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Showing papers on "Similarity solution published in 1996"


Journal ArticleDOI
TL;DR: In this article, it was shown that there is a countably infinite family of similarity solutions for viscous thread pinching with an inertial-viscous-capillary balance in an inviscid environment.
Abstract: The dynamics of capillary pinching of a fluid thread are described by similarity solutions of the Navier–Stokes equations. Eggers [Phys. Rev. Lett. 71, 3458 (1993)] recently proposed a single universal similarity solution for a viscous thread pinching with an inertial–viscous–capillary balance in an inviscid environment. In this paper it is shown that there is actually a countably infinite family of such similarity solutions which are each an asymptotic solution to the Navier–Stokes equations. The solutions all have axial scale t′1/2 and radial scale t′, where t′ is the time to pinching. The solution obtained by Eggers appears to be special in that it is selected by the dynamics for most initial conditions by virtue of being less susceptible to finite‐amplitude instabilities. The analogous problem of a thread pinching in the absence of inertia is also investigated and it is shown that there is a countably infinite family of similarity solutions with axial scale t′β and radial scale t′, where each solution...

130 citations


Journal ArticleDOI
TL;DR: In this article, the similarity form of a non-Boussinesq gaseous plume was developed by obtaining the equation for the conservation of enthalpy flux, and using it and the continuity equation to demonstrate that the flux of density deficiency is conserved, and not the flux flux of buoyancy as is the case for Boussineq plumes.
Abstract: We develop herein the similarity form of a non-Boussinesq gaseous plume. This is done by obtaining the equation for the conservation of enthalpy flux, and using it and the continuity equation to demonstrate that the flux of density deficiency is conserved, and not the flux of buoyancy as is the case for Boussinesq plumes. We then use this conservation relation to describe the form of the similarity solution in the non-Boussinesq case. The similarity solution is then used to derive the theoretical form of the entrainment velocity across the plume edge, which is seen to be in agreement with the ‘modified entrainment assumption’ suggested empirically from experiments by Ricou & Spalding (1961).

110 citations



Journal ArticleDOI
TL;DR: In this article, the similarity equations for mixed-convection boundary-layer flow past a wedge having one of its surfaces parallel to the horizontal are derived for the latter surface.
Abstract: The similarity equations for mixed-convection boundary-layer flow past a wedge having one of its surfaces parallel to the horizontal are derived for the latter surface. Both cases of prescribed wall temperature and heat flux are considered. It is shown that non-unique solutions exist for aiding (α > 0) as well as opposing flows (α 0 in addition to those already reported in the literature for α < 0. Namely, these correspond to vertical flat plate and vertical cylinder problems.

88 citations


Journal ArticleDOI
TL;DR: In this paper, the issue of convergence for initial value solutions and similarity solutions of the compressible Euler equations in two dimensions in the presence of vortex sheets (slip lines) was examined.
Abstract: We examine numerically the issue of convergence for initial‐value solutions and similarity solutions of the compressible Euler equations in two dimensions in the presence of vortex sheets (slip lines). We consider the problem of a normal shock wave impacting an inclined density discontinuity in the presence of a solid boundary. Two solution techniques are examined: the first solves the Euler equations by a Godunov method as an initial‐value problem and the second as a boundary value problem, after invoking self‐similarity. Our results indicate nonconvergence of the initial‐value calculation at fixed time, with increasing spatial‐temporal resolution. The similarity solution appears to converge to the weak ‘zero‐temperature’ solution of the Euler equations in the presence of the slip line. Some speculations on the geometric character of solutions of the initial‐value problem are presented.

