Showing papers on "Similarity solution published in 2006"

The flow of an elastico-viscous fluid past a stretching sheet with partial slip

Abstract: An analysis is carried out to study the flow characteristics in an elastico-viscous fluid (Walters' liquid-B model) over a stretching sheet with partial slip. The flow is generated due to linear stretching of the sheet. Using suitable similarity transformations on the highly non-linear partial differential equations we derive exact analytical solution with appropriate boundary conditions. The important finding in this communication is the effect of partial slip on the velocity and skin friction coefficient.

Stagnation-point flow of upper-convected Maxwell fluids

Abstract: Two-dimensional stagnation-point flow of viscoelastic fluids is studied theoretically assuming that the fluid obeys the upper-convected Maxwell (UCM) model. Boundary-layer theory is used to simplify the equations of motion which are further reduced to a single non-linear third-order ODE using the concept of stream function coupled with the technique of the similarity solution. The equation so obtained was solved using Chebyshev pseudo-spectral collocation-point method. Based on the results obtained in the present work, it is concluded that the well-established but controversial prediction that in stagnation-point flows of viscoelastic fluids the velocity inside the boundary layer may exceed that outside the layer may just be an artifact of the rheological model used in previous studies (namely, the second-grade model). No such peculiarity is predicted to exist for the Maxwell model. For a UCM fluid, a thickening of the boundary layer and a drop in wall skin friction coefficient is predicted to occur the higher the elasticity number. These predictions are in direct contradiction with those reported in the literature for a second-grade fluid.

Time-dependent plumes and jets with decreasing source strengths

Abstract: The classical bulk model for isolated jets and plumes due to Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234, 1956, p. 1) is generalized to allow for time-dependence in the various fluxes driving the flow. This new system models the spatio-temporal evolution of jets in a homogeneous ambient fluid and Boussinesq and non-Boussinesq plumes in stratified and unstratified ambient fluids.Separable time-dependent similarity solutions for plumes and jets are found in an unstratified ambient fluid, and proved to be linearly stable to perturbations propagating at the velocity of the ascending plume fluid. These similarity solutions are characterized by having time-independent plume or jet radii, with appreciably smaller spreading angles ( is the conventional entrainment coefficient. These new similarity solutions are closely related to the similarity solutions identified by Batchelor (Q. J. R. Met. Soc., vol. 80, 1954, p. 339) in a statically unstable ambient, in particular those associated with a linear increase in ambient density with height.If the source buoyancy flux (for a rising plume) or source momentum flux (for a rising jet) is decreased generically from an initial to a final value, numerical solutions of the governing equations exhibit three qualitatively different regions of behaviour. The upper region, furthest from the source, remains largely unaffected by the change in buoyancy flux or momentum flux at the source. The lower region, closest to the source, is an effectively steady plume or jet based on the final (lower) buoyancy flux or momentum flux. The transitional region, in which the plume or jet adjusts between the states in the lower and upper regions, appears to converge very closely to the newly identified stable similarity solutions. Significantly, the predicted narrowing of the plume or jet is observed. The size of the narrowing region can be determined from the source conditions of the plume or jet. Minimum narrowing widths are considered with a view to predicting pinch-off into rising thermals or puffs.

Thermal boundary layer on an exponentially stretching continuous surface in the presence of magnetic field effect

Abstract: The thermal boundary layer on an exponentially stretching continuous surface with an exponential temperature distribution in the presence of the magnetic field effect is investigated numerically. The local similarity solution is applied to the governing equations. Comparisons with previously published work are made and the results are found to be in excellent agreement. Numerical results for temperature distribution and the local Nusselt number have been presented for different values of the governing parameters. In particular, it has been found that the magnetic field decreases the temperature difference at the wall of the stretching surface, while the Nusselt number decreases with it.

