similarity transformation. The effect of magnetic field on the now characteristics is explored numerically, and it is concluded that the magnetic field tends to make the boundary layer thinner, thereby increasing the wall friction.

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.

A similarity solution for two‐phase water, air, and heat flow near a linear heat source in a porous medium

Abstract: Placement of a heat source in a partially saturated geologic medium causes strongly coupled thermal and hydrologic behavior. To study this problem, a recently developed semianalytical solution for two-phase flow of water and heat in a porous medium has been extended to include an air component and to incorporate several physical effects that broaden its range of applicability. The problem considered is the placement of a constant-strength linear heat source in an infinite homogeneous medium with uniform initial conditions. Under these conditions the governing partial differential equations in radial distance r and time t reduce to ordinary differential equations through the introduction of a similarity variable η = r/t1/2. The resulting equations are coupled and nonlinear, necessitating a numerical integration. The similarity solution developed here is used to investigate various physical phenomena related to partially saturated flow in low-permeability rock, such as vapor pressure lowering, pore level phase change effects, and an effective continuum representation of fractured/porous media. Application to several illustrative problems arising in the context of high-level nuclear waste disposal at Yucca Mountain, Nevada, indicates that fluid flow, phase changes, and latent heat transfer may have a significant impact on conditions at the repository.

Some similarity temperature profiles for the microwave heating of a half-space

Abstract: There is presently considerable interest in the utilisation of microwave heating in areas such as cooking, sterilising, melting, smelting, sintering and drying. In general, such problems involve Maxwell's equations coupled with the heat equation, for which all thermal, electrical and magnetic properties of the material are nonlinear. The heat source arising from microwaves is proportional to the square of the modulus of the electric field intensity, and is known to increase with increasing temperature. In an attempt to find a simple model of microwave heating, we examine here simple transient temperature profiles corresponding to a heat source with spatial exponential decay but increasing with temperature, for which we assume either a power-law dependence or an exponential dependence. The spatial exponential decay is known to apply exactly when the electrical and magnetic properties of the material are assumed constant. A number of transient temperature profiles for this model are examined which arise from the invariance of the governing heat equation under simple one-parameter transformation groups. Some closed analytical expressions are obtained, but in general the resulting ordinary differential equations need to be solved numerically, and extensive numerical results are presented. For both models, these results indicate the appearance of moving fronts.

Nonclassical symmetry reductions for the dispersive wave equations in shallow water

Abstract: All (six types of) the similarity reductions of the dispersive wave equations in shallow water are obtained by the direct method. The group explanation of the results is also given. It is pointed out that, in order to find all the results by nonclassical Lie approach, two additional condition equations should be satisfied at the same time together with two original equations. The logarithmic branch points for time t can be included in two types of the similarity solutions, though the model can be cast into the canonical Hamiltonian form.

Similarity solutions for viscous vortex cores

Abstract: Results are presented for a class of self-similar solutions of the steady, axisymmetric Navier–Stokes equations, representing the flows in slender (quasi-cylindrical) vortices. Effects of vortex strength, axial gradients and compressibility are studied. The presence of viscosity is shown to couple the parameters describing the core growth rate and the external flow field, and numerical solutions show that the presence of an axial pressure gradient has a strong effect on the axial flow in the core. For the viscous compressible vortex, near-zero densities and pressures and low temperatures are seen on the vortex axis as the strength of the vortex increases. Compressibility is also shown to have a significant influence upon the distribution of vorticity in the vortex core.

The boundary layers due to shear flow over a still fluid

Abstract: A constant shear flow passes on top of a heavier, still fluid. Similarity solutions are found for the boundary layers on both sides of the interface.

Properties of strongly nonlinear vortex/Tollmien–Schlichting-wave interactions

Abstract: An analytical and computational study is presented on solution properties of strongly nonlinear vortex/wave interactions involving Tollmien/Schlichting waves, in boundary-layer transition. The longitudinal vortex part, i.e. the total mean flow, is governed by a three-dimensional vortex system but coupled, through an effective spanwise slip condition at the surface, with the accompanying wave part, so that both the vortex and the wave parts are unknowns. Terminal forms of the space-marching or time-marching problem are proposed first, yielding either a lift-off separation singularity or a strong-attachment singularity. Second, a similarity version of the complete system is addressed numerically and analytically. This leads to a number of interesting solution features as the typical wave pressure is increased into the strongly nonlinear regime. In particular, lift-off separation and attachment forms seem to emerge which are analogous with those proposed above. The flow developments beyond the terminal forms are discussed, together with the links of the work with recent computational results and, tentatively, with experimental observations including the creation of lambda vortices (as a form of lift-off separation).

