Analytical study of Cattaneo–Christov heat flux model for a boundary layer flow of Oldroyd-B fluid

We investigate the Cattaneo–Christov heat flux model for a two-dimensional laminar boundary layer flow of an incompressible Oldroyd-B fluid over a linearly stretching sheet. Mathematical formulation of the boundary layer problems is given. The nonlinear partial differential equations are converted into the ordinary differential equations using similarity transformations. The dimensionless velocity and temperature profiles are obtained through optimal homotopy analysis method(OHAM). The influences of the physical parameters on the velocity and the temperature are pointed out. The results show that the temperature and the thermal boundary layer thickness are smaller in the Cattaneo–Christov heat flux model than those in the Fourier's law of heat conduction.     

Analysis of Heat Transfer Phenomenon in Magnetohydrodynamic Casson Fluid Flow Through Cattaneo–Christov Heat Diffusion Theory

Heat transport phenomenon of two-dimensional magnetohydrodynamic Casson fluid flow by employing Cattaneo–Christov heat diffusion theory is described in this work. The term of heat absorption/generation is incorporated in the mathematical modeling of present flow problem. The governing mathematical expressions are solved for velocity and temperature profiles using RKF 45 method along with shooting technique. The importance of arising nonlinear quantities namely velocity, temperature, skin-friction and temperature gradient are elaborated via plots. It is explored that the Casson parameter retarded the liquid velocity while it enhances the fluid temperature. Further, we noted that temperature and thickness of temperature boundary layer are weaker in case of Cattaneo–Christov heat diffusion model when matched with the profiles obtained for Fourier's theory of heat flux.     

Boundary layer flow and heat transfer of a Casson fluid past a symmetric porous wedge with surface heat flux

The aim of this paper is to investigate numerically the boundary layer forced convection flow of a Casson fluid past a symmetric porous wedge. Similarity transformations are used to convert the governing partial differential equations into ordinary ones. With the help of the shooting method, the reduced equations are then solved numerically. Comparisons are made with the previously published results in some special cases and they are found to be in excellent agreement with each other. The results obtained in this study are illustrated graphically and discussed in detail. The velocity is found to increase with an increasing Falkner-Skan exponent whereas the temperature decreases. With the rise of the Casson fluid parameter, the fluid velocity increases but the temperature is found to decrease in this case. Fluid velocity is suppressed with the increase of suction. The skin friction decreases with the increasing value of Casson fluid parameter. It is found that the temperature decreases as the Prandtl number increases and thermal boundary layer thickness decreases with the increasing value of Prandtl number. A significant finding of this investigation is that flow separation can be controlled by increasing the value of the Casson fluid parameter as well as by increasing the amount of suction.     

Effects of Slip on Unsteady Mixed Convective Flow and Heat Transfer Past a Stretching Surface

Unsteady mixed convective boundary layer flow and heat transfer over a stretching vertical surface in the presence of slip axe investigated. It is noted that fluid velocity decreases due to the increasing velocity slip parameter resulting in an increase in the temperature field. The rate of heat transfer decreases with the velocity slip parameter while it increases with unsteadiness parameter. The same feature is also noticed for thermal slip parameter.     

MHD boundary layer flow of Casson fluid passing through an exponentially stretching permeable surface with thermal radiation

This article numerically examines the boundary layer flow due to an exponentially stretching surface in the presence of an applied magnetic field. Casson fluid model is used to characterize the non-Newtonian fluid behavior. The flow is subjected to suction/blowing at the surface. Analysis is carded out in presence of thermal radiation and prescribed surface heat flux. In this study, an exponential order stretching velocity and prescribed exponential order surface heat flux are accorded with each other. The governing partial differential equations are first converted into nonlinear ordinary differential equations by using appropriate transformations and then solved numerically. The effect of increasing values of the Casson parameter is to suppress the velocity field. However the temperature is enhanced when Casson parameter increases. It is found that the skin-friction coefficient increases with increasing values of suction parameter. Temperature also increases for large values of power index n in both suction and blowing cases at the boundary. It is observed that the thermal radiation enhances the effective thermal diffusivity and hence the temperature rises.     

