Showing papers on "Similarity solution published in 2012"
TL;DR: The boundary layers over a continuously stretching sheet with a power law surface velocity were revisited for a sheet with variable thickness and the non-flatness of the stretching surface has significant impacts on the boundary layer development along the wall, on the velocity profiles, and on the shear stress distribution in the fluid.
228 citations
TL;DR: In this article, heat and mass transfer analysis for boundary layer stagnation-point flow over a stretching sheet in a porous medium saturated by a nanofluid with internal heat generation/absorption and suction/blowing is investigated.
175 citations
TL;DR: In this paper, the steady two-dimensional stagnation point flow of a water-based nanofluid over an exponentially stretching/shrinking sheet in its own plane is investigated. And the effects of the solid volume fraction φ and the stretching parameter λ on the fluid flow and heat transfer characteristics are thoroughly examined.
132 citations
TL;DR: In this paper, the boundary-layer flow and heat transfer in a viscous fluid containing metallic nanoparticles over a nonlinear stretching sheet are analyzed, where the stretching velocity is assumed to vary as a power function of the distance from the origin.
Abstract: The boundary-layer flow and heat transfer in a viscous fluid containing metallic nanoparticles over a nonlinear stretching sheet are analyzed. The stretching velocity is assumed to vary as a power function of the distance from the origin. The governing partial differential equation and auxiliary conditions are reduced to coupled nonlinear ordinary differential equations with the appropriate corresponding auxiliary conditions. The resulting nonlinear ordinary differential equations (ODEs) are solved numerically. The effects of various relevant parameters, namely, the Eckert number Ec, the solid volume fraction of the nanoparticles ϕ, and the nonlinear stretching parameter n are discussed. The comparison with published results is also presented. Different types of nanoparticles are studied. It is shown that the behavior of the fluid flow changes with the change of the nanoparticles type.
86 citations
TL;DR: In this article, the combined effects of Navier slip and Newtonian heating on an unsteady hydromagnetic boundary layer stagnation point flow towards a flat plate in the presence of a magnetic field are studied.
Abstract: The combined effects of Navier slip and Newtonian heating on an unsteady hydromagnetic boundary layer stagnation point flow towards a flat plate in the presence of a magnetic field are studied. The self-similar equations are obtained using similarity transformations and solved numerically by a shooting algorithm with a Runge-Kutta Fehlberg integration scheme. The velocity profiles, temperature profiles, the local skin friction coefficient, and the local Nusselt number are computed and discussed in details for various values of the different parameters. Numerical results are presented both in tabular and graphical forms, illustrating the effects of these parameters on the thermal and concentration boundary layers. It is revealed that the thermal boundary layer thickens with a rise in the flow unsteadiness and as Newtonian heating intensifies, while the local skin friction and the rate of heat transfer at the plate surface change significantly due to the slip parameter.
81 citations
TL;DR: In this article, the laminar unsteady flow over a stretchable rotating disk with deceleration is investigated, and the three dimensional Navier-Stokes (NS) equations are reduced into a similarity ordinary differential equation group, which is solved numerically using a shooting method.
49 citations
TL;DR: Similarity analysis of diffusion of chemically reactive solute distribution in MHD boundary layer flow of an electrically conducting incompressible fluid over a porous flat plate is presented in this article, where the reaction rate of the solute is considered inversely proportional along the plate.
Abstract: Similarity analysis of diffusion of chemically reactive solute distribution in MHD boundary layer flow of an electrically conducting incompressible fluid over a porous flat plate is presented. The reaction rate of the solute is considered inversely proportional along the plate. Adopting the similarity transformation technique the governing equations are converted into the self-similar ordinary differential equations which are solved by shooting procedure using Runge-Kutta method. For increase of the Schmidt number the solute boundary layer thickness is reduced. Most importantly, the effects of reaction rate and order of reaction on concentration field are of conflicting natures, due to increasing reaction rate parameter the concentration decreases, but for the increase in order of reaction it increases. In presence of chemical reaction, the concentration profiles attain negative value when Schmidt number is large.
