Showing papers on "Similarity solution published in 2011"

Blasius and Sakiadis problems in nanofluids

Abstract: The classical problems of forced convection boundary layer flow and heat transfer past a semi-infinite static flat plate (Blasius problem) and past a moving semi-infinite flat plate (Sakiadis problem) using nanofluids are theoretically studied. The similarity equations are solved numerically for three types of metallic or nonmetallic nanoparticles such as copper (Cu), alumina (Al2O3), and titania (TiO2) in the base fluid of water with the Prandtl number Pr = 6.2 to investigate the effect of the solid volume fraction parameter φ of the nanofluids. Also, the case of conventional or regular fluid (φ = 0) with Pr = 0.7 is considered for comparison with known results from the open literature. The comparison shows excellent agreement. The skin friction coefficient, Nusselt number, and the velocity and temperature profiles are presented and discussed in detail. It is found that the solid volume fraction affects the fluid flow and heat transfer characteristics.

Second law analysis for variable viscosity hydromagnetic boundary layer flow with thermal radiation and Newtonian heating

Abstract: The present paper is concerned with the analysis of inherent irreversibility in hydromagnetic boundary layer flow of variable viscosity fluid over a semi-infinite flat plate under the influence of thermal radiation and Newtonian heating. Using local similarity solution technique and shooting quadrature, the velocity and temperature profiles are obtained numerically and utilized to compute the entropy generation number. The effects of magnetic field parameter, Brinkmann number, the Prandtl number, variable viscosity parameter, radiation parameter and local Biot number on the fluid velocity profiles, temperature profiles, local skin friction and local Nusselt number are presented. The influences of the same parameters and the dimensionless group parameter on the entropy generation rate in the flow regime and Bejan number are calculated, depicted graphically and discussed quantitatively. It is observed that the peak of entropy generation rate is attained within the boundary layer region and plate surface act as a strong source of entropy generation and heat transfer irreversibility.

Gravothermal collapse of isolated self-interacting dark matter haloes: N-body simulation versus the fluid model

Abstract: We make direct comparisons between Monte Carlo N-body simulations and analytic and numerical solutions of a conduction fluid (gaseous) model, for various isolated selfinteracting dark matter (SIDM) haloes. There is a disagreement between two methods on the sucient strength of collisionality to solve cuspy core problem, but we show that the two agree within 20% for isolated haloes. The N-body agrees very well with the analytical self-similar solution of gravothermal collapse in the fluid model by Balberg et al. (2002) when one free parameter, the coecient of thermal conduction C, is chosen to be 0.75. The density profile evolves self-similarly and the central density and velocity dispersion match the analytical solution perfectly with the predicted exponent fi = 2.19, which is also an asymptotic slope of the density profile. We also initialize the simulation and the 1D numerical calculation of the conducting fluid model with the Plummer’s model, the Hernquist profile and the NFW profile to show that the fluid model is applicable to more realistic density profiles. The central density at maximum core expansion and the collapse time agree within 20% in the long mean free path regime. As the mean free path become comparable to the system size, we see the delay in collapse rate as predicted. In this transitional regime, gravothermal collapse simulation agree with the fluid model if another prefactor of thermal conduction, b, in the short mean free path is set to 0.25. Monte Carlo N-body simulation and conducting fluid model agree with each other if two prefactors of thermal conduction C and b are calibrated by the N-body simulation. Our results validate the use of these two methods for collisional self-gravitating systems. Throughout the paper, the collision is assumed to be isotropic and velocity independent.