Unsteady flow between two disks with heat transfer in the presence of a magnetic field

Abstract: The motion of an electrically conducting fluid film squeezed between two parallel disks with heat transfer in the presence of a magnetic field applied perpendicular to the disks is studied. Attention is given to the case where a similarity solution can be obtained. Approximate analytic solutions are given and a numerical solution to the resulting nonlinear ordinary differential equations for a wide range of values of the three independent parameters which influence the motion is presented. Special attention is given to the effects of squeezing and magnetic forces on heat transfer. The results establish that while squeezing has considerable effects on heat transfer, the effects of magnetic forces are negligible.

Local transformations between some nonlinear diffusion equations

Abstract: Abstract We derive local transformations mapping radially symmetric nonlinear diffusion equations with power law or exponential diffusivities into themselves or into other equations of a similar form. Both discrete and continuous transformations are considered. For the cases in which a continuous transformation exists, many additional forms of group-invariant solution may be constructed; some of these solutions may be written in closed form. Related invariance properties of some multidimensional diffusion equations are also exploited.

The three-dimensional turbulent boundary layer near a plane of symmetry

Abstract: The asymptotic structure of the three-dimensional turbulent boundary layer near a plane of symmetry is considered in the limit of large Reynolds number A self-consistent two-layer structure is shown to exist wherein the streamwise velocity is brought to rest through an outer defect layer and an inner wall layer in a manner similar to that in two-dimensional boundary layers The cross-stream velocity distribution is more complex and two terms in the asymptotic expansion are required to yield a complete profile which is shown to exhibit a logarithmic region The flow in the inner wall layer is demonstrated to be collateral to leading order; pressure-gradient effects are formally of higher order but can cause the velocity profile to skew substantially near the wall at the large but finite Reynolds numbers encountered in practice The governing set of ordinary differential equations describing a self-similar flow is derived The calculated numerical solutions of these equations are matched asymptotically to an inner wall-layer solution and the results show trends that are consistent with experimental observations

A nonlinear model of heat conduction

Abstract: A nonlinear model of heat propagation is presented, from which a new heat conduction equation is derived. An exact similarity solution in closed form of this equation is obtained, which reveals the travelling wave characteristics for the transient temperature distribution. It is shown that the temperature disturbances propagate with finite velocity, which is a monotonically decreasing function of time.

Sink flows of a suspension of rigid rods : the failure of a similarity solution

Abstract: The flow of a suspension of rigid rods into a sink of finite size which may be either planar or axisymmetric is considered. For low rod concentrations the flow at large distances from the sink is well described by similarity solutions derived by Evans. For larger concentrations, however, these similarity solutions predict unrealistic flows with many regions of inflow and outflow. We show that the flow field far upstream of the sink is then dominated by an eigensolution involving zero net volume flux. This eigensolution is determined. The prediction is compared with the results of a full numerical solution for the flow of a suspension of rods from an infinite reservoir into a finite sink. It is also used to estimate the extent of the vortex enhancement generated by flow through a large but finite contraction. The results are compared with the numerical solutions and experiments of Lipscomb et al. This example shows that the parameter range for which a self-similar flow field is appropriate can be substantially smaller for a non-Newtonian fluid than is the corresponding range in the Newtonian case.

Unified treatment of free convection adjacent to a vertical plate with three thermal boundary conditions

Abstract: Free convection adjacent to a vertical plate is considered with three different boundary conditions, namely, the plate is subjected to a prescribed temperature, a prescribed heat flux or a prescribed heat transfer coefficient. By a unified treatment of similarity analysis, the governing equations of free convection are reduced to identical system of ordinary differential equations for all the cases. It is shown that these equations are invariant under a certain transformation group and solution for one case can be used to obtain the solutions for the other two cases by a simple method. The critical cases are found for which the solutions for all the three cases are identical.

Stability of a non-orthogonal stagnation flow to three dimensional disturbances

Abstract: A similarity solution for a low Mach number nonorthogonal flow impinging on a hot or cold plate is presented. For the constant density case, it is known that the stagnation point shifts in the direction of the incoming flow and that this shift increases as the angle of attack decreases. When the effects of density variations are included, a critical plate temperature exists; above this temperature the stagnation point shifts away from the incoming stream as the angle is decreased. This flow field is believed to have application to the reattachment zone of certain separated flows or to a lifting body at a high angle of attack. Finally, the stability of this nonorthogonal flow to self similar, 3-D disturbances is examined. Stability properties of the flow are given as a function of the parameters of this study; ratio of the plate temperature to that of the outer potential flow and angle of attack. In particular, it is shown that the angle of attack can be scaled out by a suitable definition of an equivalent wavenumber and temporal growth rate, and the stability problem for the nonorthogonal case is identical to the stability problem for the orthogonal case.