47 citations
TL;DR: Huang et al. as mentioned in this paper considered the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K*ρ) in Rd, d ⩾ 2, where K(r) = rγ/γ with γ > 2.
Abstract: In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K * ρ) in Rd, d ⩾ 2, where K(r) = rγ/γ with γ > 2. It was previously observed [Y. Huang and A. L. Bertozzi, “Self-similar blowup solutions to an aggregation equation in Rn,” J. SIAM Appl. Math. 70, 2582–2603 (2010)]10.1137/090774495 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ > 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 4, we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor. We then classify the stability of these simplex solutions and prove that two-dimensional (in-)stability implies n-dimensional (in-)stability.
42 citations
TL;DR: In this paper, the governing Navier-Stokes equations are transformed into a similarity equation, which is solved by a shooting method, and the solution is an exact solution to the unsteady Navier Stokes equations.
35 citations
TL;DR: In this article, a relatively heavy, non-Newtonian power-law fluid of flow behavior index n is released from a point source into a saturated porous medium above an horizontal bed; the intruding volume increases with time as tα.
Abstract: A relatively heavy, non-Newtonian power-law fluid of flow behavior index n is released from a point source into a saturated porous medium above an horizontal bed; the intruding volume increases with time as tα. Spreading of the resulting axisymmetric gravity current is governed by a non-linear equation amenable to a similarity solution, yielding an asymptotic rate of spreading proportional to t(α+n)/(3+n). The current shape factor is derived in closed-form for an instantaneous release (α = 0), and numerically for time-dependent injection (α ≠ 0). For the general case α ≠ 0, the differential problem shows a singularity near the tip of the current and in the origin; the shape factor has an asymptote in the origin for n ⩾ 1 and α ≠ 0. Different kinds of analytical approximations to the general problem are developed near the origin and for the entire domain (a Frobenius series and one based on a recursive integration procedure). The behavior of the solutions is discussed for different values of n and α. The shape of the current is mostly sensitive to α and moderately to n; the case α = 3 acts as a transition between decelerating and accelerating currents.
33 citations
TL;DR: In this paper, the effects of the material parameter and the convective parameter on the fluid flow and heat transfer characteristics are disscussed, and dual solutions are found for the shrinking case, while for the stretching case, the solution is unique.
Abstract: The problem of a steady laminar two-dimensional stagnation point flow towards a stretching/shrinking sheet in a micropolar fluid with a convective surface boundary condition is studied. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically using the Runge–Kutta–Fehlberg method with shooting technique. The effects of the material parameter and the convective parameter on the fluid flow and heat transfer characteristics are disscussed. It is found that the skin friction coefficient and the heat transfer rate at the surface decrease with increasing values of the material parameter. Moreover, dual solutions are found to exist for the shrinking case, while for the stretching case, the solution is unique. © 2011 Canadian Society for Chemical Engineering
TL;DR: In this article, a general formulation and solution of Navier-Stokes and energy equations are sought in the study of two-dimensional unsteady stagnation-point flow and heat transfer impinging on a flat plate when the plate is moving with variable velocity and acceleration toward main stream or away from it.
Abstract: General formulation and solution of Navier–Stokes and energy equations are sought in the study of two-dimensional unsteady stagnation-point flow and heat transfer impinging on a flat plate when the plate is moving with variable velocity and acceleration toward main stream or away from it. As an application, among others, this accelerated plate can be assumed as a solidification front which is being formed with variable velocity. An external fluid, along z-direction, with strain rate a impinges on this flat plate and produces an unsteady two-dimensional flow in which the plate moves along z-direction with variable velocity and acceleration in general. A reduction of Navier–Stokes and energy equations is obtained by use of appropriate similarity transformations. Velocity and pressure profiles, boundary layer thickness, and surface stress-tensors along with temperature profiles are presented for different examples of impinging fluid strain rate, selected values of plate velocity, and Prandtl number parameter.