Scaling Transformations for Boundary Layer Flow near the Stagnation-Point on a Heated Permeable Stretching Surface in a Porous Medium Saturated with a Nanofluid and Heat Generation/Absorption Effects

Abstract: In this article, a similarity solution of the steady boundary layer flow near the stagnation-point flow on a permeable stretching sheet in a porous medium saturated with a nanofluid and in the presence of internal heat generation/absorption is theoretically studied. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary differential equations with the appropriate boundary conditions via Lie-group analysis. Copper (Cu) with water as its base fluid has been considered and representative results have been obtained for the nanoparticle volume fraction parameter \({\phi}\) in the range \({0\leq \phi \leq 0.2}\) with the Prandtl number of Pr = 6.8 for the water working fluid. Velocity and temperature profiles as well as the skin friction coefficient and the local Nusselt number are determined numerically. The influence of pertinent parameters such as nanofluid volume fraction parameter, the ratio of free stream velocity and stretching velocity parameter, the permeability parameter, suction/blowing parameter, and heat source/sink parameter on the flow and heat transfer characteristics is discussed. Comparisons with published results are also presented. It is shown that the inclusion of a nanoparticle into the base fluid of this problem is capable to change the flow pattern.

Dual Solutions in Unsteady Stagnation-Point Flow over a Shrinking Sheet

Abstract: An analysis is made to study the dual nature of solution of unsteady stagnation-point flow due to a shrinking sheet Using similarity transformations, the governing boundary layer equations are transformed into the self-similar nonlinear ordinary differential equations The transformed equations are solved numerically using a very efficient shooting method The study reveals the conditions of existence, uniqueness and non-existence of unsteady similarity solution The dual solutions for velocity distribution exist for certain values of velocity ratio parameter (c/a), and the increment in the unsteadiness parameter A increases the range of c/a where solution exists Also, with increasing A, the skin friction coefficient increases for the first solution and decreases for the second

Stretching liquid bridges with moving contact lines: The role of inertia

Abstract: Liquid bridges with moving contact lines are found in a variety of settings such as capillary feeders and high-speed printing. Although it is often assumed that the length scale for these flows is small enough that inertial effects can be neglected, this is not the case in certain applications. To address this issue, we solve the Navier-Stokes equations with the finite element method for the stretching of a liquid drop between two surfaces for non-zero Reynolds numbers. We consider an axisymmetric liquid bridge between a moving flat plate and either a stationary flat plate or a cavity. The contact lines are allowed to slip, and we evaluate the effect of the Reynolds number and contact angles on the transfer of liquid to the moving plate. In the case of two flat plates, we find that inertia forces the interface to map onto a similarity solution in a manner that shifts the breakup point toward the more wettable surface. Inertia and wettability are thus competing effects, with inertia driving fluid toward th...

Explicit and Approximated Solutions for Heat and Mass Transfer Problems with a Moving Interface

Abstract: The goal of this chapter is firstly to give a survey of some explicit and approximated solutions for heat and mass transfer problems in which a free or moving interface is involved. Secondly, we show simultaneously some new recent problems for heat and mass transfer, in which a free or moving interface is also involved. We will consider the following problems: 1. Phase-change process (Lame-Clapeyron-Stefan problem) for a semi-infinite material: i. The Lame-Clapeyron solution for the one-phase solidification problem (modeling the solidification of the Earth with a square root law of time); ii. The pseudo-steady-state approximation for the one-phase problem; iii. The heat balance integral method (Goodman method) and the approximate solution for the one-phase problem; iv. The Stefan solution for the planar phase-change surface moving with constant speed; v. The Solomon-Wilson-Alexiades model for the phase-change process with a mushy region and its similarity solution for the one-phase case; vi. The Cho-Sunderland solution for the one-phase problem with temperature-dependent thermal conductivity; vii. The Neumann solution for the two-phase problem for prescribed surface temperature at the fixed face; viii. The Neumann-type solution for the two-phase problem for a particular prescribed heat flux at the fixed face, and the necessary and sufficient condition to have an instantaneous phase-change process; ix. The Neumann-type solution for the two-phase problem for a particular prescribed convective condition (Newton law) at the fixed face, and the necessary and sufficient condition to have an instantaneous phase-change process; x. The similarity solution for the two-phase Lame-Clapeyron-Stefan problem with a mushy region. xi. The similarity solution for the phase-change problem by considering a density jump; xii. The determination of one or two unknown thermal coefficients through an overspecified condition at the fixed face for one or two-phase cases. xiii. A similarity solution for the thawing in a saturated porous medium by considering a density jump and the influence of the pressure on the melting temperature.