75 citations


Journal ArticleDOI
TL;DR: In this paper, strong spherical shock waves are studied for an imperfect gas, here modelled by a van der Waals equation of state, and similar solutions of Guderley type are shown, in which the radius of the shock is proportional to (-t) α, where t is time measured from the moment at which the shock focuses.
Abstract: Strong spherical shock waves are studied for an imperfect gas, here modelled by a van der Waals equation of state. Similarity solutions of Guderley type are shown to exist, in which the radius of the shock is proportional to (-t) α , where t is time measured from the moment at which the shock focuses. The exponent a depends on both the ratio of specific heats, γ, and on the van der Waals excluded volume, b. For small b, the solution resembles the Guderley solution and is well described by the Chester-Chisnell-Whitham (CCW) approximation. A new branch of solutions, which the CCW approximation fails to locate, is shown to exist for larger b. The linear stability of the similarity solutions is examined directly, without the use of the CCW approximation. Normal modes grow (Re (β) 0) as (-t) αβ , where the 'growth rate', β, is a function of γ, b and n, the spherical harmonic wavenumber. No physically meaningful, discrete, spherically symmetric (n = 0) modes were found. This case was examined numerically and nonlinearly for shocks launched by a spherical 'piston'; no evidence of instability was discovered.

75 citations


Journal ArticleDOI
TL;DR: In this paper, the migration of an isolated gas bubble in an immiscible liquid possessing a temperature gradient is analyzed in the absence of gravity, where the driving force for the bubble motion is the shear stress at the interface which is a consequence of the temperature dependence of the surface tension.

63 citations


Journal ArticleDOI
TL;DR: In this paper, a system of equations which describe the dynamics of a thin two-dimensional free film is analyzed in the vicinity of a rupture, which contain the effects of surface tension and van der Waals forces.

49 citations


Journal ArticleDOI
TL;DR: In this paper, the uniqueness of the solution of the title problem has been examined through a simple mathematical procedure, and the solution has been shown to be unique in a number of applications.
Abstract: The uniqueness of the solution of the title problem has been examined through a simple mathematical procedure.

39 citations


Journal ArticleDOI
TL;DR: In this paper, a cold, thin film of liquid impinging on an isothermal hot, horizontal surface is modelled as a two-dimensional jet of prescribed uniform velocity, film thickness and temperature.

33 citations


Journal ArticleDOI
TL;DR: In this article, the mixed convection boundary-layer flow in the vicinity of the median plane of symmetry of a finite span wedge is considered and the similarity equations are derived for situations where one of the wedge surfaces is kept vertical.

Journal ArticleDOI
TL;DR: In this paper, a class of unsteady boundary layers that form on flat extensible surfaces of finite but increasing length in otherwise stagnant surroundings is considered, and the problem is cast into similarity variables and the governing parabolic differential equation shown to exhibit, for various combinations of n and p, regions of mixed mathematical diffusivity and reversals in the direction of convection of vorticity.
Abstract: A class of unsteady boundary layers that form on flat extensible surfaces of finite but increasing length in otherwise stagnant surroundings is considered. The surface length R is assumed to grow with time as t p where p > 0 and the velocity at any location 0 ≤ r ≤ R on the surface as t p-np-1 r n , where n ≥ 0. The problem is cast into similarity variables and the governing parabolic differential equation shown to exhibit, for various combinations of n and p, regions of mixed mathematical diffusivity and reversals in the direction of convection of vorticity. Equations depicting such behaviour are usually termed singular parabolic and are here classified as follows : type-0, in which the mathematical diffusivity may be either positive or mixed but in which there are no reversals in the direction of convection of vorticity ; type-1, in which the mathematical diffusivity may be either positive or mixed but in which there are reversals in the direction of convection of vorticity.

Journal ArticleDOI
TL;DR: In this paper, the similarity solution for free convection boundary-layer flow above a permeable, horizontal surface in a fluid-saturated porous medium is considered, and numerical solutions are presented for a wide range of values of γ and m.
Abstract: The equations which govern the similarity solution for free convection boundary-layer flow above a permeable, horizontal surface in a fluid-saturated porous medium are considered. These are seen to depend on the dimensionless parameters γ and m measuring mass transport rate and the wall temperature variation, respectively. Numerical solutions are presented for a wide range of values of γ and m. Asymptotic solutions are obtained for ¦γ¦ large (for both fluid injection, γ > 0, and fluid withdrawal, γ < 0) and for m large. These are compared with the numerical solutions.

Journal ArticleDOI
TL;DR: In this article, a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x2)μ), where μ is a constant andx is the distance along the surface.
Abstract: This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x2)μ), where μ is a constant andx is the distance along the surface. It is shown that for μ > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when μ ⩽ -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely μ < -1/2 and μ = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of μ.