TL;DR: In this paper, the shape of the interface between viscous fluids varies rapidly close to the upper boundary and depends weakly on the viscosity ratio within the interior of the flow.
Abstract: Gravitationally driven exchange flows of viscous fluids with different densities are analysed theoretically and investigated experimentally within a horizontal channel Following initiation from rest when there is a vertical boundary dividing the two fluids, the denser fluid slumps under the less dense along the underlying boundary, while the less dense fluid intrudes along the upper boundary The motion is driven by the pressure gradients associated with the density differences between the two fluids, resisted by viscous stresses, and mathematically modelled by a similarity solution that depends on the ratio of the viscosities of the two fluids When the viscosity of the less dense fluid is much smaller than the viscosity of the denser fluid, the shape of the interface between the fluids varies rapidly close to the upper boundary and depends weakly on the viscosity ratio within the interior of the flow Matched asymptotic expansions are employed in this regime to determine the shape of the interface and the rates of its propagation along the boundaries The similarity solutions are shown to be linearly stable and thus are expected to represent the intermediate asymptotics of the flow Experiments confirm the similarity form of solutions and demonstrate close agreement with the theoretical predictions when the viscosities of the fluids are comparable, but exhibit some discrepancies when the viscosities differ more substantially It is suggested that these discrepancies may be due to mixing between the fluids close to the boundaries, which is induced by the no-slip boundary condition Exchange flows within porous domains are also investigated to determine the shape of the interface as a function of the ratio of the viscosities of the two fluids and using asymptotic analysis, this shape is determined when this ratio is much larger, or smaller, than unity
TL;DR: In this paper, the pressure field during the early development of turbulent vortex rings at two Reynolds numbers is determined using temporally resolved two-dimensional and stereoscopic particle image velocimetry (PIV) and the pressure gradient terms are obtained by solving the incompressible Euler equation so that the drag coefficients of the vortex rings can be evaluated.
Abstract: In this paper the pressure field during the early development of turbulent vortex rings at two Reynolds numbers is determined using temporally resolved two-dimensional and stereoscopic particle image velocimetry (PIV). The pressure gradient terms are obtained by solving the incompressible Euler equation so that the drag coefficients of the vortex rings can be evaluated. Maxworthy (J. Fluid Mech., vol. 64, 1974, pp. 227–239) and Glezer & Coles (J. Fluid Mech., vol. 211, 1990, pp. 243–283) each developed models to describe the long-term physics of turbulent vortex rings: the former developed a semi-empirical model which permits loss of impulse via the shedding of vorticity into the wake whereas the latter developed a similarity model based on invariance of the hydrodynamic impulse. Maxworthy’s model implies that a significant correction to the similarity solution is required to account for the drag on the vortex ring bubble. We show that during the early development of the turbulent vortex rings the drag is very small and the similarity scaling can basically be retained.
TL;DR: In this article, an analytical solution of the Falkner-Skan equation with mass transfer and wall movement is obtained for a special case, namely a velocity power index of −1/3, with an algebraically decaying velocity profile.
Journal Article•
TL;DR: In this article, the effects of thermophoresis on MHD Mixed convection, heat, and mass transfer about an isothermal vertical flat plate embedded in a fluid-saturated porous medium in the presence of viscous dissipation were investigated.
Abstract: An analysis is presented to investigate the effects of thermophoresis on MHD Mixed convection, heat, and mass transfer about an isothermal vertical flat plate embedded in a fluid-saturated porous medium in the presence of viscous dissipation . The similarity solution is used to transform the problem under consideration into a boundary value problem of coupled ordinary differential equations, which are solved numerically by using the finite difference method. Numerical computations are carried out for the non-dimensional physical parameter. The results are analyzed for the effect of different physical parameters such as thermophoretic , MHD, mixed convection, Eckert number, inertia parameter, buoyancy ratio, and Schmid number on the flow, heat, and mass transfer characteristics.