Mhd boundary layer flow due to an exponentially shrinking sheet

Abstract: This investigation deals with the MHD boundary layer flow due to an exponentially shrinking sheet. The sheet is assumed to be porous and a variable wall mass transfer is considered. The governing equations of motion are transformed into a nonlinear selfsimilar ordinary differential equation. Then the converted equation is solved numerically using the shooting method. The numerical computations yield that the requirement of wall mass suction for the steady flow is reduced when the magnetic field is imposed, i.e. the magnetic field itself delays the boundary layer separation and preserves the steady flow. In addition, dual solutions for the steady MHD flow are found for certain conditions. With the increasing magnetic parameter, the velocity increases for the first solution and decreases for the second solution. 1. Introduction. The viscous incompressible flow of Newtonian fluid due to a linearly stretching sheet was first investigated by Crane [1], who obtained an exact similarity solution. The work of Crane [1] was extended by many researchers. In this regard, Magyari and Keller [2] introduced a new type of the stretching sheet problem by considering the flow due to a sheet stretched exponentially in its own plane, and they investigated also the heat transfer characteristics for the flow taking the exponentially varying wall temperature. Elbashbeshy [3] also discussed the flow and heat transfer over an exponentially stretching surface in the presence of wall mass suction. Partha et al. [4] reported a similarity solution for mixed convective flow due to an exponentially stretching surface by taking into consideration the influence of viscous dissipation on convective transport. Later, Khan and

Similarity solution for natural convection from a moving vertical plate with internal heat generation and a convective boundary condition

Abstract: Steady laminar natural convection flow over a semi-infinite moving vertical plate in the presence of internal heat generation and a convective surface boundary condition is examined in this paper. It is assumed that the left surface of the plate is in contact with a hot fluid while the cold fluid on the right surface of the plate contains a heat source that decays exponentially with the classical similarity variable. The governing non-linear partial differential equations have been transformed by a similarity transformation into a system of ordinary differential equations, which are solved numerically by applying shooting iteration technique together with fourth order Runge-Kutta integration scheme. The effects of physical parameters on the dimensionless velocity and temperature profiles are depicted graphically and analyzed in detail. Finally, numerical values of physical quantities, such as the local skin-friction coefficient and the local Nusselt number are presented in tabular form.

Similarity Solution for Unsteady MHD Flow Near a Stagnation Point of a Three-Dimensional Porous Body with Heat and Mass Transfer, Heat Generation/Absorption and Chemical Reaction

Abstract: The problem of unsteady mixed convection heat and mass transfer near the stagnation point of a three-dimensional porous body in the presence of magnetic field, chemical reaction and heat source or sink is analyzed. An efficient, iterative, tri-diagonal implicit finite difference method is used to solve the transformed similarity equations in the boundary layer. Three cases were considered, namely, accelerating flow, decelerating flow and the steady-state case. The obtained results are presented in graphical and tabulated forms to illustrate the influence of the different physical parameters such as the magnetic field parameter, transpiration parameter, unsteadiness parameter, ratio of velocity gradients at the edge of the boundary layer parameter, heat generation/absorption parameter and the chemical reaction parameter on the velocity components in the x-and y- directions, temperature and concentration distributions, as well as the skin-friction coefficients and Nusselt and Sherwood numbers.

A group invariant solution for a pre-existing fluid-driven fracture in permeable rock

Abstract: The propagation of a two-dimensional pre-existing fracture in permeable rock by the injection of a viscous, incompressible Newtonian fluid is considered. The fluid flow in the fracture is laminar. By the application of lubrication theory, a partial differential equation relating the half-width of the fracture to the fluid pressure and leak-off velocity is derived. The model is closed by the adoption of the PKN formulation in which the fluid pressure is proportional to the fracture half-width. The partial differential equation admits four Lie point symmetries provided the leak-off velocity satisfies a first order linear partial differential equation. The solution of this equation yields the leak-off velocity as a function of the distance along the fracture and time. The group invariant solution is derived by considering a linear combination of the Lie point symmetries. The boundary value problem is reformulated as a pair of initial value problems. The model in which the leak-off velocity is proportional to the fracture half-width is considered. The working condition of constant pressure at the fracture entry is analysed in detail.

