Showing papers on "Similarity solution published in 1995"

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.

The effect of temperature-dependent viscosity on free-forced convective laminar boundary layer flow past a vertical isothermal flat plate

Abstract: The effect of a temperature-dependent viscosity on an incompressible fluid in steady laminar free-forced convective boundary layer flow over an isothermal vertical semi-infinite flat plate is studied. The local similarity solution is used to transform the system of partial differential equations, describing the problem under consideration, into a boundary value problem of coupled ordinary differential equations and an efficient numerical technique is implemented to solve the reduced system. Numerical calculations are carried out, for various values of the dimensionless parameters of the problem, which include a Prandtl number, a mixed convection parameter and a viscosity/temperature parameter. The results are presented graphically and the conclusion is drawn that the flow field and other quantities of physical interest are significantly influenced by these parameters. In particular, it is concluded that when the viscosity of a working fluid is sensitive to the variation of temperature, care must be taken to include this effect, otherwise considerable error can result in the heat transfer processes. In the literature, such care is not always evident.

Analytical description of the breakup of liquid jets

Abstract: A viscous or inviscid cylindrical jet with surface tension in a surrounding medium of negligible density tends to pinch owing to the mechanism of capillary instability. We construct similarity solutions which describe this phenomenon as a critical time is encountered, for three distinct cases : (i) inviscid jets governed by the Euler equations, (ii) highly viscous jets governed by the Stokes equations, and (iii) viscous jets governed by the Navier-Stokes equations. We look for singular solutions of the governing equations directly rather than by analysis of simplified models arising from slender-jet theories. For Stokes jets implicitly defined closed-form solutions are constructed which allow the scaling exponents to be fixed. Navier-Stokes pinching solutions follow rationally from the Stokes ones by bringing unsteady and nonlinear terms into the momentum equations to leading order. This balance fixes a set of universal scaling functions for the phenomenon. Finally we show how the pinching solutions can be used to provide an analytical description of the dynamics beyond breakup.

The Axisymmetric Motion of a Liquid Film on an Unsteady Stretching Surface

Abstract: The axisymmetric motion of a fluid caused by an unsteady stretching surface that has relevance in extrusion process and bioengineering has been investigated. It has been shown that if the unsteady stretching velocity is prescribed by rb/(1 − αt), then the problem admits a similarity solution which gives much insight to the character of solutions. The asymptotic and numerical solutions are obtained and they could be used in the testing of computer codes or analytical models of more realistic engineering systems. The results are governed by a nondimensional unsteady parameter S and it has been observed that no similarity solutions exist for S > 4

Similarity solutions in free convection boundary-layer flows adjacent to vertical permeable surfaces in porous media: II prescribed surface heat flux

Abstract: The equation which governs the similarity solution for free convection boundary-layer flow along a vertical permeable surface with prescribed surface heating and mass transfer rate is discussed. The solution is seen to depend on two non-dimensional parameters;m, the power-law exponent, and γ, the mass transfer parameter. It is shown that solutions exist for allm>−1 for γ>0 (fluid injection) whereas for γ m 0(γ), wherem 0 is determined as a function of γ. Solutions for large mass transfer rates are obtained, for both γ>0 and γ 0 the form of the asymptotic solution for γ large is seen to depend on the value ofm. Solutions form large are derived, these are seen to be different depending on whether γ is positive or negative.

Asymptotic analysis of laminar boundary-layer flow over finely grooved surfaces

Abstract: A set of matched asymptotic expansions is developed for laminar flow over grooved surfaces, which gives a formal mathematical justification and provides higher-order corrections to the separate study of the viscous sublayer pervading the grooves and of the essentially two-dimensional boundary layer far above. The question of whether, under various conditions, laminar drag is reduced or increased by grooves is answered with the aid of theoretical bounds concerning the protrusion height. The three-dimensional effect of the presence of grooves is reduced to an equivalent wall condition for the two-dimensional boundary layer equations

An unusual moving boundary condition arising in anomalous diffusion problems

Abstract: In the context of analyzing a new model for nonlinear diffusion in polymers, an unusual condition appears at the moving interface between the glassy and rubbery phases of the polymer. This condition, which arises from the inclusion of a viscoelastic memory term in our equations, has received very little attention in the mathematical literature. Due to the unusual form of the moving-boundary condition, further study is needed as to the existence and uniqueness of solutions satisfying such a condition. The moving boundary condition which results is not solvable by similarity solutions, but can be solved by integral equation techniques. A solution process is outlined to illustrate the unusual nature of the condition; the profiles which result are characteristic of a dissolving polymer.