Self-Similarity in the Outer Region of Adverse-Pressure-Gradient Turbulent Boundary Layers

Abstract: This paper presents a consistent theory of self-similarity and of equilibrium in the outer region of turbulent boundary layers that explains recent experimental findings on the subject, including new ones presented here. The theory is first presented in a general form where the outer scales are left unspecified and it is not assumed that the mean velocity defect and the Reynolds stresses share a common velocity scale. It is shown that the main results of the traditional similarity theory remain valid even in this case. Common outer scaling with the Zagarola-Smits length and velocity scales is then chosen. A new pressure gradient parameter is introduced to characterize the local effect of the pressure gradient in all flow conditions including strong adverse-pressure-gradient conditions. By analyzing several adverse-pressure-gradient flow cases, it is shown that self-similarity of the mean velocity defect profile is reached in all cases in localized but significant flow regions. The same is, however, not true of the Reynolds stress profiles. In agreement with the similarity analysis, the self-similar velocity defect profile is found to be a function of the pressure gradient and most flows studied here are only in an approximate state of equilibrium in the region of self-similar defect profiles despite the excellent collapse of the profiles.

Scaling Phenomena in Fatigue and Fracture

Abstract: The general classification of scaling laws will be presented and the basic concepts of modern similarity analysis - intermediate asymptotics, complete and incomplete similarity - will be introduced and discussed. The examples of scaling laws corresponding to complete similarity will be given. The Paris scaling law in fatigue will be discussed as an instructive example of incomplete similarity. It will be emphasized that in the Paris law the powers are not the material constants. Therefore, the evaluation of the life-time of structures using the data obtained from standard fatigue tests requires some precautions.

Transient elastohydrodynamic drag on a particle moving near a deformable wall

Abstract: We examine the prescribed time-dependent motion of a rigid particle (a sphere or a cylinder) moving in a viscous fluid close to a deformable wall. The fluid motion is described by a nonlinear evolution equation, derived using lubrication theory, which is solved using numerical and asymptotic methods; a local linear pressure-displacement model describes the wall. When the particle moves from rest towards the wall, fluid trapping beneath the particle leads to an overshoot in the normal force on the particle; a similarity solution is used to describe trapping at early times and a multiregion asymptotic structure describes fluid draining at late times. When the particle is pulled from rest away from the wall, a peeling process (described by a quasi-steady travelling wave) determines the rate at which fluid can enter the growing gap between the particle and the wall, leading to a transient adhesive normal force. When a cylinder moves from rest transversely over the wall, transient peeling motion is again observed (especially when the wall is initially indented), giving rise to an overshoot in the transverse drag. Simulations for a translating sphere show highly nonlinear wall deformations characterized by a localized crescent-shaped ridge. Despite generating sharp transient deformations, we found no numerical evidence of finite-time choking events.

Self-Similar Solutions for Relativistic Shocks Emerging from Stars with Polytropic Envelopes

Abstract: We consider a strong ultrarelativistic shock moving through a star whose envelope has a polytrope-like density profile. When the shock is close to the star's outer boundary, its behavior follows the self-similar solution given by Sari for implosions in planar geometry. Here we outline this solution and find the asymptotic solution as the shock reaches the star's edge. We then show that the motion after the shock breaks out of the star is described by a self-similar solution remarkably like the solution for the motion inside the star. In particular, the characteristic Lorentz factor, pressure, and density vary with time according to the same power laws both before and after the shock breaks out of the star. After emergence from the star, however, the self-similar solution's characteristic position corresponds to a point behind the leading edge of the flow rather than at the shock front, and the relevant range of values for the similarity variable changes. Our numerical integrations agree well with the analytic results both before and after the shock reaches the star's edge.

Macropterons, micropterons and similarity reductions for the regularized Ostrovsky-Grimshaw model for fluids and plasmas with symbolic computation

Abstract: The Ostrovsky model is widely used to describe mechanical and physical problems such as internal or surface waves in the oceans and magnetic sounds in plasmas. This model has recently been Grimshaw-regularized for certain continuity in the mass field, while computerized symbolic computation becomes a branch of artificial intelligence. In this paper, some similarity reductions are found for the regularized Ostrovsky-Grimshaw model with symbolic computation, to a coupled set of nonlinear ordinary differential equations. The micropterons and macropterons are analytically presented and discussed, and have been found to contain certain solitonic cores plus a number of sinusoidal ``wings''. Examples are the micropterons and macropterons for fluid velocities in the wave propagation direction and transverse direction, respectively.

