Showing papers on "Similarity solution published in 1994"

Heat transfer characteristics of a continuous stretching surface

Abstract: The similarity solutions for the governing ordinary differential equations of the boundary layer corresponding to a stretching surface have been reported. Power law velocity and temperature distribution were assumed for velocity exponent 3≥m≥−0.41176, −1.1≥m≥−3, and for temperature exponent 3≥n≥−3. Solutions have been found forn=0 and allm where heat transferred from the stretching surface to the ambient. The direction and amount of heat flow were found to be dependent on the magnitude ofn andm for the same Prandtl number. Nusselt number increases with increasingm andPr for uniform and variable surface temperature however, for uniform surface heat flux it decreases with increasingm for constantPr.

Similarity solutions for drained a undrained cavity expansions in soils

Abstract: A procedure for analysing the expansion of cylindrical and spherical cavities in hardening/softening soils is described, with the aim of providing a realistic and theoretically sound model on which to base analyses of the action of penetrometers and piles. Similarity solution methods are used to analyse the expansion of created cavities from zero initial radius. They circumvent the need to solve partial differential equations, and require the use of relatively simple computational methods involving the numerical solution of a small number of first-order ordinary differential equations. The procedure can be applied to any of the existing plasticity models used to model the deformation of soils and sands, and is illustrated here by use of a modified Cam clay critical state model. Both drained and undrained expansions are studied. Attention is concentrated on the relation between the values of the far-field, in situ and state variables and of the ‘measurable’ stress components on the cavity wall. L'article d...

Iterated instabilities during droplet fission

Abstract: Recent observations indicate that the shape of a fluid interface undergoes repeated instabilities arbitrarily close to breakoff. We interpret this behavior as the result of successive instabilities of the similarity solution of Eggers [Phys. Rev. Lett. [bold 71], 3458 (1993)]. We show that the similarity solution is unstable to finite amplitude perturbations, with critical amplitude going to zero at the singularity. Thermal fluctuations in the fluid can trigger the instabilities.

Similarity solutions for flow and heat transfer of a viscoelastic fluid over a stretching sheet

Abstract: This paper presents a study of the flow and heat transfer of an incompressible second-order fluid past a stretching sheet using Dandapat and Gupta's boundary layer solution. Two cases are studied, namely, (i) the sheet with constant surface temperature (CST case) and (ii) the sheet with prescribed surface temperature (PST case). The solutions for the temperature and the heat transfer characteristics for large Prandtl number (σ) are obtained using a Runge-Kutta method of fourth order with step size Δz = 0.01.

Similarity analysis of boundary layer equations of a class of non-newtonian fluids

Abstract: A similarity analysis of three-dimensional boundary layer equations of a class of non-Newtonian fluids in which the stress is an arbitrary function of rates of strain is made. It is shown that under scaling transformation, for an arbitrary stress function, only 90° of wedge flow leads to similarity solutions, whereas for a specific more restricted form, similarity solutions exist for arbitrary wedge angles. In the case of spiral group transformation, no similarity solutions exist if we force the stress function to remain arbitrary after the transformation, whereas for a specific more restricted form, similarity solutions exist for arbitrary wedge angles. For both transformations, similarity equations for power-law and Newtonian fluids are presented as special cases of the analysis. Finally the conditions for invariance and the form of the stress function for a two-dimensional case are also presented.

An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces

Abstract: The fourth-order nonlinear partial differential equation for surface diffusion is approximated by a new integrable nonlinear evolution equation. Exact solutions are obtained for thermal grooving, subject to boundary conditions representing a section of a grain boundary. When the slope m of the groove centre is large, the linear model grossly overestimates the groove depth. In the linear model dimensionless groove depth increases linearly with m, but in the nonlinear model it approaches an upper limit. A nontrivial similarity solution is found for the limiting case of a thermal groove whose central slope is vertical