Nonsteady Stagnation-Point Flows Over Permeable Surfaces: Explicit Solutions of the Nevier-Stokes Equations

Abstract: Some new explicit solutions of the unsteady two-dimensional Navier-Stokes equations describing nonsteady stagnation-point flows with surface suction or injection are presented. The solutions have been obtained using a new approach for finding explicit similarity solutions of partial differential equations. As distinct from the common Birkhoff's similarity transformation, which permits only one form of an unsteady potential flow field and only one form of time dependence for suction (or blowing) velocity, the transforms obtained permit consideration of a variety of special solutions differing in the forms of time dependence

Constant front speed in weakly diffusive non-Fickian systems

Abstract: In certain polymer-penetrant systems, the effects of Fickian diffusion are dominated by nonlinear viscoelastic behavior. Consequently, such systems often exhibit concentration fronts unlike those seen in classical Fickian systems. These fronts not only are sharper than in standard systems but also propagate at constant speed. The mathematical model presented is a moving boundary-value problem, where the boundary separates the polymer into two distinct states, glassy and rubbery, where different physical processes dominate. The moving boundary condition that results is not solvable by similarity solutions but can be solved by integral equation techniques. In the case under consideration, namely, one where the standard Fickian diffusion coefficient is small, asymptotic solutions where a comparatively sharp front moves with constant speed are obtained.

Development of the Fokker‐Planck Equation and its solutions for modeling transport of conservative and reactive solutes in physically heterogeneous media

Abstract: When a scale-dependent fractal dispersivity is introduced in the convective-dispersive equation of transport in subsurface flow, the Fokker-Planck equation (FPE), also known as the second diffusion equation, is derived. Similarity solutions of the one-dimensional FPE, subject to a Dirac delta function input, are also presented. The similarity solution is shown to function as a kernel in the convolution integral to yield an output on a real timescale. The input function is derived by a procedure known as the inverse problem with the aid of Laplace transform. The FPE and its solutions contain a fractal dimension D and cutoff limit ϵ, two shape parameters β and γ in the FPE, and other parameters that appear in Darcy's law. A two-dimensional FPE and its solutions have been presented elsewhere. It is reasoned that the scale-dependent dispersion in unsaturated flow is also an issue for investigation.

Directional solidification into static stability

Abstract: Consider the directional solidification of a binary alloy rejecting a heavy solute as it solidifies upward. If the solidification front is planar, the fluid melt ahead of the front is stably stratified and convection is not expected. In this paper we analyse the linear stability of planar solidification asymptotically in the limit of large solutal Rayleigh number, R. Three distinct linear modes are found which correspond to internal waves, buoyancy edge waves, or morphological modes. Of these three modes, only the morphological modes are subject to an instability. We find that for large Rayleigh number this instability first occurs at long wavelengths with wavenumbers that scale on R -1/14 . The scalings derived from the linear analysis are used to construct a nonlinear theory for the morphological instability in the large Rayleigh number limit. Similarity solutions are found which describe steadily convecting, non-planar growth reminiscent of an observed phenomenon known as steepling.

