Showing papers on "Similarity solution published in 1990"

The bifurcation of liquid bridges

Abstract: Details of the shape of the liquid bridge joining a nascent water drop to its parent body are presented for times before, after and at the instant of bifurcation when the drop is created and also when the secondary droplet is formed. After the instant of bifurcation there is ‘unbalanced’ surface tension which gives an impulse to the rest of the fluid causing strong surface deformations. The major point of this work is to draw attention to the strong up–down asymmetry at each bifurcation point. The geometric similarity at each bifurcation instant supports the conjecture that the flow converges to just one similarity solution of the type described by Keller & Miksis (1983) in which only surface tension and inertia are important. Features of the flow before and after bifurcation are discussed.

Similarity solutions of the two-dimensional unsteady boundary-layer equations

Abstract: The method of Lie group transformations is used to derive all group-invariant similarity solutions of the unsteady two-dimensional laminar boundary-layer equations. A new method of nonlinear superposition is then used to generate further similarity solutions from a group-invariant solution. Our results are shown to include all the existing solutions as special cases. A detailed analysis is given to several classes of solutions which are also solutions to the full Navier–Stokes equations and which exhibit flow separation.

Squeezing flow between parallel plates

Abstract: The flow between two parallel plates (rectangular or circular) approaching or receding from each other symmetrically is analysed. The Xavier-Stokes equations have been transformed into an ordinary differential equation using a similarity transformation and the resulting equations are solved numerically. Results for the velocity components, pressure distribution and shearing stress on the wall are presented. In the case of squeezing flow between two circular plates the load supporting capacity of the upper plate has been calculated.

Spreading and instability of a viscous fluid sheet

Abstract: Experiments by Huppert (1982) have demonstrated that a finite volume of fluid placed on an inclined plane will elongate into a thin sheet of fluid as it slides down the plane. If the fluid is initially placed uniformly across the plane, the sheet retains its two-dimensionality for some time, but when it has become sufficiently long and thin, the leading edge develops a spanwise instability. A similarity solution for this motion was derived by Huppert, without taking account of the edge regions where surface tension is important. When these regions are examined, it is found that the conditions at the edges can be satisfied, but only when the singularity associated with the moving contact line is removed. When the sheet is sufficiently elongated, the profile of the free surface shows an upward bulge near the leading edge. It is suggested that the observed instability of the shape of the leading edge is a result of the dynamics of the fluid in this bulge. The related problem of a ridge of fluid sliding down the plane is examined and found to be linearly unstable. The spanwise lengthscale of this instability is, however, dependent on the width of the channel occupied by the fluid, which is at variance with the observed nature of the instability. Preliminary numerical solutions for the nonlinear development of a small disturbance to the position of a straight leading edge show the incipient development of a finger of fluid with a width that does not depend on the channel size, in accordance with the observations.

Study of Nordgren's equation of hydraulic fracturing

Abstract: A nonlinear partial differential equation modeling the propagation of a vertical hydraulic fracture first derived by Nordgren is studied. When properly posed, Nordgren's derivation constitutes a Stefan problem and requires another boundary condition-namely, that the velocity of the fluid at the crack tip equals the velocity of crack propagation. With this addition, Nordgren's similarity solution in the no-leakoff case is perfected by a power-series solution. New shut-in solutions are derived in the large-and-no-leakoff cases where the crack of the large-leakoff shut-in solution retracts and the crack of the no-leakoff shut-in solution extends forever. This study ignores the effect of crack-tip rock strength on crack propagation.

Non-Darcian Boundary Layer Flow and Forced Convective Heat Transfer Over a Flat Plate in a Fluid-Saturated Porous Medium

Abstract: The local similarity solution procedure was successfully adopted to investigate non-Darcian flow and heat transfer through a boundary layer developed over a horizontal flat plate in a highly porous medium. The full boundary layer equations, which consider the effects of convective inertia, solid boundary, and porous inertia in addition to the Darcy flow resistance, were solved using novel transformed variables deduced from a scale analysis. The results from this local similarity solution are found to be in good agreement with those obtained from a finite difference method. The effects of the convective inertia term, boundary viscous term, and porous inertia term on the velocity and temperature fields were examined in detail. Furthermore, useful asymptotic expressions for the local Nusselt number were derived in consideration of possible physical limiting conditions.

