Showing papers on "Similarity solution published in 1988"

Fluid flow due to a stretching cylinder

Abstract: The fluid flow outside of a stretching cylinder is studied. The problem is governed by a third‐order nonlinear ordinary differential equation that leads to exact similarity solutions of the Navier–Stokes equations. Because of algebraic decay, an exponential transform is used to facilitate numerical integration. Asymptotic solutions for large Reynolds numbers compare well with numerical results. The heat transfer is determined.

Three‐Dimensional Flow Effects in Silicon CVD in Horizontal Reactors

Abstract: Numerical modeling of Si homoepitaxial deposition from in the horizontal reactor has been undertaken employing the steady‐state, fully parabolic flow approximation for the heat, momentum, and mass‐transfer equations. Reactants are assumed to be dilute in their carrier gas. The resulting set of partial differential equations are discretized using finite elements and solved using the method of lines. The effect of a third dimension, the side temperature boundary conditions, and natural convection are explored. Given recent kinetics data on the silane decomposition and silylene insertion reactions, it is shown that a quasi‐thermodynamic equilibrium exists in the heated region above the surface, at least for hydrogen gas ambients. The combination of a fast silylene surface reaction and a slow silane surface reaction implies that the effective surface reaction rate is governed by the homogeneous thermodynamic equilibrium in the gas phase and diffusion through a thin mass‐transfer boundary layer near the surface. Examples of how tilting the susceptor contributes to growth uniformity are presented, and the effectiveness of similarity solution models at predicting this enhanced axial uniformity is tested.

The infinite candle and its stability—A paradigm for flickering diffusion flames

Abstract: Large diffusion flames are known to flicker at a frequency (∼12Hz) that is remarkably insensitive to flow rate, burner size, or gas composition. We have confirmed this phenomenon and propose a theoretical explanation. We note that, in addition to the forced convection associated with a tube-burner diffusion flame, there is strong natural convection generated by the hot gases. This bouyancy-induced flow surrounds the forced component and depends only on the thermomechanical properties of the hot and cold gas, together with g , the gravitational acceleration. We argue that it is the instability (of modified Kelvin-Helmholtz type) of this annular flow that is responsible for the flickering. A paradigm for this flow is defined by the infinite candle , an ideal plane diffusion flame in which the flow field is induced solely by buoyancy. The infinite candle admits a similarity solution. An inviscid, parallel flow stability analysis of this flow-field yields a frequency for which the spatial growth of the disturbance is a maximum. This is within a factor of 2 of the observed frequency.

Forced flow near a heated rotating disk: a similarity solution

Abstract: This paper presents an analysis of the flow and heat transfer between two parallel infinite disks that are separated by a distance L. One of the disks is solid, heated, and rotated; the other disk is porous, unheated, and stationary. Fluid is injected through the porous disk normal to its surface and toward the rotating disk. The three-dimensional Navier-Stokes and energy equations have been reduced to a system of ordinary differential equations by means of a similarity transformation and have been solved over a range of values of the two Reynolds numbers Reωequals; L2ω/νL and Reν = LU/νL, for two values of the temperature ratio Tr = (Ts - TL)/TL, of 0.001 (constant property flow) and 2.33. For Tr = 0.001 the velocity profiles are independent of the temperature profile which is determined for Pr = 0.67; for Tr = 2.33, the velocity and temperature profiles are determined for helium. The assumptions made in the analysis are shown to be valid provided that the mixed convection parameter , is small. Velocity ...

Similarity solutions in axisymmetric mixed convection boundary-layer flow

Abstract: The similarity equations for mixed-convection axisymmetric boundary-layer flow are considered. The equations involve a buoyancy parameter α and a curvature parameter β. The equations are solved numerically and it is found that for large α, and β of O(1), an asymptotic solution is approached, the nature of which is discussed. When β is also large, of O(α1/4), the problem, at leading order, becomes independent of the mainstream and the free-convection limit is obtained. This problem is also discussed, including the behaviour for large values of β0, the free-convection curvature parameter. For α < 0 we find that the solution can be continued past the point where the wall heat transfer becomes zero (where previous mixed-convection similarity solutions in plane geometry were terminated) with the solution ending as α → 0−. The nature of this limit is also discussed. For α < 0 it is also found that there are solutions only in α b = α < 0 with two branches of solution bifurcating out of α = α b , and values of α b are computed for a range of β. The behaviour of the solution for large values of the curvature parameter β, and α of O(1), is discussed where it is shown that the solution proceeds in inverse powers of log β.

