A new family of unsteady boundary layers over a stretching surface

In this paper, a new family of unsteady boundary layers over a stretching flat surface was proposed and studied. This new class of unsteady boundary layers involves the flows over a constant speed stretching surface from a slot, and the slot is moving at a certain speed. Depending on the slot moving parameter, the flow can be treated as a stretching sheet problem or a shrinking sheet problem. Both the momentum and thermal boundary layers were studied. Under special conditions, the solutions reduce to the unsteady Rayleigh problem and the steady Sakiadis stretching sheet problem. Solutions only exist for a certain range of the slot moving parameter, α. Two solutions are found for −53.55° < α < −45°. There are also two solution branches for the thermal boundary layers at any given Prandtl number in this range. Compared with the upper solution branch, the lower solution branch leads to simultaneous reduction in wall drag and heat transfer rate. The results also show that the motion of the slot greatly affects the wall drag and heat transfer characteristics near the wall and the temperature and velocity distributions in the fluids.     

Axisymmetric boundary layers on non-uniformly heated horizontal surfaces

Similarity solutions for the velocity and temperature induced by axisymmetrically heated horizontal surfaces with a power law temperature distribution are derived and investigated. Two physical situations, a stationary and a radially moving temperature distribution are considered; both above and below the surface. Apart from a perturbation solution for small temperature differences for the latter, the equations have to be numerically integrated. The results include the non-existence of a solution below the surface for particularly large heat inputs. A major difference from previous work on cartesian geometries is that there must be a non-zero heat flux between the surface and fluid, which is related to the total heat flux through the fluid.     

On boundary layer flow of a Sisko fluid over a stretching sheet

Abstract

In this paper, the steady boundary layer flow of a non-Newtonian fluid over a nonlinear stretching sheet is investigated. The Sisko fluid model, which is combination of power-law and Newtonian fluids in which the fluid may exhibit shear thinning/thickening behaviors, is considered. The boundary layer equations are derived for the two-dimensional flow of an incompressible Sisko fluid. Similarity transformations are used to reduce the governing nonlinear equations and then solved analytically using the homotopy analysis method. In addition, closed form exact analytical solutions are provided for n = 0 and n = 1. Effects of the pertinent parameters on the boundary layer flow are shown and solutions are contrasted with the power-law fluid solutions.     

Laminar natural convection boundary layers on horizontal surfaces possessing a circular cut-out

Summary A first-order boundary-layer analysis, including variable properties effects, is given for laminar natural convection on a horizontal surface possessing a circular cut-out. The latter provides the leading edge of the boundary layer and it is shown that, initially, boundary layer development occurs as if the leading edge were straight. Far from the cut-out, boundary layer development is shown to become completely axially symmetric.     

New explicit approximate solution of MHD viscoelastic boundary layer flow over stretching sheet

In this paper, the study the momentum and heat transfer characteristics in an incompressible electrically conducting non‐Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly nonlinear coupled ordinary differential equations by similarity transformations. The resultant coupled highly nonlinear ordinary differential equations are solved by means of, homotopy analysis method (HAM) for constructing an approximate solution of heat transfer in magnetohydrodynamic (MHD) viscoelastic boundary layer flow over a stretching sheet with non‐uniform heat source. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiry parameter, which provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics. Copyright © 2012 John Wiley & Sons, Ltd.     

Mixed convection heat transfer in the boundary layers on an exponentially stretching surface with magnetic field

An analysis has been carried out to describe mixed convection heat transfer in the boundary layers on an exponentially stretching continuous surface with an exponential temperature distribution in the presence of magnetic field, viscous dissipation and internal heat generation/absorption. Approximate analytical similarity solutions of the highly non-linear momentum and energy equations are obtained. The present results are found to be in excellent agreement with previously published work on various special cases of the problem. Numerical results for temperature distribution and the local Nusselt number have been obtained for different values of the governing parameters. The numerical solutions are obtained by considering an exponential dependent stretching velocity and prescribed boundary temperature on the flow directional coordinate. The effects of various physical parameters like Prandtl number, Hartman number, Grashof number on dimensionless heat transfer characteristics are discussed in detail. In particular, it has been found that increase in Prandtl number decreases the skin-friction coefficient at the stretching surface, while increase in the strength of the magnetic field leads to increase in the local Nusselt number.     

