Symmetries of boundary layer equations of power-law fluids of second grade

A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.     

SIMILARITY SOLUTIONS OF BOUNDARY LAYER EQUATIONS FOR A SPECIAL NON-NEWTONIAN FLUID IN A SPECIAL COORDINATE SYSTME

Two dimensional equations of steady motion for third order fluids are expressed in a special coordinate system generated by the potential flow corresponding to an inviscid fluid. For the inviscid flow around an arbitrary object, the streamlines are the phicoordinates and velocity potential lines are psi-coordinates which form an orthogonal curvilinear set of coordinates. The outcome, boundary layer equations, is then shown to be independent of the body shape immersed into the flow. As a first approximation, assumption that second grade terms are negligible compared to viscous and third grade terms. Second grade terms spoil scaling transformation which is only transformation leading to similarity solutions for third grade fluid. By ~sing Lie group methods, infinitesimal generators of boundary layer equations are calculated. The equations are transformed into an ordinary differential system. Numerical solutions of outcoming nonlinear differential equations are found by using combination of a Runge-Kutta algorithm and shooting technique.     

Heat and mass transfer by natural convection at a stagnation point in a porous medium considering Soret and Dufour effects

Dufour and Soret effects on flow at a stagnation point in a fluid-saturated porous medium are studied in this paper. A two dimensional stagnation-point flow with suction/injection of a Darcian fluid is considered. By using an appropriate similarity transformation, the boundary layer equations of momentum, energy and concentration are reduced to a set of ordinary differential equations, which are solved numerically using the Keller-box method, which is a very efficient finite differences technique. Nusselt and Sherwood numbers are obtained, together with the velocity, temperature and concentration profiles in the boundary layer. For the large suction case, asymptotic analytical solutions of the problem are obtained, which compare favourably with the numerical solutions. A critical view of the problem is presented finally.     

Axisymmetric flow and heat transfer of the Carreau fluid due to a radially stretching sheet: Numerical study

The prime objective of this article is to study the axisymmetric flow and heat transfer of the Carreau fluid over a radially stretching sheet. The Carreau constitutive model is used to discuss the characteristics of both shear-thinning and shear-thickening fluids. The momentum equations for the two-dimensional flow field are first modeled for the Carreau fluid with the aid of the boundary layer approximations. The essential equations of the problem are reduced to a set of nonlinear ordinary differential equations by using local similarity transformations. Numerical solutions of the governing differential equations are obtained for the velocity and temperature fields by using the fifth-order Runge–Kutta method along with the shooting technique. These solutions are obtained for various values of physical parameters. The results indicate substantial reduction of the flow velocity as well as the thermal boundary layer thickness for the shear-thinning fluid with an increase in the Weissenberg number, and the opposite behavior is noted for the shear-thickening fluid. Numerical results are validated by comparisons with already published results.     

Effect of non-uniform heat source on MHD heat transfer in a liquid film over an unsteady stretching sheet

An analysis has been carried out to study the magnetohydrodynamic boundary layer flow and heat transfer characteristics of a laminar liquid film over a flat impermeable stretching sheet in the presence of a non-uniform heat source/sink. The basic unsteady boundary layer equations governing the flow and heat transfer are in the form of partial differential equations. These equations are converted to non-linear ordinary differential equations using similarity transformation. Numerical solutions of the resulting boundary value problem are obtained by the efficient shooting technique. The effects of magnetic and the non-uniform heat source/sink parameters on the dynamics are discussed. Findings of the paper reveal that non-uniform heat sinks are better suited for effective cooling of the stretching sheet. Skin friction coefficient and the local Nusselt number are also explored for typical values of magnetic and non-uniform heat source/sink parameters. The results are in excellent agreement with the earlier published works, under some limiting cases.     

Three-dimensional laminar boundary layer on a blunt body

We consider the Prandtl laminar boundary layer which occurs with stationary flow about a blunted cone at an angle of attack. The solution of the Prandtl equations is sought using a finite difference method. It is found that a smooth solution of the problem exists only in the vicinity of the rounded nose of the body, while far from the nose the solutions acquire a singularity; in the problem symmetry plane (on the downwind side) there is a discontinuity of the first derivatives of the velocity components and the density. In the study of the Prandtl boundary layer in the problem of stationary flow about a pointed cone at an angle of attack, it has been shown [1] that the self-similar solution (dependent on two independent variables) of the Prandtl equations has a discontinuity of the first derivatives in the problem symmetry plane (on the downwind side of the cone). The suggestion has been made that in the three-dimensional problem of flow about a blunt cone at an angle of attack the solutions of the Prandtl equations may also be discontinuous. The present study was carried out to clarify the nature of the behavior of the solutions of the three-dimensional Prandtl equations. To this end we considered stationary supersonic flow of an ideal gas past a blunted cone. The results of this study (as well as those of [1]) were obtained using a numerical, finite-difference method. However, an analysis of the numerical results (investigation of the scheme stability, reduction of step size, etc.) shows that the properties of the solutions of the finite-difference equations are not in this case a result of numerical effects, but reflect the behavior of the solutions of the differential equations. The mathematical problem on the boundary layer which is considered in this study will be formulated in §2; this formulation is due to K. N. Babenko.     