TL;DR: In this article, a modified similarity solution is presented that incorporates the drop capillary-pressure term, and transient simulations of corner region profiles are shown to converge onto the new similarity solution better than that of Clasen et al.
Abstract: A characteristic feature of pinch-off of fluid threads is the formation of drops connected to thinning filaments. This phenomenon is encountered in a number of widely used applications requiring the production of drops such as electronics microfabrication via inkjet printing, spray coating/drying, and microarraying. In pinch-off of viscoelastic fluid threads, the region that connects the drops to the filaments develops into a sharp corner. Recently, Clasen et al. [J. Fluid Mech. 556, 283–308 (2006)]10.1017/S0022112006009633 showed that such a corner evolves self-similarly. They, however, neglected the capillary pressure in the drop. A modified similarity solution is presented here that incorporates the drop capillary-pressure term, and transient simulations of corner region profiles are shown to converge onto the new similarity solution better than that of Clasen et al. Indeed, the new similarity solution is valid in all the three regions: the drop, the corner, and the filament regions. Similarity solutio...
TL;DR: In this article, mathematical and numerical analyses are presented to investigate self-similarity solutions of a two-dimensional MHD boundary layer flow over a permeable surface, and the required boundary conditions to obtain a similarity solution, are detailed.
Abstract: In this paper mathematical and numerical analyses are presented to investigate self-similarity solutions of a two-dimensional MHD boundary layer flow over a permeable surface. Required boundary conditions to obtain a similarity solution, are detailed. In the case of the Nonlinear Density Temperature (NDT) parameter, the self-similarity solution may be multiple, and the requirement of appropriate conditions of the model control parameters, provides a global similarity solution. We will explain the underlying conditions for the existence of a solution, which leads to multiple solutions in the general case. We also give some numerical results to show the MHD influence on the solution stability.
01 Jan 2012
TL;DR: In this article, the authors examined the hydro magnetic boundary layer flow with heat and mass transfer over a vertical plate in the presence of magnetic field, chemical reaction and a convective heat exchange at the surface with the surrounding.
Abstract: This paper examined the hydro magnetic boundary layer flow with heat and mass transfer over a vertical plate in the presence of magnetic field, chemical reaction and a convective heat exchange at the sur- face with the surrounding has been studied. The similarity solution is used to transform the system of partial differential equations and an efficient numerical technique is implemented to solve the reduced system by using the Runge-Kutta fourth order method with shooting technique. A comparison with the previous results shows a very good agreement. The results are presented graphically and the conclusion is drawn that the flow field and other quantities of physical interest are significantly influenced by these parameters.
TL;DR: New similarity solution is found by implementing the SADE package for finding nonclassical symmetries of boundary layer equations for two-dimensional and radial flows by finding new similarity solution.
Abstract: The nonclassical symmetries of boundary layer equations for two-dimensional and radial flows are considered. A number of exact solutions for problems under
consideration were found in the literature, and here we find new similarity solution by implementing the SADE package for finding nonclassical symmetries.
TL;DR: The discrete system of equations for a chain consisting of a large number of spheres interacting via the Hertz force of index 3/2 in strain is examined in the very long wavelength limit, yielding an effective medium description.
Abstract: The discrete system of equations for a chain consisting of a large number of spheres interacting via the Hertz force of index 3/2 in strain is examined in the very long wavelength limit, yielding an effective medium description. The resulting continuum second-order equation of motion possesses a subset of simple waves obeying a first-order equation of reduced index 5/4. These simple waves appear not to have examined before. For a given initial strain, the simple wave solution prescribes initial sphere centroid velocities. Together the initial strain and velocities are used in the second-order discrete system. Results for shock wave development compare very well between the second-order discrete system (minus physically valid oscillations) and the reduced first-order equation. A second-order simulation of colliding waves examines the ability of waves to pass through each other, with a phase advance accruing during the collision process. An arbitrary initial condition is shown to evolve toward a universal similarity solution proportional to (x/t)(4). A closed-form solution is given including the complete history of the waveform, shock location, and amplitude.