Hydromagnetic flow past a continuously moving semi-infinite plate for large suction

Abstract: The similarity solution for hydromagnetic flow of an incompressible viscous electrically conducting fluid past a continuously moving semi-infinite porous plate in the presence of a magnetic field has been obtained for the case of small magnetic Reynolds number. The perturbation method has been used to solve the similarity equations at large suction. The resulting equations have been solved by analytical method. The effect of the magnetic parameter is to increase the skin-friction coefficient while it has no significant effect on the Nusselt number.

Free Convection on a Horizontal Plate With Blowing and Suction

Abstract: The effects of blowing and suction on forced convection and those on free convection over a extensively. On the other hand, the problem of free convection above a horizontal plate with blowing or suction has received substantially less attention. Gill et al. presented a local similarity solution for an isothermal horizontal plate with finite transverse velocity. Clarke and Riley, allowing for variable density, obtained a similarity solution for the special case of a plate with constant temperature and a particular distribution of blowing rate. The data available are quite restrictive. There is still a shortage of accurate data for a wide range of both suction and blowing rate. In this analysis of free convection over a semi-infinite horizontal plate, both the wall temperature and transpiration rate are assumed to be power-law variations. Finite-difference solutions and local similarity and nonsimilarity solutions are obtained over a wide range of transpiration rate from very strong suction to very strong blowing. Special considerations are given to the most practical cases of an isothermal plate under the condition of uniform blowing or suction.

Similarity solutions for avalanches of granular materials down curved beds

Abstract: The present paper investigates similarity solutions for the two-dimensional flow of a mass of cohesionless granular material down rough curved beds having gradually varying slopes. The work is relevant to the motion of rockfalls and loose snow flow avalanches. The depth and velocity profiles for the moving mass are determined in analytical form and the evolution equation for the total length of the pile is integrated numerically using a Runge-Kutta technique. Although similarity solutions can occur for general bed shapes (as long as the curvature is small), specific computations are performed here for two families of bed profiles, one which is in the shape of a circular arc and the other in which the slope decays exponentially with downstream distance. The pile of granular material starts from rest, initially accelerates and then decelerates, finally coming to rest as a result of bed friction and the gradually decreasing bed slope. Depending upon the frictional parameters, the shape of the bed and the initial depth to length of the pile, it is found that the variation of total length with time can exhibit different behaviours. The pile can grow monotonically, it can asymptote to a constant length, it can grow to a maximum and then decrease or it can decrease to a minimum and then increase with time. Furthermore, there are regions in parameter space for which the pile moves as a rigid body either for the whole time of travel or for portions of it.

Gravity flow of a viscous liquid down a slope with injection

Abstract: The free‐surface lubrication equations are solved numerically for the time‐dependent three‐dimensional flow down an inclined plane produced by continuous injection through a circular orifice in the plane. The flow pattern ultimately becomes steady with stream depth and width scaling as the − (1)/(7) th power and (3)/(7) th power of the downslope distance from the orifice, in agreement with a known similarity solution. The predicted stream width gives partial agreement with a recent experimental result, even though significant thermal effects were present in the experiment. The physical prototype is flow of lava.

Jeffrey-Hamel flow of Leslie-Ericksen nematic liquids

Abstract: A similarity solution of the Leslie-Ericksen equations for nematic liquid crystals is obtained for flow between converging and diverging planar walls (Jeffrey-Hamel flow). There are three regions in the flow: extensional or compressional flow near the centerline, shear near the wall, and a wall boundary layer in which elastic stresses control the transition from the wall-induced orientation to the bulk behavior. The boundary layer thickness is obtained in closed form; the scaling with the Ericksen number depends on whether or not the boundary layer extends into the region of extensional flow. Imposition of a magnetic field with an azimuthal component in a converging flow can result in a Freedericksz-like transition from radial to transverse orientation at the center line at a critical field strength. This transition provides a new means to measure the irrotational viscosity λ 2 .

Similarity solution and wave propagation in a reactive polytropic gas

Abstract: We consider similarity solutions of the ZND model for detonation waves. Assuming as boundary condition the RH relations for the precursor shock, we obtain exact similarity solutions corresponding to reaction rates compatible with the associated stretching group of transformations which leaves invariant the governing system. The location of weak discontinuities across a similarity line and their evolution laws are determined.