Journal ArticleDOI
SH Doole1
TL;DR: In this paper, a two-dimensional, one phase Stefan-type problem is described as a model for an industrial erosion/deposition process, which includes surface tension effects and a kinetic condition at the free boundary.

Journal ArticleDOI
TL;DR: In this paper, a mathematical model for the flow and heat transfer in the free convection from an arbitrary inclined isothermal flat plate embedded in a porous medium is presented, in which the Darcy-Boussinesq approximation is adopted to account for bouyancy force.
Abstract: A mathematical model for the flow and heat transfer in the free convection from an arbitrary inclined isothermal flat plate embedded in a porous medium is presented, in which the Darcy–Boussinesq approximation is adopted to account for bouyancy force. A novel inclination parameter ξ is proposed such that all cases of the horizontal, inclined and vertical plates can be described by a single set of transformed boundary layer equations. Moreover, the similarity equations for the limiting cases of the horizontal and vertical plates are recovered from the transformed equations by setting ξ=0 and ξ=1, respectively. Detailed results for the skin friction coefficient and Nusselt number as well as for the dimensionless velocity and temperature profiles are presented for a wide range of the parameter ξ. A comparison with similarity solution shows excellent agreement.

Journal ArticleDOI
P. Astin1, Graham Wilks1
TL;DR: In this article, new solutions of the Falkner-Skan equation are presented which display jet-like behavior. But the parameter range over which the new solutions exist is bounded by a singular limit, which is closely related to the well-known plane jet similarity solution.
Abstract: New solutions of the Falkner-Skan equation are presented which display jet-like behaviour. Within non-uniform flow the parameter range over which the new solutions exist is bounded by a singular limit. The limit is shown to be closely related to the well-known plane jet similarity solution.

01 Jan 1996
TL;DR: In this article, the authors considered the special case in which the initial saturation as well as the properties of the porous medium have a single coinciding discontinuity and proposed a self-similar profile to validate numerical algorithms describing two-phase flow in porous media.
Abstract: In this paper we consider the process of one-dimensional redistribution of two immiscible and incompressible fluids in a heterogeneous porous medium. We treat in detail the special case in which the initial saturation as well as the properties of the porous medium have a single coinciding discontinuity. Then the time-dependent saturation profile is of self-similar form, i.e. depends only on $x/ sqrt{t$. This self-similar profile can be used to validate numerical algorithms describing two-phase flow in porous media with discontinuous heterogeneities. We discuss the construction of the similarity solution, in which we give special attention to the matching conditions at the interface where the medium properties are discontinuous. We also outline a numerical procedure to obtain the similarity solution and we provide applications in terms of the Brooks-Corey and the Van Genuchten model.

Journal ArticleDOI
TL;DR: The effect of time-dependent forcing on basin-scale flows is investigated in this article, where analytical and numerical solutions are considered separately and compared, and a more realistic steady solution is found numerically in a flatbottomed sector.
Abstract: The effect of time-dependent forcing on steady solutions representing basin-scale flows is investigated. Analytical and numerical solutions are considered separately and compared. We first use symmetry methods to show how any steady solution of the ideal thermocline equations can be used to generate a family of unsteady solutions, via an arbitrary function of time ®(t). The resulting time-dependent solutions correspond to distortion of the isopycnal surfaces by a velocity field which varies linearly in the three coordinate directions. Although the displacements are linear, the fluctuations can lead to a form of nonlinear streaming wherever the function ® appears nonlinearly in expressions for mass and heat fluxes. For an example steady solution, changes in internal energy caused by the time-dependence are associated with changes in thermocline depth and fluxes of energy from the western boundary, although it is unclear to what extent this behaviour is specific to the example chosen. We also describe another symmetry of the time-dependent thermocline equations which generates wave-like solutions from arbitrary steady solutions. All the time-dependent solutions are special cases of a symmetry which applies to a general advection equation. Potential vorticity advection provides another special case. With the inclusion of convective and dissipative processes, a more realistic steady solution is found numerically in a flatbottomed sector. If the surface forcing functions oscillate annually, the resulting flow resembles the analytical predictions. As the oscillation period increases, spatial variations in phase disrupt the agreement as first boundary and then diffusive effects become important. For decadal period oscillations, nonlinear streaming is found to significantly increase the meridional overturning.