Boundary layer transition studies

Abstract: A small-scale wind tunnel previously used for turbulent boundary layer experiments was modified for two sets of boundary layer transition studies. The first study concerns a laminar separation/turbulent reattachment. The pressure gradient and unit Reynolds number are the same as the fully turbulent flow of Spalart and Watmuff. Without the trip wire, a laminar layer asymptotes to a Falkner & Skan similarity solution in the FPG. Application of the APG causes the layer to separate and a highly turbulent and approximately 2D mean flow reattachment occurs downstream. In an effort to gain some physical insight into the flow processes a small impulsive disturbance was introduced at the C(sub p) minimum. The facility is totally automated and phase-averaged data are measured on a point-by-point basis using unprecedently large grids. The evolution of the disturbance has been tracked all the way into the reattachment region and beyond into the fully turbulent boundary layer. At first, the amplitude decays exponentially with streamwise distance in the APG region, where the layer remains attached, i.e. the layer is viscously stable. After separation, the rate of decay slows, and a point of minimum amplitude is reached where the contours of the wave packet exhibit dispersive characteristics. From this point, exponential growth of the amplitude of the disturbance is observed in the detached shear layer, i.e. the dominant instability mechanism is inviscid. A group of large-scale 3D vortex loops emerges in the vicinity of the reattachment. Remarkably, the second loop retains its identify far downstream in the turbulent boundary layer. The results provide a level of detail usually associated with CFD. Substantial modifications were made to the facility for the second study concerning disturbances generated by Suction Holes for laminar flow Control (LFC). The test section incorporates suction through interchangeable porous test surfaces. Detailed studies have been made using isolated holes in the impervious test plate that used to establish the Blasius base flow. The suction is perturbed harmonically and data are averaged on the basis of the phase of the disturbance, for conditions corresponding to strong suction and without suction. The technique was enhanced by using up to nine multiple probes to reduce the experimental run-time. In both cases, 3D contour surfaces in the vicinity of the hole show highly 3D TS waves which fan out in the spanwise direction forming bow-shaped waves downstream. The case without suction has proved useful for evaluating calculation methods. With suction, the perturbations on the centerline are much stronger and decay less rapidly, while the TS waves in the far field are similar to the case without suction. Downstream, the contour surfaces of the TS waves develop spanwise irregularities which eventually form into clumps. The spanwise clumping is evidence of a secondary instability that could be associated with suction vortices. Designers of porous surfaces use Goldsmith's Criterion to minimize cross-stream interactions. It is shown that partial TS wave cancellation is possible, depending on the hole spacing, disturbance frequency and free-stream velocity. New high-performance Constant Temperature Hot-Wire Anemometers were designed and built, based on a linear system theory analysis that can be extended to arbitrary order. The motivation was to achieve the highest possible frequency reponse while ensuring overall system stability. The performance is equal to or superior to commercially available instruments at about 10% of the cost. Details, such as fabrication drawings and a parts list, have been published to enable the instrument to be construced by others.

Inviscid separated flow over a non-slender delta wing

Abstract: We consider inviscid incompressible flow about an infinite non-slender flat delta wing with leading-edge separation modeled by symmetrical conical vortex sheets. A similarity solution for the three dimensional steady velocity potential Φ is sought with boundary conditions to be satisfied on the line which is the intersection of the wing sheet surface with the surface of the unit sphere. A numerical approach is developed based on the construction of a special boundary element or ‘winglet’ which is effectively a Green function for the projection of ∇2Φ = 0 onto the spherical surface under the similarity ansatz. When the wing semi-apex angle γo is fixed satisfaction of the boundary conditions of zero normal velocity on the wing and zero normal velocity and pressure continuity across the vortex sheet then leads to a nonlinear eigenvalue problem. A method of ensuring a condition of zero lateral force on a lumped model of the inner part of the rolled-up vortex sheet gives a closed set of a equations which is solved numerically by Newton's method. We present and discuss the properties of solutions for γ0 in the range 1.30 < γ <89.50. The dependencies of these solutions on γ0 differs qualitatively from predictions of slender-body theory. In particular the velocity field is in general not conical and the similarity exponent must be calculated as part of the global eigenvalue problem. This exponent, together with the detailed flow field including the position and structure of the separated vortx sheet, depend only on γ0. In the limit of small γ0, a comparison with slender-body theory is made on the basis of an effective angle of incidence.

Classification of the Extended Symmetries of the Inhomogeneous Nonlinear Diffusion Equation

Abstract: Using Lie group methods, we analyse nonlinear diffusion equations in an inhomogeneous medium f ( x ) u t =( g ( x ) D ( u ) u x ) x with arbitrary diffusion coefficient D ( u ), and arbitrary thermal coefficients f ( x ) and g ( x ), which have a wide spectrum of applications in many areas of science. The Lie-group-based similarity method leads to a classification of the diffusion and thermal coefficients according to its symmetry properties. With the help of the adjoint representation, the optimal system of similarity reductions is calculated. Exact similarity solutions of the second-order ordinary differential equatiors resulting from the reductions are demonstrated by examples.