TL;DR: In this article, a simplified model for two-phase flows of immiscible displacement processes in porous media is proposed to capture the interfacial fronts, which is given by explicit expressions and satisfies the continuity conditions of pressure and normal velocity across the interface.
Abstract: For two-phase flows of immiscible displacement processes in porous media, we proposed a simplified model to capture the interfacial fronts, which is given by explicit expressions and satisfies the continuity conditions of pressure and normal velocity across the interface. A new similarity solution for the interfacial evolution in the rectangular coordinate system was derived by postulating a first-order approximation of the velocity distribution in the region that the two-phase fluids co-exist. The interfacial evolution equation can be explicitly expressed as a linear function, where the slope of the interfacial equation is simply related to the mobility ratio of two-phase fluids in porous media. The application of the proposed solutions to predictions of interfacial evolutions in carbon dioxide injected into saline aquifers was illustrated under different mobility ratios and operational parameters. For the purpose of comparison, the numerical solutions obtained by level set method and the similarity solutions based on the Dupuit assumptions were presented. The results show that the proposed solution can give a better approximation of interfacial evolution than the currently available similarity solutions, especially in the situation that the mobility ratio is large. The proposed approximate solutions can provide physical insight into the interfacial phenomenon and be readily used for rapidly screening carbon dioxide storage capacity in subsurface formations and monitoring the migration of carbon dioxide plume.
01 Jan 2012
TL;DR: In this article, a similar analysis of the three dimensional incompressible laminar boundary layer flow of general class of non-Newtonian fluids is presented. But the analysis is restricted to the case of flow over wedge.
Abstract: Similarity analysis is made of the three dimensional incompressible laminar boundary layer flow of general class of non-Newtonian fluids. This work is an extension of previous analysis by Na and Hansen (1967), where the similarity solution of laminar three dimensional boundary layer equations of Power-law fluids was investigated. For the present flow situation, it is observed that the similarly solutions exist only for the case of flow over wedge. Further, it is also observed that for the more general case of the boundary layer flow of non-Newtonian fluids over anybody shapes yields non –similar solutions. Present similarity equations are well agreed with those available in literature.
TL;DR: In this article, the authors investigated the nature of the development of two-dimensional laminar flow of an incompressible fluid at the reversed stagnation point and obtained an asymptotic solution by neglecting the viscous terms.
Abstract: This paper investigates the nature of the development of two-dimensional laminar flow of an incompressible fluid at the reversed stagnation-point. Proudman and Johnson (1962) first studied the flow and obtained an asymptotic solution by neglecting the viscous terms. Robins and Howarth (1972) stated that this is not true in neglecting the viscous terms within the total flow field. Viscous terms in this analysis are now included, and a similarity solution of two-dimensional reversed stagnation-point flow is investigated by solving the full Navier-Stokes equations.
Journal Article•
TL;DR: In this article, a new jet linear approximation method is incorporated into the complex variable boundary element solver that describes the jet generated by the water entry of wedges to nearly simulate the free surface shape,jet and pressure distribution along the wetted area of the wedge contour.
Abstract: The water entry problem of the 2-D wedge with a constant speed is numerically analyzed based on the velocity potential theory.A new jet linear approximation method is incorporated into the complex variable boundary element solver that describes the jet generated by the water entry of wedges to nearly simulate the free surface shape,jet and pressure distribution along the wetted area of the wedge contour.And the same time Cauchy integral theorem is used as the integration equation,and the similarity solution as the initial condition.Some important numerical techniques are discussed in detail such as time marching solution,jet treatment,grid mesh generation and the free surface deformation updating.Finally,the pressure distribution and wave elevation for the wedges with different deadrise angles are gotten.The proposed method is validated through comparisons with the similarity solution and good approximations are found.