On one-dimensional planar and nonplanar shock waves in a relaxing gas

Abstract: The paper examines the evolutionary behavior of shock waves of arbitrary strength propagating through a relaxing gas in a duct with spatially varying cross section. An infinite system of transport equations, governing the strength of a shock wave and the induced discontinuities behind it, are derived in order to study the kinematics of the shock front. The infinite system of transport equations, when subjected to a truncation approximation, provides an efficient system of only finite number of ordinary differential equations describing the shock propagation problem. The analysis, which accounts for the dynamical coupling between the shock fronts and the flow behind them, describes correctly the nonlinear steepening effects of the flow behind the shocks. Effects of relaxation on the evolutionary behavior of shocks are discussed. The first‐order truncation approximation accurately describes the decay behavior of weak shocks; the usual decay laws for weak shocks in a nonrelaxing gas are exactly recovered. The results concerning shocks of arbitrary strength are compared with the characteristic rule. In the limit of vanishing shock strength, the transport equation for the first‐order induced discontinuity leads to an exact description of an acceleration wave. In the strong shock limit, the second‐order truncation criterion leads to a propagation law for imploding shocks which is in agreement (within 5% error) with the Guderley’s exact similarity solution.

Solution of the boundary layer equation for a power law fluid in magneto-hydrodynamics

Abstract: The problem of a non-Newtonian power law conducting fluid past a semi-finite plate, in the presence of an external electromagnetic field, is studied. The electric conductivity is taken as a function of the velocity. The method of expansion for a small parameter is used to obtain the solution and for a special form of the free stream velocity, the method of similarity solutions is used to obtain the exact solutions of the boundary layer equations associated with this problem in a closed form. It was found that the presence of the electromagnetic field reduces the velocity of the fluid.

The effect of viscosity on the height of disks floating above an air table

Abstract: A similarity solution is used to analyse the flow in the thin gap between a floating disk and an air table, the similarity solution being that for an axisymmetric stagnation point but with an upper boundary condition within the boundary layer. Viscous effects increase the height of the floating disk by 20% even at a Reynolds number of 50. Theoretical predictions are compared with experimental observations. The effect of the pressure distribution under the disk on the air flux through the table is examined.

Flow and reactive transport in porous media induced by well injection: Similarity solution

Abstract: The authors study concentration profiles of solutes undergoing equilibrium absorption in the vicinity of a water well. For the case of a contamination event, the limit problem of vanishing well radius, which is of self-similar nature, is analysed in detail. Existence, uniqueness, and qualitative properties of solutions of the corresponding ordinary differential equations are shown. Some numerical examples are also presented.

A similarity solution for the Direct Interaction Approximation and its relationship to renormalization‐group analyses of turbulence

Abstract: The application of Kraichnan’s Direct Interaction Approximation [J. Fluid Mech. 5, 497 (1959)] to the inertial range of Navier–Stokes turbulence is examined. A similarity form valid in the inertial range is presented, along with several approximations permitting explicit solutions. The relationship of the present results to some recent renormalization‐group calculations is discussed. It is shown that these renormalization‐group calculations misrepresent the Direct Interaction Approximation’s nonlinear interaction and that the addition of an artificial stirring force is unnecessary.

On nonlinear diffusion in fractal structures

Abstract: An exact similarity solution for nonlinear diffusion in fractal sructures, in the presence of absorption, is presented and disscused. The concentration distribution in a spherical symmetry geometry for the Cauchy problem, corresponding to an instantaneous point source solution, reveals the occurence of traveling wave characteristics. The conditions for the existence of these diffusive waves are shown in terms of nonlinear and fractal effects. The absorption effect gives rise to a spatial localization of the moving concentration front.

Novel similarity solutions of the sonic small-disturbance equation with applications to airfoil transonic aerodynamics

Abstract: Similarity solutions of the small-disturbance equation for a two-dimensional near-sonic potential flow are studied The analysis uses basic similarity solutions of the problem in the hodograph plane The first-order similarity solution of the velocity potential in the physical plane is described analytically by a parametric representation in terms of the hodograph-similarity variable and function The second-order similarity term is governed by a linear equation and is described analytically as a product of two functions: one is the power of the basic hodograph-similarity function, and the other is a solution of the basic hodograph-hypergeometric equation with a different constant The application of the present solutions to the far-field approximation of a sonic flow about a thin airfoil results in a relation between Frankl's special similarity parameter and the hodograph-similarity variable The new solutions are also applied to the problem of a near-sonic small-disturbance flow in the surroundings of a

Flow In Porous Media

Abstract: The flow in porous media is hard to describe. Based on a representative elementary volume, we use conservation laws of porous media to construct one-phase models (which models the flow through a porous medium where only one liquid or gas is present) and two-phase models. Both lead to the same partial differential equation for the saturation of a phase, which has an equilibrium and a non-equilibrium form. We analytically solve the equilibrium form using similarity solutions, this gives us useful results. For the non-equilibrium form we use a numerical approach to find a similarity solution. With the results we can say how the water distributes in some porous media.