Journal ArticleDOI
TL;DR: In this paper, the authors analyzed transformation for boundary layer equations for two-dimensional steady natural convection along a vertical flat plate embedded in porous media, and found that similarity solution exists for the whole flow region as the wall temperature distribution is in linear variation and the inertia resistance is without consideration.

Journal ArticleDOI
TL;DR: In this article, the authors derived the transformation of boundary layer equations for two-dimensional steady natural convection on a vertical wall embedded in porous media, and three different kinds of thermal boundary conditions are prescribed for wall heat flux: uniform distribution, power law variation, and exponential variation.

Journal ArticleDOI
TL;DR: The similarity solution ansatz that Burde applied to adiabatic boundary-layer flow over flat plates and slender bodies of revolution is employed in this article to investigate mixed convection flow adjacent to heated inclined flat plates.
Abstract: The similarity solution ansatz that Burde applied to adiabatic boundary-layer flow over flat plates and slender bodies of revolution is employed here to investigate mixed convection flow adjacent to heated inclined flat plates. Both Newtonian and Darcian fluids are considered under the assumption that they obey the Boussinesq approximation. New results fall into two distinct categories for heated plates with oblique wall suction. Class I problems correspond to radial source/sink flows interior to a wedge and class II problems pertain to uniform rectilinear flow over flat plates. Except in special cases, solutions of the class I equations must be obtained numerically while all class II equations possess explicit analytical solutions encompassing natural, mixed, or forced convection depending on the magnitude of the free-stream velocity. Numerical integration of prototype class I problems reveals single or dual solutions for radial inflow and an infinity of oscillatory solutions for radial outflow. The similarity solutions reported here describe flows with continuous distributions of temperature and suction angle along the plate, and in some cases the variation of the temperature, suction strength, or suction angle may be chosen freely.

Journal ArticleDOI
TL;DR: In this article, the spatio-temporal evolution of an active scalar in surface-tension-driven flow is studied using a recently developed self-consistent nonlinear model.
Abstract: The spatio-temporal evolution of an active scalar in surface-tension-driven flow is studied using a recently developed self-consistent nonlinear model. In the one-dimensional case an exact similarity solution is found which, depending on the initial conditions, describes the spreading or the finite-time collapse of the scalar. The time-dependence of the width of the surfactant distribution can be qualitatively understood from basic fluid-dynamical principles. The possibility of experimental verification of the theoretical prediction is briefly discussed.

Journal ArticleDOI
TL;DR: In this paper, a front dynamics approach is developed to study the evolution of planar curves whose normal speed depends on curvature, which is similar to Whitham's shock dynamics theory for the propagation of shock wave in gases but assumes a different propagation rule.
Abstract: A front dynamics approach is developed to study the evolution of planar curves whose normal speed depends on curvature. The formulation is similar to Whitham’s shock dynamics theory for the propagation of shock wave in gases but assumes a different propagation rule. Equations that describe the motion of the front are obtained, and these are evolution equations for the normal direction and local arc length of the front. The solution of these equations leads to the front positions using an appropriate integration along rays. A similarity solution of the equations is found for the evolution of an initial corner. Free-boundary problems for the motion of a junction connecting front segments are discussed. A numerical method is presented to calculate the evolution of any number of front segments. The segments can be closed or open, connected to wall boundaries or not, or connected to other segments at 3-segment junctions. Several sample problems are considered to illustrate the method. An extension of the metho...