On penetrative convection at low Peclet number

Abstract: A new theoretical model is developed for the growth of a convecting fluid layer at the base of a stable, thermally stratified layer when heated from below. The imposed convective heat flux is taken to be comparable to the heat flux conducted down the background gradient so that diffusion ahead of the interface between the convecting and stable layers makes a significant contribution to the interfacial heat flux and to the rate of rise of the interface. Closure of the diffusion problem in the stable region requires the interfacial heat flux to be specified, and it is argued that this is determined by the ability of convective eddies to mix warmed fluid below the interface downwards. The interfacial velocity, which may be positive or negative, is then determined by the joint requirements of continuity of heat flux and temperature. A similarity solution is derived for the case of an initially linear temperature gradient and uniform heating. Solutions are also given for a heat flux that undergoes a step change and for a heat flux determined from a four-thirds power law with a fixed base temperature. Numerical calculations show that the predictions of the model are in good agreement with previously reported experimental measurements. Similar calculations are applicable to a wide range of geophysical problems in which the tendency for diffusive restratification is comparable to that for mixed-layer deepening by entrainment.

Natural convection beneath a downward facing heated plate in a porous medium

Abstract: The boundary layer singularity appearing at the edge of a downward facing heated plate buried in a fluid-saturated porous medium is analyzed, and a boundary condition at the edge is deduced. Both constant temperature and constant heat flux conditions on the downward facing surface of an infinite strip and a circular disk are considered. The boundary layer equations are shown to possess a similarity solution for the constant temperature boundary condition and results for both the infinite strip and circular plate geometries are obtained. In the case of constant heat flux no similarity solution exists and this problem is solved by numerical integration of the governing partial differential equation. Solutions are also given for a slightly inclined plate maintained at constant temperature

Problems in nonlinear diffusion

Abstract: A variety of effects can occur from different forms of nonlinear diffusion or from coupling of diffusion to other physical processes. I consider two such classes of problems; first, the analysis of behavior of diffusive solutions of the generalized porous media equation, and second, the study of stress-driven diffusion in solids. The porous media equation is a nonlinear diffusion equation that has applications to numerous physical problems. By combining classical techniques for the study of similarity solutions with perturbation methods, I have examined some new initial-boundary value problems for the porous media equation, including "stopping" and "merging" problems. Using matched asymptotic expansions and boundary layer analysis, I have shown that the initial deviations from similarity solution form in these problems are asymptotic beyond all orders. Applications of these studies to the Cahn-Hilliard and Fisher's equations are also considered. In my examination of stress-driven diffusion, I consider models for the behavior of systems in the emerging technological field of viscoelastic diffusion in polymer materials. Using asymptotic analysis, I studied some of the non-traditional effects, shock formation in particular, that occur in initial-boundary value problems for these models. Phase-interface traveling waves for "Case II" diffusive transport were also studied, using phase plane techniques.

Forced convection and sedimentation past a flat plate

Abstract: The steady laminar flow of a well-mixed suspension of monodisperse solid spheres, convected steadily past a horizontal flat plate and sedimenting under the action of gravity, is examined. It is shown that, in the limit as Re approaches infinity and epsilon approaches 0, where Re is the bulk Reynolds number and epsilon is the ratio of the particle radius a to the characteristic length scale L, the analysis for determining the particle concentration profile has several aspects in common with that of obtaining the temperature profile in forced-convection heat transfer from a wall to a fluid stream moving at high Reynolds and Prandtl numbers. Specifically, it is found that the particle concentration remains uniform throughout the O(Re(exp -1/2)) thick Blasius boundary layer except for two O(epsilon(exp 2/3)) thin regions on either side of the plate, where the concentration profile becomes non-uniform owing to the presence of shear-induced particle diffusion which balances the particle flux due to convection and sedimentation. The system of equations within this concentration boundary layer admits a similarity solution near the leading edge of the plate, according to which the particle concentration along the top surface of the plate increases from its value in the free stream by an amount proportional to X(exp 5/6), with X measuring the distance along the plate, and decreases in a similar fashion along the underside. But, unlike the case of gravity settling on an inclined plate in the absence of a bulk flow at infinity considered earlier, here the concentration profile remains continuous everywhere. For values of X beyond the region near the leading edge, the particle concentration profile is obtained through the numerical solution of the relevant equations. It is found that, as predicted from the similarity solution, there exists a value of X at which the particle concentration along the top side of the plate attains its maximum value phi(sub m) and that, beyond this point, a stagnant sediment layer will form that grows steadily in time. This critical value of X is computed as a function of phi(sub s), the particle volume fraction in the free stream. In contrast, but again in conformity with the similarity solution, for values of X sufficiently far removed from the leading edge along the underside of the plate, a particle-free region is predicted to form adjacent to the plate. This model, with minor modifications, can be used to describe particle migration in other shear flows, as, for example, in the case of crossflow microfiltration.