Effects of longitudinal heat conduction of a vertical thin plate in a natural convective cooling process

Abstract: This paper analyzes the cooling process of a vertical thin plate caused by a free convective flow, taking into account the effects of both longitudinal and transversal heat conduction in the plate Due to the finite thermal conductivity of the plate, a longitudinal temperature gradient arises within it, which prevents any similarity solution in the boundary layer, changing the mathematical character of the problem from parabolic to elliptic, for large values of the Rayleigh number The energy balance equations are reduced to a system of three differential equations with two parameters: the Prandtl number and a non-dimensional plate thermal conductivity α In order to obtain the evolution of the temperature of the plate as a function of time and position, the coupled balance equations are integrated numerically for several values of the parameters, including the cases of very good and poor conducting plates The results obtained, are compared with an asymptotic analysis based on the multiple scales technique carried out for the case of a very good conducting plate There is at the beginning a fast transient in non-dimensional time scale of order α−1 followed by a slow non-dimensional time scale of order unity, which gives the evolution of the cooling process Good agreement is achieved even for values of the conduction parameter α of order unity The asymptotic solution allows us to give closed form analytical solution for the plate temperature evolution in time and space The overall thermal energy of the plate decreases faster for smaller values of α

Inverse method for the investigation of nonlinear instabilities associated with negative-energy modes

Abstract: Earlier work on explosively unstable similarity solutions of Hamilton's equations with Hamiltonians homogeneous of degree [ital N] and satisfying resonance conditions is applied to study the nonlinear stability of linearly stable equilibria with neighboring positive- and negative-energy waves. A multiple-time-scale expansion near equilibrium yields a Hamiltonian system of the assumed structure. In the inverse method an explosively unstable similarity solution is assumed and one solves for the coefficients of the terms in a Hamiltonian of some given structure. Through some general arguments and many examples one concludes that explosively unstable solutions occur generally for wide ranges of coefficient values. Hence the original equilibrium is nonlinearly unstable for wide ranges of interaction parameters.

Free convection in a saturated porous medium beyond the similarity solution

Abstract: The non-similar problems associated with a non-isothermal vertical flat plate embedded in a fluid-saturated porous medium were considered to assess the performances of the two distinctive boundary layer solution methods, namely, the local similarity solution and the integral method. The results generated from these two approximate solution methods are compared against the results from a two-point finite difference and those based on a Merk-type series expansion. Comparison of the results reveals that both integral and local similarity methods perform excellently. Especially, the accuracy acquired by the local similarity solution is so high that the difference between the results from the local similarity solution and those from the two-point finite difference and local non-similarity solution methods is hardly discernible for the case of monotonic increasing wall temperature.

Similarity solutions to rotating Couette flow with power law fluids

Abstract: The nonlinear rheological effects of a power law fluid in rotating Couette flow are addressed. The governing equations are derived and exact similarity solutions to the Cauchy problem for the angular velocity and shear stress distributions are shown and discussed. The case when the flow is generated by a line impulse of angular momentum is considered. The existence of a travelling wave solution for a shear thickening fluid is shown.

Self‐similar solutions for axisymmetric jets using two turbulence models

Abstract: Similarity solutions for incompressible axisymmetric jets using a k–ϵ and a constant eddy diffusivity turbulence models are considered. For the k–ϵ model, the governing equations are very complex. Therefore, a transformation that simplifies these equations and makes them amenable to efficient numerical solution is used. Results for the velocity, turbulent kinetic energy and dissipation rate are obtained. Also, velocity decay rate, growth rate, entrainment and kinetic energy decay rate are determined. A comparison with experimental data and other works utilizing a parabolic marching asymptotic solution to the full partial differential equations is made. This comparison shows that similarity solutions are more accurate than solutions using numerical marching procedures.