Heat transfer over an unsteady stretching surface

Similarity solution of the laminar boundary layer equations corresponding to an unsteady stretching surface have been studied. The governing time-dependent boundary layer are transformed to ordinary differential equations containg Prandtl number and unsteadiness parameter. The effect of various govern-ing parameters such as Prandtl number and unsteadiness param-eter which determine the velocity and temperature profiles and heat transfer coefficient are studied.     

Boundary Layer for the Navier-Stokes-alpha Model of Fluid Turbulence

We study boundary-layer turbulence using the Navier-Stokes-alpha model obtaining an extension of the Prandtl equations for the averaged flow in a turbulent boundary layer. In the case of a zero pressure gradient flow along a flat plate, we derive a nonlinear fifth-order ordinary differential equation, which is an extension of the Blasius equation. We study it analytically and prove the existence of a two-parameter family of solutions satisfying physical boundary conditions. Matching these parameters with the skin-friction coefficient and the Reynolds number based on momentum thickness, we get an agreement of the solutions with experimental data in the laminar and transitional boundary layers, as well as in the turbulent boundary layer for moderately large Reynolds numbers.     

Equations of an Elastic Anisotropic Layer

Differential equations of an elastic orthotropic layer are constructed on the basis of expansion of the solutions of the elasticity theory in terms of the Legendre polynomials. The order of the system of differential equations is independent of the form of the boundary conditions on the layer surfaces, which allows a correct formulation of conditions on contact surfaces.     

Free convection heat and mass transfer with Hall current, Joule heating and thermal diffusion

A boundary layer analysis has been presented to study the combined effects of viscous dissipation, Joule heating, transpiration, heat source, thermal diffusion and Hall current on the hydromagnetic free convection and mass transfer flow of an electrically conducting, viscous, homogeneous, incompressible fluid past an infinite vertical porous plate. The governing partial differential equations of the hydromagnetic free convective boundary layer flow are reduced to non-linear ordinary differential equations and solutions for primary velocity, secondary velocity, temperature and concentration field are obtained for large suction. The expressions for the skin-friction, the heat transfer and the mass transfer are also derived. The results of the study are discussed through graphs and tables for different numerical values of the parameters entered into the equations governing the flow.     

A class of solutions of the boundary layer equations

Exact solutions of the boundary layer equations can be obtained in closed form only in rare cases. These generally involve self-similar solutions for which the corresponding ordinary differential equation can be integrated exactly. In this paper solutions of more general form, containing additive functions of the longitudinal x coordinate in the expression's for the stream function and the self-similar variable, are considered in relation to two-dimensional steady boundary layers. This makes it possible to enlarge the class of problems whose solutions are analytic expressions and in a number of cases can be obtained in the form of expressions containing arbitrary functions of x, which makes possible various interpretations of the solution. In order to introduce arbitrary functions into the solutions of the equations of the axisymmetric boundary layer the problem is reduced to an overdetermined system of ordinary differential equations. This method is also capable of being applied more widely.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 45–51, March–April, 1990.     

Analytical solution to stagnation-point flow and heat transfer over a stretching sheet based on homotopy analysis

This paper is concerned with two-dimensional stagnation-point steady flow of an incompressible viscous fluid towards a stretching sheet whose velocity is proportional to the distance from the slit. The governing system of partial differential equations is first transformed into a system of dimensionless ordinary differential equations. Analytical solutions of the velocity distribution and dimensionless temperature profiles are obtained for different ratios of free stream velocity and stretching velocity, Prandtl number, Eckert number and dimensionality index in series forms using homotopy analysis method（HAM）. It is shown that a boundary layer is formed when the free stream velocity exceeds the stretching velocity, and an inverted boundary layer is formed when the free stream velocity is less than the stretching velocity. Graphs are presented to show the effects of different parameters.     

Oblique stagnation slip flow of a micropolar fluid

This paper considers the problem of steady two-dimensional boundary layer flow of a micropolar fluid near an oblique stagnation point on a fixed surface with Navier’s slip condition. It is shown that the governing nonlinear partial differential equations admit similarity solutions. The resulting nonlinear ordinary differential equations are solved numerically using the Keller box method for some values of the governing parameters. It is found that the flow characteristics depend strongly on the micropolar and slip parameters.     

Series solutions for the stagnation flow of a second-grade fluid over a shrinking sheet

This study derives the analytic solutions of boundary layer flows bounded by a shrinking sheet. With the similarity transformations, the partial differential equations are reduced into the ordinary differential equations which are then solved by the homotopy analysis method （HAM）. Two-dimensional and axisymmetric shrinking flow cases are discussed.     