On the flow past a sphere in a cylindrical tube: Limiting Weissenberg number

Abstract: The paper is concerned with the steady flow generated by a sphere falling along the centreline of a cylindrical tube containing a viscoelastic fluid which is modelled by the Oldroyd-B constitutive relation. By exploiting the similarity solution in the neighbourhood of the centreline of the tube it is found that there is a limiting Weissenberg number above which no steady state axisymmetric solution can exist. The full numerical solution to the problem using a boundary element method is reported and compared with results obtained by other numerical methods. We find an overall agreement between different sets of results pointing to the existence of the limiting Weissenberg number.

Mixed convection on a vertical needle with heated tip

Abstract: The mixed convection on a vertical slender adiabatic paraboloid with a tip heat source is studied. The boundary layer equations admit similarity solutions that are governed by a nondimensional free‐stream parameter γ and a heat source parameter α. Numerical results show for aiding flow (α>0) the solutions are unique and for opposing flow (α<0) the solutions may be unique, dual, or nonexistent. Velocity and temperature profiles are obtained.

Similarity solutions for nonlinear diffusion — further exact solutions

Abstract: Although the nonlinear diffusion equation has been extensively studied and there exists substantial literature in many diverse areas of science and technology, the number of exact concentration profiles is nevertheless limited. In a recent article in this journal (Hill [1]) a brief review of known exact results is given, as well as an elementary integration procedure which appears to be a general device for obtaining integrals associated with similarity solutions. This paper extends the results given in [1] and for particular power law diffusivitiescm (such asm = −/12, −1, −/32 and −2) presents a number of new exact solutions obtained by fully integrating the ordinary differential equations derived in [1]. In addition new results are found for a general nonlinear diffusion equation which includes one-dimensional diffusion with an inhomogenouus and nonlinear diffusivitycmxmas well as symmetric nonlinear diffusion in cylinders and spheres. Moreover by a separate and ad-hoc procedure a new solution is obtained of the travelling wave type but with a variable wave speed. Some of the new exact solutions obtained for one-dimensional nonlinear diffusion with power law diffusivitiescmare illustrated graphically.

On the free convection boundary layer on a vertical plate with prescribed surface heat flux

Abstract: The free convection boundary layer on a vertical plate with a prescribed surface heat flux proportional to (1 +x2)µ (µ a constant) is discussed. For µ > −1―2 the boundary-layer solution develops from a similarity solution valid forx small to the one valid forx large. However, with µ ⩽ −1―2 the similarity equations forx large are not solvable and the behaviour for largex in this case is discussed. It is found that there are two cases to consider, namely µ < −1―2 and µ = −1―2. In both cases the leading-order problem is homogeneous involving an arbitrary constant which is determined from an integral property of the full boundary-layer problem. However, in the former case the asymptotic behaviour is algebraic, with the perturbation to the leading-order solution, arising from the heat flux boundary condition, being ofO[x1+2µ]. The latter case also involves logarithmic terms, with the perturbation to be leading-order solution now being ofO[(logx)−1].

Similarity solutions for a Voellmy model of snow avalanches with finite mass

Abstract: This paper, though independently written, continues an analysis of similarity solutions for the two-dimensional flow of a mass of cohesionless granular material down rough, flat and curved beds, see Savage and Nohguchi [12], Nohguchi, Hutter and Savage [7]. The basal friction force is assumed to be composed of a Mohr-Coulomb type component with a bed friction angle that is position dependent plus a viscous Voellmy-type resistive stress, that is proportional to the velocity squared. This granular flow model is conjectured to adequately model the motion and dispersion of flow avalanches of snow whose air borne powder component can be ignored. The depth and velocities relative to those of the centre of mass of the moving pile are determined analytically, and it is shown that the pile has a parabolic cap shape and the difference velocity varies linearly with distance from the centre of mass. The length and the position and velocity of the centre mass are calculated numerically. We explicitly show:

Mixed convection of non-newtonian fluids from a vertical plate embedded in a porous medium

Abstract: This paper investigates mixed free and forced convection of non-Newtonian fluids from a vertical isothermal plate embedded in a homogenous porous medium. A mathematical model is developed based on the modified Darcy's law and boundary-layer approximations, and the exact similarity solution is obtained as well as an integral solution. These two solutions agree within 3% for aiding flows and 10% for opposing flows. It is found that, non-Newtonian characteristics of fluids have appreciable influences on velocity profiles, temperature distributions and flow regimes.

Natural-convection boundary-layer flow over a heated plate with arbitrary inclination

Abstract: The natural-convection boundary-layer flow over a semi-infinite heated plate of arbitrary inclination is studied by first identifying a set of combined boundary-layer variables and then casting the governing equations into a universal form. The generalized problem yields the existing similarity solutions for the limiting cases of horizontal and vertical plates, and describes the gradual transition of the flow pattern between these two limits at distances from the leading edge which depend on the inclination angle. Near the leading edge the buoyancy force acting normal to the plate causes an ‘impulsive’ driving force to the fluid motion along the plate, while the ‘regular’ driving force exerted by the tangential buoyancy force becomes dominating downstream. Both the exact and the locally-similar solutions are obtained and are found to agree well with each other.