The boundary layers of an unsteady incompressible stagnation-point flow with mass transfer

Abstract: In the current work, the boundary layers of an unsteady incompressible stagnation-point flow with mass transfer were further investigated. Similarity transformation technique was used and the similarity equation group was solved using numerical methods. Interesting observation is that there are multiple solutions seen for negative unsteadiness parameters, β. The influences of mass transfer, unsteadiness parameter, and Prandtl numbers on velocity and temperature profiles, wall drag, and wall heat fluxes were investigated and analyzed. The asymptotic behaviors for the similarity equations in limiting situations were theoretically analyzed. It is found that solutions exist for all mass transfer parameters for β≥−1. For a certain mass transfer parameter, there are two solutions when βc<β<0; there is one solution for (β=βc)∪(β≥0); there is no solution for β<βc, where βc is a critical unsteadiness parameter dependent on mass transfer parameter.

The influence of thermal radiation on hydromagnetic Darcy-Forchheimer mixed convection flow past a stretching sheet embedded in a porous medium

Abstract: A boundary layer analysis is performed to study the influence of thermal radiation and buoyancy force on two-dimensional magnetohydrodynamic flow of an incompressible viscous and electrically conducting fluid over a vertical stretching sheet embedded in a porous medium in the presence of inertia effect. The governing system of partial differential equations is first transformed into system of ordinary differential equations using self-similarity transformation. A special form for magnetic field is chosen to obtain the similarity solution. The transformed boundary layer equations are solved numerically for some important values of the physical parameters. The present results are compared with the previously published papers and the results are found to be in excellent agreement. The important features of the flow, heat and mass transfer characteristics for different values of thermal radiation, porous permeability, magnetic field and buoyancy parameters are analyzed and discussed. The effects of various physical parameters on the skin friction coefficient, local Nusselt number and local Sherwood number are also presented. It is found that increase in the value of thermal radiation parameter R 1 increases the skin friction coefficient and Sherwood number whereas reverse trend is seen for the local Nusselt number.

Non-spherical similarity solutions for dark halo formation

Abstract: We carry out fully three-dimensional simulations of evolution from self-similar, spherically symmetric linear perturbations of a cold dark matter (CDM)-dominated Einstein-de Sitter universe. As a result of the radial orbit instability, the haloes which grow from such initial conditions are triaxial with major-to-minor axial ratios of the order of 3: 1. They nevertheless grow approximately self-similarly in time. In all cases, they have power-law density profiles and near-constant velocity anisotropy in their inner regions. Both the power-law index and the value of the velocity anisotropy depend on the similarity index of the initial conditions, the former as expected from simple scaling arguments. Halo structure is thus not 'universal' but remembers the initial conditions. On larger scales the density and anisotropy profiles show two characteristic scales, corresponding to particles at the first pericentre and at the first apocentre after infall. They are well approximated by the Navarro―Frenk―White model only for one value of the similarity index. In contrast, at all radii within the outer caustic the pseudo-phase-space density can be fitted by a single power law with an index which depends only very weakly on the similarity index of the initial conditions. This behaviour is very similar to that found for haloes formed from ACDM initial conditions and so can be considered approximately universal.