Pressure diffusion in unsteady non-Darcian flows through porous media

Abstract: The pressure diffusion in unsteady non-Darcian flows through a porous medium is studied analytically. The exact similarity solutions for the pressure and velocity distributions for the Cauchy problem, (i.e. the case of source type solution) are presented and graphically illustrated. Two classes on non-Darcian flows are investigated. The class of unsteady turbulent gas flows with polytropic thermodynamic evolution, and the class of non-Newtonian fluid flows of power law behavior. The derived governing pressure-diffusion equations belong to a class of nonlinear degenerate parabolic equations having solutions with compact support. It is shown that due to the nonlinear effects associated with non-Darcian flows, the pressure and velocity distributions exhibit traveling wave characteristics. The conditions for the existence of these diffusive waves are found, and expressed in terms of the properties of the fluid and the porous medium. (authors). 11 refs., 5 figs.

Non-steady MHD flow for a non-Newtonian fluid with variable conductivity

Abstract: This paper deals with a similarity solution for the unsteady flow of a conducting non-Newtonian power-law in-compressible fluid, when a porous plate is moving uniformly in the presence of a transverse magnetic field, assuming that the electrical conductivity is a function of the velocity. The aim of this analysis is to determine the velocity and the effect of variation of the electrical conductivity on the solution. The basic equations have been solved by applying the perturbation method for small and large values of the magnetic interaction parameterN. The main features of the exact solution is that it represents shear flow.

New similarity solutions for hypersonic boundary layers with applications to inlet flows

Abstract: A re-examination of self-similar solution possibilities for steady laminar boundary layers is given for nonadiabatic strongly hypersonic flows of an ideal gas. Two apparently new solutions are found, pertaining to adverse pressure gradients, both of which involve no separation with distance into the pressure rise. The first involves an exponential pressure rise on a cooled similarly shaped body, whereas the second involves a modified ramp type pressure rise. The skin friction, heat transfer, displacement thickness, and weak viscous interaction-induced pressure change properties of these two solution families are analyzed in detail, including their application to high-speed inlet type of flows.

Different similarity regimes for the turbulent burst equation

Abstract: On etudie l'existence de differents regimes auto-similaires pour une equation de reaction-diffusion intervenant dans la description des explosions turbulentes dans un fluide. On s'interesse specialement a la variation du regime auto-similaire avec le parametre fondamental

Viscous-Gravity Spreading of Liquid Drops on Solids

Abstract: This study is concerned with viscous-gravity spreading of liquid drops on a solid surface. The effect of the initial shape on spreading is investigated. Numerical results are in good agreement with the similarity solution for both cases of axisymmetric and cylindrical drops for sufficiently long times. In addition to the results obtained for radius as a function of time, changes in the potential energy and shapes are also studied. Shapes are shown to undergo two or three phases depending on the initial shape. On the other hand, losses of potential energy are shown to be converted into viscous dissipation.

Buoyant Plume through a Permeable Porous Layer Located above a Line Heat Source in an Infinite Fluid Space

Abstract: An experimental and analytical study is conducted on the effects of a horizontal porous layer on the development of the buoyant plume arising from a line heat source in an infinite fluid space. Visual observations and the experimental temperature distributions show the expansion and contraction of the plume at the lower and upper interfaces of the permeable layer. Similarity solutions assuming their proper virtual origins can approximately predict the changes of the plume width. Detailed numerical examination indicates that the lateral flow induced near the interfaces causes deviation from the similarity solutions. The vicinity of the interface is understood to be a transition region between the similarity solutions. The numerical analysis adopts the Beavers-Joseph slip boundary condition with the slip coefficient α estimated by α=1/√(e), where e is the porosity of the porous layer. The fairly good agreement between the experimental and numerical results confirms the validity of the slip coefficient.