Upstream‐advancing waves generated by three‐dimensional moving disturbances

Abstract: The wave field resulting from a surface pressure or a bottom topography in a horizontally unbounded domain is studied. Upstream‐advancing waves successively generated by various forcing disturbances moving with near‐resonant speeds are found by numerically solving a forced Kadomtsev–Petviashvili (fKP) equation, which shows in its simplest form the interplay of a basic linear wave operator, longitudinal and transverse dispersion, nonlinearity, and forcing. Curved solitary waves are found as a slowly varying similarity solution of the Kadomtsev–Petviashvili (KP) equation, and are favorably compared with the upstream‐advancing waves numerically obtained.

Thermophoretic deposition in crossflow over a cylinder

Abstract: tions.4 The pressure gradient to be used in the momentum equation is found either from the potential flow solution over the cylinder or from the best fit for the experimental data.4 The temperature gradient established in the thermal boundary layer drives the particles either toward or away from the cylinder surface. The velocity acquired by the small parti- cles relative to the gas velocity is known as the thermophoretic velocity vt. Following the standard assumptions,5 the conser- vation law for particle concentration, with the help of the continuity equation, reduces to HERMOPHORESIS causes small particles to be driven away from a hot surface and toward a cold one. This phenomenon has many practical applications. It affects the removal of small particles from gas streams, determines ex- haust gas particle trajectories from combustion devices, and helps in studying the particulate material deposition on turbine blades. It is also of importance in the manufacture of fumed silica, carbon black, and titania particles for the paint industry. All studies of thermophoreti c deposition in external flow (except Homsy et al.1 and Alam and Mehrotra2) are either for a zero pressure gradient in the boundary layer or for cases for which similarity solution is possible. Practical applications of thermophoresis in external flow, however, involve a nonzero pressure gradient for which no similarity solution is possible. Therefore, we study the thermophoretic deposition of aerosol particles in crossflow over a circular cylinder. Both theoretical and experimental pressure distributions in the hydrodynamic boundary layer over the cylinder are considered, unlike those in Homsy et al. 1 and Alam and Mehrotra,2 where only the former is studied. A finite-differe nce method is used for the solution. The working fluid is taken to be air.

Viscoelastic flow in a curved channel: a similarity solution for the Oldroyd-B fluid

Abstract: A similarity solution is used to analyse the flow of the Oldroyd fluid B, which includes the Newtonian and Maxwell fluids, in a curved channel modelled by the narrow annular region between two circular concentric cylinders of large radius. The solution is exact, including inertial forces. It is found that the non-Netonian kinematics are very similar to the Newtonian ones, although some stress components can become very large. At high Reynolds number a boundary layer is developed at the inner cylinder. The structure of this boundary layer is asymptotically analysed for the Newtonian fluid. Non-Newtonian stress boundary layers are also developed at the inner cylinder at large Reynolds numbers.

The Stability of a Collisionless Cosmological Shell

Abstract: The P3 M technique is used here to simulate the evolution of collisionless shells in an Omega = 1 universe. Starting from the spherical similarity solution, a bootstrap technique is used to follow the evolution over very large expansion factors. It is found that the overall structure follows the similarity solution for a long period during which bound clumps grow within the shell. At late times the growth of structure depends on induced velocity perturbations in material outside the shell. If such perturbations are suppressed, structure on the shell becomes self-similar. When induced motions in the background medium are included, the evolution at late times is dominated by large-scale modes as predicted by linear stability analysis. The stable final state appears to consist of one or two massive clumps on the edge of a spherical void. The possible application of these results to the origin of galaxies and large-scale structure is discussed.

Shocks in a model for stress-driven diffusion

Abstract: Diffusion of a solvent into a polymer slab is sometimes characterized by a sharp front that moves into the medium and lasts for a long time. The behavior of such a front cannot be explained by a standard diffusion equation. Model equations are proposed that include solvent flux due to stress gradients in the polymer in addition to the Fickian flux. The stress in turn obeys a concentration-dependent evolution equation. In the limit of small diffusivity, the solution to a penetration problem is shown to have a steep front that progresses into the medium. The equations that govern the front are derived. A method for approximating the front is presented that uses a piece of the long time similarity solution to represent the concentration profile behind the front. The addition of a convective term to the equations is shown to raise the possibility of a traveling wave solution.