One‐dimensional freezing of nonheaving unsaturated soils: Model formulation and similarity solution

Abstract: [1] Freezing of unsaturated soils is associated with the formation of a moving freezing zone and liquid water flow toward the zone. An equilibrium thermodynamic formulation of coupled flow and heat transport in variably saturated partially frozen porous media is developed and a self-similar solution is derived for the case of a semi-infinite horizontal porous media column with a constant freezing temperature on one boundary. Solutions to the self-similar equations are derived using a Runge-Kutta solution procedure. The solution is found to yield two possible modes distinguished by zones composed of different combinations of ice, liquid water, and air. One of the modes contains three zones: a frozen zone (WI) with just ice and liquid water; a transition zone (AWI) with ice, liquid water, and air; and an unsaturated zone (AW) with liquid water and air. The second mode contains only the WI zone and the AW zone. It is found that the WI zone is a quintessential part of the solution. The AWI zone is found to exist when the advancement of the freezing zone is relatively fast, while it is absent when the zone advances slowly. Predictions of ice saturation and liquid water saturation with the self-similar solution are compared to published experimental data. Pore pressure is calculated as a linear combination of ice pressure and liquid water pressure, and the calculated figures are used to provide a condition for model limitation in the case of incipient ice lens formation. The developed similarity solution provides insight into the mechanics of liquid water movement and pore filling with ice and the conditions for incipient heaving.

A theoretical model of the pressure field arising from asymmetric intraglottal flows applied to a two-mass model of the vocal folds.

Abstract: A theoretical flow solution is presented for predicting the pressure distribution along the vocal fold walls arising from asymmetric flow that forms during the closing phases of speech. The resultant wall jet was analyzed using boundary layer methods in a non-inertial reference frame attached to the moving wall. A solution for the near-wall velocity profiles on the flow wall was developed based on a Falkner-Skan similarity solution and it was demonstrated that the pressure distribution along the flow wall is imposed by the velocity in the inviscid core of the wall jet. The method was validated with experimental velocity data from 7.5 times life-size vocal fold models, acquired for varying flow rates and glottal divergence angles. The solution for the asymmetric pressures was incorporated into a widely used two-mass model of vocal fold oscillation with a coupled acoustical model of sound propagation. Asymmetric pressure loading was found to facilitate glottal closure, which yielded only slightly higher values of maximum flow declination rate and radiated sound, and a small decrease in the slope of the spectral tilt. While the impact on symmetrically tensioned vocal folds was small, results indicate the effect becomes more significant for asymmetrically tensioned vocal folds.

Non-isobaric Marangoni boundary layer flow for Cu, Al2O3 and TiO2 nanoparticles in a water based fluid

Abstract: In this paper, a non-isobaric Marangoni boundary layer flow that can be formed along the interface of immiscible nanofluids in surface driven flows due to an imposed temperature gradient, is considered. The solution is determined using a similarity solution for both the momentum and energy equations and assuming developing boundary layer flow along the interface of the immiscible nanofluids. The resulting system of nonlinear ordinary differential equations is solved numerically using the shooting method along with the Runge-Kutta-Fehlberg method. Numerical results are obtained for the interface velocity, the surface temperature gradient as well as the velocity and temperature profiles for some values of the governing parameters, namely the nanoparticle volume fraction φ (0≤φ≤0.2) and the constant exponent β. Three different types of nanoparticles, namely Cu, Al2O3 and TiO2 are considered by using water-based fluid with Prandtl number Pr =6.2. It was found that nanoparticles with low thermal conductivity, TiO2, have better enhancement on heat transfer compared to Al2O3 and Cu. The results also indicate that dual solutions exist when β<0.5. The paper complements also the work by Golia and Viviani (Meccanica 21:200–204, 1986) concerning the dual solutions in the case of adverse pressure gradient.

Turbulent diffusion in tall tubes. I. Models for Rayleigh-Taylor instability

Abstract: Rayleigh-Taylor instability in high-aspect-ratio domains has been studied experimentally and a hierarchy of modelling approaches has been used to understand the dynamics of the problem. Part I examines the simplest case of initially homogenous layers above and below the Rayleigh-Taylor unstable interface. Part II examines the more complex case where one layer is stably stratified in density. Here, in Part I, we develop models for turbulent mixing induced by Rayleigh-Taylor instability based on a diffusion equation for density. By considering the force balance in the problem, and using Prandtl’s mixing length hypothesis, we compute a non-constant turbulent diffusivity, and this leads to a non-linear diffusion equation. We reiterate a h~t25 scaling and use this to develop a new similarity solution to the nonlinear diffusion equation in an infinite domain. To match experimental boundary conditions of a finite domain, we use numerical integration, and finally, we compare with implicit large eddy simulation.