Book ChapterDOI
01 Jan 1996
TL;DR: In the theory of thermal conduction, a basic type of problem is the evolution of the temperature field due to a sudden change in boundary temperature (Carslaw and Jaeger 1959) as discussed by the authors.
Abstract: In the theory of thermal conduction, a basic type of problem is the evolution of the temperature field due to a sudden change in boundary temperature (Carslaw and Jaeger 1959). Asymptotically for small times, an error-function similarity solution is valid near an arbitrary smooth wall

Journal ArticleDOI
TL;DR: In this article, the authors presented the results of calculations based on the similarity solution and compared these results with experimental data and 2-D numerical computations obtained with a commercial package, and they followed essentially the method by Paullay et al and derived a similarity solution for the turbulent, axisymmetric round jet, without using any assumptions about the shape of the velocity profile.
Abstract: Turbulent round jets are encountered in a variety of forms in engineering and natural systems. Among the similarity solutions, Wollmers and Rotta used the self-preserving property of a turbulent jet and developed a k-kl model to solve the turbulence closure problem. So obtained similarity solutions for a round jet by assuming a velocity profile in the form of a gaussian error function. This assumption is very restrictive because it does not allow the shape of the velocity profile to evolve from the governing equations, but compels them to accommodate the assumed velocity profile. Paullay et al. used the principle of moving equilibrium, to derive a similarity solution for plane and radial jets. Their solution is not general enough to include the round axisymmetric jets, which are most commonly met in engineering practice, because the governing equations used are not applicable to round jets. The authors followed essentially the method by Paullay et al. and derived a similarity solution for the turbulent, axisymmetric round jet, without using any assumptions about the shape of the velocity profile. In this short paper, they present the results of calculations based on the similarity solution and compare these results with experimental data and 2-D numericalmore » computations obtained with a commercial package. The agreement of the results and experiments shows that the similarity solution yields accurate results.« less


Journal Article
TL;DR: In this article, a flat horizontal plate aligned parallel to a uniform flow with velocity U∞ and temperature T∞ is assumed that a heat source (or sink) of strength Q is at the leading edge of the plate and that the plate is adiabatic everywhere else.

Journal ArticleDOI
TL;DR: In this paper, a series of transient problems in the flows of concentrated suspensions were investigated to test the effects of particle migration on the evolution of concentration and velocity profiles, and the authors reported a similarity solution to a Rayleigh problem, where the boundary of the infinite half space is given a velocity proportional to the square root of time.
Abstract: We have investigated a series of transient problems in the flows of concentrated suspensions to test the effects of particle migration on the evolution of concentration and velocity profiles. First, we report a similarity solution to a Rayleigh problem, where the boundary of the infinite half space is given a velocity proportional to the square root of time. Next, the classical Rayleigh problem, where the boundary is impulsively started initially at a constant velocity, is examined. The structure of the kinematics resembles that obtained in the first problem, but the concentration does not have a similarity form, and tends asymptotically to a uniform profile at large time. Finally, we solve the flow of a suspension past a semi-infinite plate, and discuss its connection to the Rayleigh problem. In all three cases, our calculations reveal Newtonian kinematics in the practical limit of a L ⪡ 1 , where a is the particle size, and L is a viscous diffusion length scale. In addition we see vastly different time and length scales in the evolution of the velocity and the concentration profiles. The velocity develops faster in time (by O( a L ) 2 ), and extends further in space (by O( L a ) ) than the concentration profile.

Book ChapterDOI
01 Jan 1996
TL;DR: In this article, the two-point velocity correlation in the far fields of the axisymmetric and planar jets was analyzed using Direct Numerical Simulations of the temporally evolving wake (Moser and Rogers, 1994).
Abstract: It has long been recognized (e.g. Towsend, 1956) that the governing equations for the single-point moments in several free-shear flows admit similarity solutions if the flows evolve from virtual sources. These similarity solutions, however, do not provide information about many important features of the flow, such as how the turbulent kinetic energy is distributed amongst the different scales of motion. In order to gain this information it is necessary to consider more complex statistical measures of the flow, such as the two-point correlations.Most of the previous analyses of the two-point equations have considered decaying isotropic turbulence (e.g., Batchelor, 1948 or George, 1992) or homogeneous shear turbulence (George and Gibson, 1992). Ewing (1995) later demonstrated that the equations for the two-point velocity correlation in the far fields of the axisymmetric and planar jets admit similarity solution. However, no attempt was made to test the hypotheses using data from these non- homogeneous flows. Here, data from two Direct Numerical Simulations of the temporally evolving wake (Moser and Rogers, 1994) are used to test the similar hypothesis for the two-point correlations in this flow.