Isothermal magnetostatic atmospheres. II. Similarity solutions with current proportional to the magnetic potential cubed

Abstract: The paper presents a family of isothermal magnetostatic atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry. The distributed current in the model J is directed along the x-axis, where x is the horizontal ignorable coordinate. The current J is taken to be proportional to the cube of the magnetostatic potential A and falls off exponentially with distance vertical to the base with an e-folding distance equal to the gravitational scale height. A range of similarity solution examples are displayed depending on the values of the similarity parameters. Each similarity parameter corresponds to a symmetry of the underlying nonlinear elliptic equation for A. The similarity parameters also determine the source currents for the potential field solution of the family. The solutions show the interplay between the gravitational force, the J and B force (B, magnetic field induction) and the gas pressure gradient. 21 references.

The stability of granular flow in converging hoppers

Abstract: This paper investigates the steady state flow of an incompressible granular material in a converging hopper, in two and three dimensions. The material is modeled as a continuum which is in plastic yield throughout the bin. Jenike [Utah Engrg. Experimental Station Bull., 108 (1961)] adopted such a model and found a similarity solution of the governing equations. In this paper, the stability of Jenike’s solution to time-independent perturbations is examined, with attention given to the effect of material parameters on stability. Specifically, a linearized stability analysis is used to show that flows in three dimensions are usually stable to small perturbations, while many flows are unstable in two dimensions. It is also shown that an often used one-dimensional “slice analysis” gives misleading stability results. Finally, the governing equations are solved numerically, thereby exploring the nonlinear behavior of solutions.

Stability of Collapsing Isothermal Spheres

Abstract: The stability of collapsing isothermal spheres to radial perturbations is studied under the assumption of self-similar flow. A linear perturbation equation for the growth of density contrast is derived in the similarity coordinate, and eigenfunctions are obtained in the asymptotic regimes of the Larson-Penston and the expansion wave solutions. In either solution, fragmentation occurs only very slowly and is almost negligible within an initial free-fall time. 9 references.

A model for stress-driven diffusion in polymers

Abstract: Penetration of solvents into polymers is sometimes characterized by steep concentration gradients that move into the polymer and last for long times. The behavior of these fronts cannot be explained by standard diffusion equations, even with concentration dependent diffusion coefficients. The addition of stress terms to the diffusive flux can produce such progressive fronts. Model equations are proposed that include solvent flux due to stress gradients in addition to the Fickian flux. The stress in turn obeys an concentration dependent evolution equation. The model equations are analyzed in the limit of small diffusivity for the problem of penetration into a semi-infinite medium. Provided that the coefficient functions obey certain monotonicity conditions, the solvent concentration profile is shown to have a steep front that progresses into the medium. A formula governing the progression of the front is developed. After the front decays away, the long time behavior of the solution is shown to be a similarity solution. Two techniques for approximating the solvent concentration and the front position are presented. The first approximation method is a series expansion; formulas are given for the initial speed and deceleration of the front. The second approximation method uses a portion of the long time similarity solution to represent the short time solution behind the front. The addition of a convective term to the solvent flux is shown to raise the possibility of a traveling wave solution. The existence of the traveling wave solution is shown for certain types of coefficient functions. The way the initial front speed evolves onto the traveling wave speed is sketched out.

Heat transfer to an underlying fluid due to the axisymmetric spreading of material on the surface

Abstract: Similarity boundary layer solutions are found for the fluid underlying an axisymmetric spreading material layer. Three thermal boundary conditions for the fluid-material interface are considered, corresponding to constant temperature interface, adiabatic interface with heat source at origin, and constant heat flux interface. The boundary layer thickness is proportional to the distance from origin. Physical significance is discussed.

Hydrodynamic arrest of a flat body moving towards a parallel surface at arbitrary Reynolds number

Abstract: The motion of a flat body towards a parallel plane surface in incompressible fluid is considered both in the presence and absence of an applied force for a non-vanishing initial velocity. In the inviscid limit, a first integral of the equations is obtained and analytic solutions are presented for the cases of finite body inertia with zero applied force and finite applied force with negligible body inertia. In the former case when the ratio of body inertia to fluid inertia is large, a singular behaviour is observed in the arrest of the body before impact wherein the time-dependent pressure and radial velocity of the fluid exhibit a sharp peak and there is a large transfer of kinetic energy from the body to the thin fluid layer. For a real fluid, a general procedure is described to obtain solutions at arbitrary Reynolds number for naturally occurring initial velocity conditions. Solutions to the full Navier-Stokes equations are obtained for an arbitrary Reynolds number based on gap height which are valid provided the flow remains laminar and the gap height is small. In general, the equations of motion of the body and fluid are both dynamically and kinematically coupled. The dynamic coupling, however, is removed when the body inertia is neglected. In particular, the cases of hydrodynamic arrest with zero applied force, and draining of the fluid under a constant applied force are considered. The natural initial conditions lead to a new exact similarity solution of the Navier-Stokes equations which is valid for an instantaneous time-dependent Re based on gap height of greater than approximately 100, wherein the top and bottom boundary layers remain distinct. The longer time portions of the motion and the final arrest are described by a numerical calculation for intermediate Reynolds number and a low-Reynolds-number analysis.