Chemically reactive solute transfer in a boundary layer slip flow along a stretching cylinder

Abstract: This paper presents the distribution of a solute undergoing a first order chemical reaction in an axisymmetric laminar boundary layer flow along a stretching cylinder. Velocity slip condition at the boundary is used instead of no-slip condition. Similarity transformations are used to convert the partial differential equations corresponding to momentum and concentration into highly nonlinear ordinary differential equations. Numerical solutions of these equations are obtained by the shooting method. The velocity decreases with increasing slip parameter. The skin friction as well as the mass transfer rate at the surface is larger for a cylinder than for a flat plate.

Approximate solutions to MHD Falkner-Skan flow over permeable wall

Abstract: The magnetohydrodynamic (MHD) Falkner-Skan boundary layer flow over a permeable wall in the presence of a transverse magnetic field is examined. The approximate solutions and skin friction coefficients of the MHD boundary layer flow are obtained by using a method that couples the differential transform method (DTM) with the Pade approximation called DTM-Pade. The approximate solutions are expressed in the form of a power series that can be easily computed with an iterative procedure. The approximate solutions are tabulated, plotted for the values of different parameters and compared with the numerical ones obtained by employing the shooting technique. It is found that the approximate solution agrees very well with the numerical solution, showing the reliability and validity of the present work. Moreover, the effects of various physical parameters on the boundary layer flow are presented graphically and discussed.

Exact Solution of Three-Dimensional Unsteady Stagnation Flow on a Heated Plate

Abstract: XACT solutions of Navier–Stokes and energy equations regardingtheproblemofstagnation-point flowandheattransfer inthevicinityofa flatplateoracylinderarefoundintheliteraturefor many cases. Fundamental studies in which flows are readily superposed and/or the axisymmetric case were considered include the following papers presented in the literature: two-dimensional stagnation-point flow [1]; three-dimensional stagnation-point flow [2]; and axisymmetric stagnation flow on a circular cylinder [3]. Further exact solutions to the Navier–Stokes equations are obtained by superposition of the uniform shear flow and/or stagnation flow on a body oscillating or rotating in its own plane or cylinder, with or without suction. The examples are: superposition of twodimensional and three-dimensional stagnation-point flows [4]; superposition ofstagnation-point flowon a flatplateoscillating in its own plate, and also consideration of the case where the plate is stationary and the stagnation stream is made to oscillate done [5]; heat Transfer in an axisymmetric stagnation flow on a cylinder [6]; nonsimilar axisymmetric stagnation flow on a moving cylinder [7]; unsteady viscous flow in the vicinity of an axisymmetric stagnationpoint on a cylinder [8]; axisymmetric stagnation flow towards a movingplate[9];axisymmetricstagnation-point flowimpingingona transversely oscillating plate with suction [10]; axisymmetric stagnation-point flowandheattransferofaviscous fluidonamoving cylinder with time-dependent axial velocity and uniform transpiration [11] and also [12–14] can be mentioned. In this study the three-dimensional unsteady viscous stagnation flow and heat transfer in the vicinity of an accelerated flat plate are investigated by solving Navier–Stokes equations in Cartesian coordinate system. The external fluid, along z-direction, with strain rate a impinges on this flat plate while the plate is moving with variable velocity and acceleration along z-direction. A self-similar solution for the Navier–Stokes equations and energy equation is derivedinthisproblem.Areductionoftheseequationsisobtainedby use of these appropriate similarity transformations. The obtained ordinary differential equations are solved using numerical techniques. Velocity and pressure profiles, boundary-layer thickness and surface stress-tensors along with temperature profiles are presented for different values of impinging fluid strain rate, different plate velocities and Prandtl-number parameters for the steady state case.