TL;DR: In this paper, the boundary value problems governing a similarity reduction for two situations involving unsteady laminar boundary layer flow due to a stretching surface in a quiescent viscous incompressible fluid were investigated and the existence of a solution for all relevant values of physical parameters was proved.
Abstract: This article considers two situations involving unsteady laminar boundary layer flow due to a stretching surface in a quiescent viscous incompressible fluid. In one configuration, the surface is impermeable with prescribed heat flux, in the other, the surface is permeable with prescribed temperature. The boundary value problems governing a similarity reduction for each of these situations are investigated and the existence of a solution is proved for all relevant values of physical parameters. The uniqueness of the solution is also proved for some (but not all) values of the parameters. Finally, a priori bounds are obtained for the skin friction coefficient and local Nusselt number.
TL;DR: In this article, a generalized three-dimensional similarity solution was proposed to describe the swept Hiemenz flow, which can impinge boundary layers at arbitrary wall-normal suction velocities using a rescaled similarity coordinate.
Abstract: The plane stagnation flow onto (Hiemenz boundary layer, HBL) and the asymptotic suction boundary layer flow over a flat wall (ASBL) are two boundary layer flows for which the incompressible Navier-Stokes equations are amenable to exact similarity solutions. The Hiemenz solution has been extended to swept Hiemenz flows by superposition of a third, spanwise-homogeneous sweep velocity. This solution becomes singular as the chordwise, tangential base flow component vanishes. In this limit, the homogeneous ASBL solution is valid, which however cannot describe the swept Hiemenz flow, because it does not contain any chordwise velocity. This work presents a generalized three-dimensional similarity solution which describes three-dimensional spanwise homogeneously impinging boundary layers at arbitrary wall-normal suction velocities, using a rescaled similarity coordinate. The HBL and the ASBL are shown to be two limits of this solution. Further extensions consist of oblique impingement or different boundary suction directions, such as slip or stretching walls. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
01 Jan 2012
TL;DR: In this article, the effects of physical parameters on the dimensionless velocity and temperature profiles are depicted graphically and analyzed in detail, and numerical values of physical quantities, such as the local skinfriction coefficient and the local Nusselt number are presented in tabular form.
Abstract: Steady laminar natural convection over a semi-infinite moving vertical plate in the presence of internal heat generation, viscous dissipation and a convective surface boundary condition in a porous medium 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 nonlinear partial differential equations have been transformed by a similarity transformation into a system of ordinary differential equations, which are solved numerically by using the shooting techniques with the forth order Runga-Kutta method. 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 skinfriction coefficient and the local Nusselt number are presented in tabular form.
26 Sep 2012
TL;DR: In this paper, a non-classical similarity transformation is employed to transform the Navier-Stokes equation into a nonlinear ordinary differential equation with specific boundary conditions, and the homptopy analysis method is employed for solving the resulting nonlinear differential equation to study the effects of the nanoparticle volume fraction and wall velocity on the flow velocity profile, the boundary layer thickness and the local skin friction coefficient.
Abstract: The present article discusses the characteristics of Newtonian water-base-Copper nano-fluid flowing over an infinite flat plate moving with a constant velocity in the direction of the flow. The non-classical similarity transformation is employed to transform the Navier-Stokes equation into a nonlinear ordinary differential equation with specific boundary conditions. The Homptopy analysis method (HAM) is employed to solve the resulting nonlinear differential equation to study the effects of the nanoparticle volume fraction and wall velocity on the flow velocity profile, the boundary layer thickness and the local skin friction coefficient. The existence and non-uniqueness of the solution as a function of the wall velocity will be also discussed.