Isothermal magnetostatic atmospheres. III - Similarity solutions with current proportional to the magnetic potential squared

Abstract: An investigation of a family of isothermal magneto-static atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry is carried out The distributed current in the model J is directed along the x-axis where x is the horizontal ignorable coordinate The current J is taken proportional to the square of the magneto-static potential A and falls off exponentially with distance vertical to the base with an e-folding distance equal to the gravitational scale height A range of solution examples are displayed depending on the four free parameters in the solutions Three of these parameters determine the source currents for the potential field solution of the family, whereas the fourth parameter determines the magnitude of the distributed current The solutions illustrate the contribution of the anisotropic J × B force (B, magnetic field induction), the gravitational force, and gas pressure gradient to the force balance

Similarity solutions in space dependent extended kinetic theory

Abstract: An exact analytical solution to a simple problem of extended kinetic theory for a spatially inhomogeneous binary gas mixture is determined in this paper by resorting to Lie group theoretical methods. It is shown that there are cases in which the nonlinear evolution equations for the densities allow transformation groups of infinite order, where the corresponding similarity solution actually represents the general solution to the problem. Results are then briefly commented on.

Free Shear Flows

Abstract: In this chapter we discuss the momentum and heat-transfer properties of uncoupled laminar and turbulent free shear flows, far from solid walls. As in the case of flows over walls, the thin-shear-layer equations admit similarity solutions for some laminar free shear flows, and the corresponding similarity variables can be found by a number of methods. In Section 4.1 we discussed the group-theoretic method. Here in Sections 8.1 and 8.2 we shall use a different approach to find the similarity variables of a two-dimensional laminar jet and a mixing layer between two uniform streams at different temperatures. Later, in Sections 8.3 and 8.4, we shall use the same approach for similar turbulent free shear flows. It should be noted that the similarity solutions become valid only at large distances from the origin because the initial conditions at, for example, a jet nozzle will not match the similarity solution. To illustrate the slow approach to similarity, we shall obtain the solutions of free shear layers for nonsimilar flows and compare them with similarity solutions.

Similarity solution for low Mach number spherical shocks

Abstract: A nonlinear progressive wave equation (NPE) describes the evolution of a low Mach number shock wave. The NPE is the nonlinear time domain counterpart of the frequency domain linear parabolic wave equation (PE) for small angle propagation. The NPE in spherical symmetry admits a similarity solution that specifies both the shape of the pulse and the shock strength as a function of range. For finite amplitude spherical waves whether self‐similar or not, the theory predicts constancy of an impulse integral corresponding to that of linear theory. For the self‐similar waves, theory and available data are in qualitative agreement in the following areas: (1) The shock strength decreases with range as an approximate power law; (2) the temporal behavior of the solution at fixed range is a shock discontinuity followed by a roughly exponential decay; and (3) the effective relaxation time behind the shock is in reasonable agreement with data for slightly more than the first decade in range.

MHD Laminar wall jet over a curved surface in a variable normal magnetic field

Abstract: The effects of a magnetic field on jets of an electrically conducting fluid flowing over a curved surface, in the presence of a variable normal magnetic field, are studied. It is noted that the convex surface curvature h and the magnetic interaction parameter m, in small perturbation similarity solution have qualitatively similar effects. They increase the mass flux, decrease the momentum flux, decrease the wall shear and cause an adverse pressure gradient in the flow field.

Multiple similarity solutions of buoyancy induced flows for ice melting in cold pure water

Abstract: A model of laminar buoyancy induced plane flows driven by the thermal transport near a vertical ice wall melting in cold pure water is studied. Computed results show that the convective inversions are found with a temperature ratio parameter R varying in [0, 1 2 ]. Numerical solutions are obtained in two disjoint intervals of R which are separated by a gap on which no solution exists. Multiple solutions are found at some R in these regions. With R near the lower edge of the gap, the solutions are similar. However, pairs of solutions with R varying above the upper limit of the gap are found drastically different. They indicate physically the potential existence of a large amount of energy for any trend arising that drives one flow state to another.