Boundary layer flow of Reiner-Philippoff fluids

An analysis is made of the boundary layer flow of Reiner-Philippoff fluids. This work is an extension of a previous analysis by Hansen and Na [A.G. Hansen and T.Y. Na, Similarity solutions of laminar, incompressible boundary layer equations of non-Newtonian fluids. ASME 67-WA/FE-2, presented at the ASME Winter Annual Meeting, November (1967)], where the existence of similar solutions of the boundary layer equations of a class of general non-Newtonian fluids were investigated. It was found that similarity solutions exist only for the case of flow over a 90° wedge and, being similar, the solution of the non-linear boundary layer equations can be reduced to the solution of non-linear ordinary differential equations. In this paper, the more general case of the boundary layer flow of Reiner-Philippoff fluids over other body shapes will be considered. A general formulation is given which makes it possible to solve the boundary layer equations for any body shape by a finite-difference technique. As an example, the classical solution of the boundary layer flow over a flat plate, known as the Blasius solution, will be considered. Numerical results are generated for a series of values of the parameters in the Reiner-Philippoff model.     

Effects of variable fluid viscosity on flow past a heated stretching sheet embedded in a porous medium in presence of heat source/sink

The boundary layer flow and heat transfer of a fluid through a porous medium towards a stretching sheet in presence of heat generation or absorption is considered in this analysis. Fluid viscosity is assumed to vary as a linear function of temperature. The symmetry groups admitted by the corresponding boundary value problem are obtained by using a special form of Lie group transformations viz. scaling group of transformations. These transformations are used to convert the partial differential equations corresponding to the momentum and the energy equations into highly non-linear ordinary differential equations. Numerical solutions of these equations are obtained by shooting method. It is found that the horizontal velocity decreases with increasing temperature-dependent fluid viscosity parameter up to the crossing-over point but increases after that point and the temperature decreases in this case. With the increase of permeability parameter of the porous medium the fluid velocity decreases but the temperature increases at a particular point of the sheet. Effects of Prandtl number on the velocity boundary layer and on the thermal boundary layer are studied and plotted.     

Solution of boundary layer flow and heat transfer of an electrically conducting micropolar fluid in a non-Darcian porous medium

In this paper, we have studied the effects of radiation on the boundary layer flow and heat transfer of an electrically conducting micropolar fluid over a continuously moving stretching surface embedded in a non-Darcian porous medium with a uniform magnetic field has been analyzed analytically. The governing fundamental equations are approximated by a system of nonlinear locally similar ordinary differential equations which are solved analytically by applying homotopy analysis method (HAM). The effects of Darcy number, heat generation parameter and inertia coefficient parameter are determined on the flow. Convergence of the obtained series solution is discussed. The homotopy analysis method provides us with a new way to obtain series solutions of such problems. This method contains the auxiliary parameter which provides us with a simple way to adjust and control the convergence region of series solution. By suitable choice of the auxiliary parameter, we can obtain reasonable solutions for large modulus.     

A quasi-similarity analysis of the turbulent boundary layer on a flat plate

The partial differential equation of the boundary layer on a flat plate are simplified by using the universal variables for turbulent flow. For laminar flow this gives boundary layer having a finite thickness and a friction coefficient differing by a few percent from the Blasius value. For a turbulent flow a differential equation for the velocity distribution is obtained with a parameter which varies slowly with the streamwise coordinate. The numerical value of this parameter is determined as an eigenvalue of the differential equations giving a velocity profile which evolves as the boundary layer thickens. Numerical calculations using a simple eddy viscosity model gave results in very good agreement with experiment.     

Use of the generalized similarity method for calculating the inner zone of a turbulent boundary layer

The use of the generalized similarity method for calculating laminar boundary layers has been fully justified (see [1, §113, 114, 148]). The replacement of the partial differential equations by ordinary differential equations, their universality and the possibility of physically interpreting the solutions in the first, parametric stage of the calculations, which distinguish the generalized similarity method from direct numerical integration methods, are preserved in the case of a turbulent boundary layer also. A comparison of the calculated and experimental velocity profiles in the inner zone of the turbulent boundary layer suggests that the generalized similarity method could be used for calculating the turbulent layer as a whole.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 25–34, September–October, 1990.     

Effect of thermal radiation on heat transfer in an unsteady copper–water nanofluid flow over an exponentially shrinking porous sheet

The effect of thermal radiation on an unsteady boundary layer flow and heat transfer in a copper–water nanofluid over an exponentially shrinking porous sheet is investigated. With the use of suitable transformations, the governing equations are transformed into ordinary differential equations. Dual non-similarity solutions are obtained for certain values of some parameters. Owing to the presence of thermal radiation, the heat transfer rate is greatly enhanced, and the thermal boundary layer thickness decreases.