The uniqueness and stability of the solution of the Riemann problem of a system of conservation laws of mixed type

Abstract: We establish the uniqueness and stability of the similarity solution of the Riemann problem for a 2 × 2 system of conservation laws of mixed type, with initial data separated by the elliptic region, which satisfies the viscosity- capillarity travelling wave admissibility criterion

The boundary-layer flow past a suddenly heated vertical surface in a saturated porous medium

Abstract: The boundary-layer flow generated on an impermeable vertical surface in a saturated porous medium is considered in the case when wall heating at a rate proportional toxλ is switched on at timet=0, (x measures distance along the wall and λ is a constant). The similarity equations which hold in the limit of larget are discussed and are shown to have a solution only for λ>−1. The behaviour of the solution as λ → −1 and as λ → ∞ is obtained. Numerical solutions of the initial value problem are then obtained for a range of values of λ. A direct numerical integration is possible for λ≥1, while an iterative procedure is required for λ<1, with the numerical scheme becoming unstable for λ=−0.5.

Similarity solutions for axisymmetric plane radial power law fluid flows through a porous medium

Abstract: This paper investigates the unsteady flow of non-Newtonian fluids of power low behavior through a porous medium in a plane radial geometry. The equation governing the flow is a nonlinear parabolic partial differential equation with a source term whose solution satisfies certain fixed and moving boundary conditions. The attention is focused on the finding of similarity solution when the fixed boundary condition and the source term satisfy certain restrictions. In this case similarity transformations are determined and the resulting ordinary differential equations are deduced. For shear thinning fluids the existence of a pressure disturbance front moving with finite velocity is shown and expression for its location as a function of time is determined. The solutions in closed form have been given for certain particular cases where the resulting differential equations can be analytically solved. A numerical procedure has also been presented.

A comparison between experimental measurements and numerical calculations of the structure of premixed rotating counterflow methane-air flames

Abstract: In this paper we investigate computationally the structure and extinction of rotating counterflow premixed methane-air flames. By seeking a similarity solution of the governing two-dimensional conservation equations, we reduce the problem to the solution of a two-point boundary value problem along the axial spatial coordinate. We apply arclength continuation methods to investigate the equivalence ratio at extinction versus the rotation rate for four flames with ejection velocities equivalent to the experimental conditions considered by Chen et al. The results of our study indicate that by counterrotating the gases from the burners the lean extinction limit can be lowered. This is due to the lowering of the value of the local strain rate in the neighborhood of the stagnation plane.

Numerical simulation of cylindrical converging shocks in solids

Abstract: Cylindrically converging shock waves in solids have been analyzed by the random choice method (RCM). A Riemann solver for fluidlike solids with the Gruneisen‐type equation of state is constructed and incorporated into the RCM. It is then applied to the cylindrical shock tube problems for solid copper with driving pressures of 20 and 200 GPa. The numerical results are compared with those of the finite difference method (FDM). The shock speed is smaller and the discontinuity at the shock front is smeared out due to the artificial viscosity in the FDM calculation. Spatial distributions of pressure, density, and particle velocity calculated by the RCM show that the steepness at the shock front is maintained both in the converging and reflecting stages. It is shown that the pressure on the shock front and the total energy contained in the central circular area are much larger in the reflecting stage than in the focusing stage. The dimensional analysis has shown that the similarity solution exists; however the numerical result shows that the flow does not fall within the similarity regime in the region of calculation. It suggests that the self‐similar flow is only limited in extreme proximity to the axis.

Similarity solutions for multi-component flows

Abstract: A one-dimensional shock-reflection test problem in the case of slab, cylindrical or spherical symmetry is discussed for multi-component flows. The differential equations for a similarity solution are derived and then solved numerically in conjunction with the Rankine-Hugoniot shock relations.

Painlevé test and exact similarity solutions of a class of nonlinear diffusion equations

Abstract: For (1+1)‐dimensional diffusion equations where the diffusion coefficient is an arbitrary power of the dependent variable, a Painleve test is performed on both the partial differential equation and the similarity reductions. The invariance of the Painleve property under a large class of similarity reductions is shown. For the reductions with the Painleve property, the explicit similarity solutions are given. It is demonstrated how results of a Painleve analysis can be used to obtain further exact similarity solutions, even in cases without the Painleve property.

Linear stability of supersonic cone boundary layers

Abstract: The effect of the variable surface geometry of a cone on the linear stability of a supersonic boundary layer flowing over it is investigated subject to different quasiparallel flow approximations. It is shown that, if a suitable set of disturbance state variables is chosen for the normal mode analysis, these effects can accurately be accounted for. In fact, a planar coordinate system can be used for the stability analysis of the cone boundarylayer profiles and a simple "correction" can subsequently be applied to obtain an accurate approximation to the spatial growth rates.