A series method for the solution of laminar boundary layers on moving surfaces

Summary A general series method is reviewed for the solution of laminar, two-dimensional boundary layers on moving surfaces. Various methods for the determination of the radius of convergence of such series are discussed. Three illustrative examples are given, namely the vapour boundary layer induced by film condensation on a vertical plate and the outer layer boundary layers produced by natural convection at high Prandtl number on horizontal and vertical plates.

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Zusammenfassung Eine allgemeine Reihenmethode wird für die Lösung der laminaren zweidimensionalen Grenzschichten entlang einer bewegten Fläche geprüft. Verschiedene Methoden werden für die Bestimmung des Konvergenzradius solcher Reihen diskutiert. Drei Beispiele werden angegeben, nämlich die Dampfgrenzschicht, die durch Filmkondensation auf einer senkrechten Platte erzeugt wird, und die äussere Grenzschicht, die durch natürliche Konvektion bei höheren Prandtl-Zahlen auf vertikalen und horizontalen Platten erzeugt wird.

     

On boundary layers

The concept of boundary layers, introduced by A. Volberg in [7], is generalized from subsets of the unit disk to subsets of general non-tangentially accessible (NTA) domains. Capacitary conditions of Wiener type series of both necessary and sufficient type for boundary layers are presented and the connection between boundary layers and minimally thin sets is studied.     

Approximate analytical solutions of a class of boundary layer equations over nonlinear stretching surface

Third order nonlinear ordinary differential equations, subject to appropriate boundary conditions arising in fluid dynamics, are solved using three different methods viz., the Dirichlet series, method of stretching of variables, and asymptotic function method. Similarity transformations are used to convert the governing partial differential equations into nonlinear ordinary differential equations. The numerical results obtained from the above methods for various problems are given in terms of skin friction. Our study revealed that the results obtained from these methods agree well with those of direct numerical simulation of ordinary differential equations. Also, these methods have advantages over pure numerical methods in obtaining derived quantities such as velocity profile accurately for various values of the parameters at a stretch.     

Analytical solutions of the boundary layer flow of power-law fluid over a power-law stretching surface

This article discusses analytical solutions for a nonlinear problem arising in the boundary layer flow of power-law fluid over a power-law stretching surface. Using perturbation method analytical solution is presented for linear stretching surface. This solution covers large range of shear thinning and shear thickening fluids and matches excellently with the numerical solution. Furthermore, some new exact solutions are found for particular combination of m (power-law stretching index) and n (power-law fluid index). This leads to generalize the case of linear stretching to nonlinear stretching surface. The effects of fluid index n on the boundary layer thickness and the skin friction for nonlinear stretching surface is analyzed and discussed. It is observed that the boundary layer thickness and the skin friction coefficient increase as non-linear parameter n decreases. This study gives a new dimension to obtain analytical solutions asymptotically for highly nonlinear problems which to the best of our knowledge has not been examined so far.     

Finite element solution of double-diffusive boundary layer flow of viscoelastic nanofluids over a stretching sheet

This paper deals with the double-diffusive boundary layer flow of non-Newtonian nanofluid over a stretching sheet. In this model, where binary nanofluid is used, the Brownian motion and thermophoresis are classified as the main mechanisms which are responsible for the enhancement of the convection features of the nanofluid. The boundary layer equations governed by the partial differential equations are transformed into a set of ordinary differential equations with the help of group theory transformations. The variational finite element method (FEM) is used to solve these ordinary differential equations. We have examined the effects of different controlling parameters, namely, the Brownian motion parameter, the thermophoresis parameter, modified Dufour number, viscoelastic parameter, Prandtl number, regular Lewis number, Dufour Lewis number, and nanofluid Lewis number on the flow field and heat transfer characteristics. Graphical display of the numerical examine are performed to illustrate the influence of various flow parameters on the velocity, temperature, concentration, reduced Nusselt, reduced Sherwood and reduced nanofluid Sherwood number distributions. The present study has many applications in coating and suspensions, movement of biological fluids, cooling of metallic plate, melt-spinning, heat exchangers technology, and oceanography.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