Mixed convection heat transfer from a vertical heated plate embedded in a sparsely packed porous medium

Abstract: An improved numerical study on mixed convection from a heated vertical plate embedded in a Newtonian fluid saturated sparsely packed porous medium is undertaken by considering the variation of permeability, porosity and thermal conductivity. The boundary layer flow in the porous medium is governed by the LapwoodForchheimer-Brinkman extended Darcy model. Similarity transformations are employed and the resulting ordinary differential equations are solved numerically by using a shooting algorithm with the Runge-KuttaFehlberg integration scheme to obtain velocity and temperature distributions. Besides, the skin friction and Nusselt number are also computed for various physical parameters governing the problem under consideration. It is found that the inertial parameter has a significant influence on decreasing the flow field, whereas its influence is reversed on the rate of heat transfer for all values of permeability parameter considered. Further, the results under the limiting conditions were found to be in good agreement with the existing ones.

Thermal radiation effects on magnetohydrodynamic mixed convection flow of a micropolar fluid past a continuously moving semi-infinite plate for high temperature differences

Abstract: An analysis is presented to study the effect of radiation on magnetohydrodynamic mixed convective steady laminar boundary layer flow of an optically thick electrically conducting viscous micropolar fluid past a moving semi-infinite vertical plate for high temperature differences. A uniform magnetic field is applied perpendicular to the moving plate. The density of the micropolar fluid is assumed to reduce exponentially with temperature. The usual Boussinesq approximation is neglected because of the high temperature differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting governing equations are transformed using a similarity transformation and then solved numerically by applying an efficient technique. The effects of radiation parameter R, magnetic parameter M, couple parameter Δ and density/temperature parameter n on the velocity, angular velocity and temperature profiles as well as the local skin friction coefficient, wall couple stress and the local Nusselt number are presented graphically and in tabular form.

Dam-Break Flow for Arbitrary Slopes of the Bottom

Abstract: The dam-break flow problem in the shallow-water approximation on an inclined bed for arbitrary slopes of the bottom is considered. An analytical solution for the spreading of the water fronts at the initial stages is given. A self-similar solution asymptotically valid at large time is also found. For intermediate times the problem is solved numerically by the method of characteristics.

Group method analysis of mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder

Abstract: The transformation group theoretic approach is applied to the system of equations governing the unsteady mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder. The application of a two-parameter group reduces the number of independent variables by two, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The possible forms of surface-temperature Tw, potential velocity U and sin Open image in new window with position and time are derived in steady and unsteady cases. New formulae of dimensionless temperature are presented using the group method analysis. Hiemenz and Falkner-Skan equations are obtained as special cases. The new similarity representations and similarity transformations in steady/unsteady states are obtained. The family of ordinary differential equations has been solved numerically using a fourth-order Runge-Kutta algorithm with the shooting technique. The effect of varying parameters governing the problem is studied.

Effect of slip on existence, uniqueness, and behavior of similarity solutions for steady incompressible laminar flow in porous tubes and channels

Abstract: The existence and multiplicity of similarity solutions for steady, fully developed, incompressible laminar flow in uniformly porous tubes and channels with one or two permeable walls is investigated from first principles. A fourth-order ordinary differential equation obtained by simplifying the Navier-Stokes equations by introducing Berman’s stream function [A. S. Berman, J. Appl. Phys. 24, 1232 (1953)] and Terrill’s transformation [R. M. Terrill, Aeronaut. Q. 15, 299 (1964)] is probed analytically. In this work that considers only symmetric flows for symmetric ducts; the no-slip boundary condition at porous walls is relaxed to account for momentum transfer within the porous walls. By employing the Saffman [P. G. Saffman, Stud. Appl. Math. 50, 93 (1971)] form of the slip boundary condition, the uniqueness of similarity solutions is investigated theoretically in terms of the signs of the guesses for the missing initial conditions. Solutions were obtained for all wall Reynolds numbers for channel flows whereas no solutions existed for intermediate values for tube flows. Introducing slip did not fundamentally change the number or the character of solutions corresponding to different sections. However, the range of wall Reynolds numbers for which similarity solutions are theoretically impossible in tube flows was found to be a weak function of the slip coefficient. Slip also weakly influenced the transition wall Reynolds number corresponding to flow in the direction of a favorable axial pressure gradient to one in the direction of an adverse pressure gradient. Momentum transfer from the longitudinal axis to the walls appears to occur more efficiently in porous channels compared to porous tubes even in the presence of slip.