Analysis of Inclined Wall Plume by the k-ε 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 vertical wall plume as the special case and the nearly horizontal wall plume. The velocity and concentration profiles of the inclined wall plume are explained well by the similarity solutions.

Similarity solutions to three-dimensional nonlinear diffusion equations

Abstract: The similarity transformation method has been used to solve three-dimensional nonlinear diffusion equations. The general Lie group is calculated. Exact solutions to the cylindrical and spherical symmetry cases are found, and their relations to some real physical processes are discussed.

Laminar Wind-Aided Flame Spread Across a Thick Horizontal Fuel Slab

Abstract: An explicit expression for the parametric dependence of the rate of wind-aided flame spread across a thick horizontal fuel slab is obtained by approximate analysis. The slab, semi-infinite in length and depth, is taken to gasify by sublimation upon heating from ambient temperature to pyrolysis temperature. The evolved fuel vapor is burned in a vigorous gas-phase diffusion flame with oxygen from the laminar air stream. The hot combusion-product gases flow over the downwind portions of the surface of the slab, to preheat more of the slab to the pyrolysis temperature. The rate at which the pyrolysis-front position propagates downwind is identified with the rate of the flame spread; this rate is the key information sought from solution of this nonstandard Stefan-type problem. The two-dimensional, unsteady phenomenon admits a similarity solution, such that the solution is expressible in terms of (1) a Blasius-type independent variable (familiar from boundary-layer studies), and (2) the ratio of stream...

The Thermal Capacity Effect Upon Transient Natural Convection in a Rectangular Cavity

Abstract: Transient natural convection in a rectangular enclosure is analyzed using a finite difference scheme. The enclosure is adiabatic and filled with water. The buoyancy induced flow is generated by a flat vertical uniform flux surface that has a finite thermal capacity. The full two-dimensional equations representing conservation of mass, momentum, and energy are solved in their time-dependent form. The solution technique used is a modified finite difference procedure very similar to Simple Arbitrary Lagrangian Eulerian (SALE) technique. Two values of surface thermal capacity are investigated, each resulting in a different flow regime during the transient. At short times a simple one-dimensional conduction regime is found to occur. As the leading edge effects arrive at any downstream location the conduction regime is terminated and true convection effects set in. At intermediate times a different flow regime is detected, namely a steady two-dimensional regime that approaches the steady state similarity solution for a similarly heated surface immersed in an infinite fluid medium. Excellent agreement is found with previous analyses and measurements during the early and intermediate transients.

Some generalized similarity solutions for plastic flow

Abstract: In kinematically determined regimes, the velocity equations for axially symmetric flow of a perfectly plastic solid which obeys Tresca's yield condition and associated flow rule possess many symmetries. These equations apply also to the flow of granular materials according to the ‘double-shearing’ theory of Spencer [1], [2]. Using Lie group methods, five classes of generalized self-similar solutions are identified. Special cases are the two types of solution due to Lippmann [3] [4] and the flow past a cone found by Spencer [5]. For each class of solution, determination of the stream function requires the solution of a second-order ordinary differential equation, which can in each case be reduced to the analysis of a first-order equation. Examples of the flow fields and corresponding streamlines for three of the four newly determined cases are computed numerically.

Isogroup and 'general' similarity solution of a nonlinear diffusion equation

Abstract: For a class of one-dimensional nonlinear diffusion equations, where the diffusion coefficient varies as some power of the dependent variable, the invariance group and its Lie algebra are given. The isovector fields which generate the isogroup are then used to derive a 'general' similarity solution. For a particular case the solution can be reduced to a one-parameter group solution which is in full agreement with previously published results.

A Comparison of Approximate and Exact Solutions for Homogeneous Irreversible Chemical Reaction in the Laminar Boundary Layer

Abstract: A comparison of approximate and exact solutions for homogeneous irreversible chemical reaction in the laminar boundary layer flow has been made. By using the Method of Weighted Residuals, approximate analytical expressions for the velocity and concentration profiles were developed for the case of a laminar boundary layer flow over a flat plate at zero incidence angle, where isothermal, homogeneous, nth order chemical reaction takes place. Both the Subdomain and Galerkin methods were employed to examine the influence of the choice of the weighting function on the predictions, and to provide a means for improving the solutions systematically. The problem was also solved numerically for the case of first order reaction by using a similarity solution for the hydrodynamic flow and a power series expansion method for the mass transfer. The analytical results were compared with the exact solutions in order to evaluate the accuracy of the approximate analytical solutions. The Method of Weighted Residuals provided...