Nano boundary layer equation with nonlinear Navier boundary condition

At the micro and nano scale the standard no slip boundary condition of classical fluid mechanics does not apply and must be replaced by a boundary condition that allows some degree of tangential slip. In this study the classical laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition replaced by a nonlinear Navier boundary condition. This boundary condition contains an arbitrary index parameter, denoted by n>0, which appears in the coefficients of the ordinary differential equation to be solved. The case of a boundary layer formed in a convergent channel with a sink, which corresponds to n=1/2, is solved analytically. Another analytical but non-unique solution is found corresponding to the value n=1/3, while other values of n for n>1/2 correspond to the boundary layer formed in the flow past a wedge and are solved numerically. It is found that for fixed slip length the velocity components are reduced in magnitude as n increases, while for fixed n the velocity components are increased in magnitude as the slip length is increased.     

Three-dimensional periodic boundary layers

The method of successive approximations used by Eichelbrenner and A?kovíc¹ for the study of unsteady three-dimensional boundary layer flow has been extended to analyse the periodic boundary layers in three dimensions. The analysis, which is valid for oscillations of small amplitude, shows some special features such as “steady streaming” flow in the first-order cross-flow similar to the one that has been predicted and observed by Schlichting for two-dimensional periodic boundary layers.     

On magneto-aerodynamic boundary layers

Zusammenfassung Der Einfluss eines senkrecht zur Schicht stehenden magnetischen Feldes auf die Geschwindigkeits- und Temperaturverteilung in einer Grenzschicht wird für einen Sonderfall der ?nderung der elektrischen Leitf?higkeit von Luft untersucht. Die Wirkung des magnetischen Feldes auf den Reibungswiderstand und den W?rmeübergang wird bei relativ zur Platte und zur Flüssigkeit in grosser Entfernung von der Platte festgehaltenem Magnetfeld ermittelt.      

Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink

This paper presents the study of momentum and heat transfer characteristics in a hydromagnetic flow of viscoelastic liquid over a stretching sheet with non-uniform heat source, where the flow is generated due to a linear stretching of the sheet and influenced by uniform magnetic field applied vertically. Here an analysis has been carried out to study the effect of magnetic field on the visco-elastic liquid flow and heat transfer over a stretching sheet with non-uniform heat source. The non-linear boundary layer equation for momentum is converted into ordinary differential equation by means of similarity transformation and is solved exactly. Heat transfer differential equation is also solved analytically. The effect of magnetic field on velocity, skin friction and temperature profiles are presented graphically and discussed.     

Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface

In the present investigation we have analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) Prescribed exponential order surface temperature (PEST) and (2) Prescribed exponential order heat flux (PEHF). The highly nonlinear coupled partial differential equations of Jeffrey fluid flow along with the energy equation are simplified by using similarity transformation techniques based on boundary layer assumptions. The reduced similarity equations are then solved analytically by the homotopy analysis method (HAM). The convergence of the HAM series solution is obtained by plotting (^h/_2p)\hbar-curves for velocity and temperature. The effects of physical parameters on the velocity and temperature profiles are examined by plotting graphs.     

Thin-film flow over a nonlinear stretching sheet

A thin viscous liquid film flow is developed over a stretching sheet under different nonlinear stretching velocities. An evolution equation for the film thickness, is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. A comparison is made with the analytic solution obtained in [B. S. Dandapat, A. Kitamura, B. Santra, “Transient film profile of thin liquid film flow on a stretching surface”, ZAMP, 57, 623-635 (2006)]. It is observed that all types of stretching produce film thinning but non-monotonic stretching produces faster thinning at small distance from the origin. The velocity u along the stretching direction strongly depends on the distance along the stretching direction and the Froude number.