Viscoelastic inertial flow driven by an axisymmetric accelerated surface

Abstract: A similarity transform is used to analyse the flow of an upper-convected Maxwell fluid in an infinitely long cylinder whose surface has a velocity that increases in magnitude linearly with axial coordinate. Two types of problem are considered, the accelerated surface flow - when the surface velocity is outward towards the tube ends, and the decelerated surface flow - when it is inward. For the accelerated surface flow, the introduction of elasticity prevents the loss of similarity solution that occurs without elasticity at a Reynolds number ( Re ) of 10.25: with elasticity, solutions up to a Reynolds number of 95 were computed. As elasticity is introduced, normal stress gradients in an elastic boundary layer near the accelerated surface help offset inertially generated negative axial pressure gradients; with sufficient elasticity the turning point in the non-elastic solution family at Re = 10.25 disappears. For the decelerated surface flow, solutions could not be computed beyond a critical Re that depends on the level of elasticity considered, because at this critical Re , the axial velocity profile at the centreline becomes infinitely blunt.

Structure and behavior of diffusion flames in a pressure gradient

Abstract: The structure of a diffusion flame embedded in a flow field parallel to the flame is studied under conditions where this external flow imposes an adverse pressure gradient. It is convenient to think of the physical problem as a flame lying along the flow direction of a divergent channel. The mathematical problem is reduced to a set of ordinary differential equations by (i) employing the Howarth transformation to eliminate the variable density and (ii) introducing a similarity solution somewhat in the manner of the Falkner-Skan treatment of boundary layer flows. Because the low-density gas near the flame responds more readily to the pressure gradient than does the higher density gas, a reverse flow develops in the low density region which severely affects both the structure of the flame and the fuel consumption rate. For a flame with unit stoichiometry, the reverse flow eventually leads to extinction of the flame by separating the two shear layers that bound the fuel and oxidizer streams. For stoichiometry corresponding to methane-air, the flame situates itself near the oxidizer side of the reverse flow and has no tendency toward extinguishment.

A Semi-Analytic Approach to Cooling Flow Evolution

Abstract: Piecewise similarity solutions are used to model the evolution of a cooling flow in the absence of star formation, heating, or heat conduction. In these solutions the cooling radius R c increases as a power of time and the character of the X-ray emission changes qualitatively across the cooling radius. Application to M87 shows that time-dependence does not change the conclusion obtained by others that gas drops out of the flow; it is shown that inside ≈ 0.2R c local thermal instabilities can plausibly lead to star formation. The similarity solution fit to M87 implies that the cooling flow was more vigorous in the past.

A similarity transformation for subcooled mixed convection film boiling in a porous medium

Abstract: The classical two-phase boundary layer concept was adopted to analyse the problem of combined free and forced convection film boiling over an impermeable wall embedded in a porous medium. The governing equations, namely, the equations of continuity, energy and Darcy's model, were written for both vapor and liquid layers, and were solved simultaneously by means of the similarity transformation. Similarity solutions are found for a vertical flat plate, a horizontal circular cylinder and a sphere. Numerical integrations were carried out for given sets of the parameters associated with the vapor superheating Sup, the liquid subcooling Sub, the vapor mass flow rate R and the ratio of the liquid Rayleigh number to Peclet number Ra xf /Pe xf . It is found that a significant simplification of the problem is possible by setting the liquid stream function at the interface to zero. This simplification also reveals that the solution of the problem virtually depends on only three parameters, namely, Sup, Ra xf /Pe xf and the lumped parameter Sub/R.

Analysis of inclined wall plume by turbulence model

Abstract: The similarity solution of inclined wall plume is obtained analytically. The mathematical model used herein consists of the continuity equation of flow, the momentum balance equation in the flow direction, the diffusion equation of concentration, the equation of kinetic energy of turbulence and the equation of viscous dissipation rate of turbulence. It is shown that this set of equations has the similarity solution which can be solved numerically for each angle of the inclined wall. This numerical model is applied to the wide range of the slope angle, which includes the plume along the vertical wall for the special case and along the nearly horizontal wall. The velocity and concentration profiles of the inclined wall plume are explained well by the similarity solution.