MHD Flow and Heat Transfer of a Generalized Burgers' Fluid Due to an Exponential Accelerating Plate with Effects of the Second Order Slip and Viscous Dissipation

In classical study on generalized viscoelastic fluid, the momentum equation was derived by considering the fractional constitutive model, while the energy equation was ignored its effect. This paper presents an investigation for the magnetohydrodynamic(MHD) flow and heat transfer of an incompressible generalized Burgers' fluid due to an exponential accelerating plate with the effect of the second order velocity slip. The energy equation and momentum equation are coupled by the fractional Burgers' fluid constitutive model. Numerical solutions for velocity, temperature and shear stress are obtained using the modified implicit finite difference method combined with the G1-algorithm,whose validity is confirmed by the comparison with the analytical solution. Our results show that the influences of the fractional parameters α and β on the flow are opposite each other, which is just like the effects of the two parameters on the temperature. Moreover, the impact trends of the relaxation time λ_1 and retardation time λ_3 on the velocity are opposite each other. Increasing the boundary parameter will promote the temperature, but has little effect on the temperature boundary layer thickness.     

Heat transfer for boundary layers with cross flow

An analysis is presented to study the dual nature of solutions for the forced convective boundary layer flow and heat transfer in a cross flow with viscous dissipation terms in the energy equation. The governing equations are transformed into a set of three self-similar ordinary differential equations by similarity transformations. These equations are solved numerically using the very efficient shooting method. This study reveals that the dual solutions of the transformed similarity equations for velocity and temperature distributions exist for certain values of the moving parameter, Prandtl number, and Eckert numbers. The reverse heat flux is observed for larger Eckert numbers; that is, heat absorption at the wall occurs.     

Three-dimensional magnetohydrodynamics axisymmetric stagnation flow and heat transfer due to an axisymmetric shrinking/stretching sheet with viscous dissipation and heat source/sink

The present work is concerned with the effects of viscous dissipation and heat source/sink on a three-dimensional magnetohydrodynamic boundary layer axisymmetric stagnation flow, and the heat transfer of an electrically conducting fluid over a sheet, which shrinks or stretches axisymmetrically in its own plane where the line of the symmetry of the stagnation flow and that of the shrinking （stretching） sheet are, in general, not aligned. The governing equations are transformed into ordinary differential equations by using suitable similarity transformations and then solved numerically by a shooting technique. This investigation explores the conditions of the non-existence, existence and uniqueness of the solutions of the similar equations numerically. It is noted that the range of the velocity ratio parameter, where the similarity solution exists, is increased with the increase of the value of the magnetic parameter. Furthermore, the study reveals that the non-alignment function affects the shrinking sheet more than the stretching sheet. In addition, the numerical results of the velocity profile, temperature profile, skin-friction coefficient, and rate of heat transfer at the sheet are discussed in detail with different parameters.     

Flow and heat transfer of a nanofluid over a hyperbolically stretching sheet

This article explores the boundary layer flow and heat transfer of a viscous nanofluid bounded by a hyperbolically stretching sheet. Effects of Brownian and thermophoretic diffusions on heat transfer and concentration of nanoparticles are given due attention. The resulting nonlinear problems are computed for analytic and numerical solutions. The effects of Brownian motion and thermophoretic property are found to increase the temperature of the medium and reduce the heat transfer rate. The thermophoretic property thus enriches the concentration while the Brownian motion reduces the concentration of the nanoparticles in the fluid. Opposite effects of these properties are observed on the Sherwood number.     

Variable fluid properties and variable heat flux effects on the flow and heat transfer in a non-Newtonian Maxwell fluid over an unsteady stretching sheet with slip velocity

The effects of variable fluid properties and variable heat flux on the flow and heat transfer of a non-Newtonian Maxwell fluid over an unsteady stretching sheet in the presence of slip velocity have been studied. The governing differential equations are transformed into a set of coupled non-linear ordinary differential equations and then solved with a numerical technique using appropriate boundary conditions for various physical parameters. The numerical solution for the governing non-linear boundary value problem is based on applying the fourth-order Runge-Kutta method coupled with the shooting technique over the entire range of physical parameters. The effects of various parameters like the viscosity parameter, thermal conductivity parameter, unsteadiness parameter, slip velocity parameter, the Deborah number, and the Prandtl number on the flow and temperature profiles as well as on the local skin-friction coefficient and the local Nusselt number are presented and discussed. Comparison of numerical results is made with the earlier published results under limiting cases.     

Dual solutions in boundary layer flow of a moving fluid over a moving permeable surface in presence of prescribed surface temperature and thermal radiation

An analysis of the heat transfer for a boundary layer forced convective flow past a moving permeable flat surface parallel to a moving fluid is presented. Prescribed surface temperature at the boundary is considered. A thermal radiation term in the energy equation is considered. The similarity solutions for the problem are obtained and the reduced ordinary differential equations are solved numerically. To support the validity of the numerical results, a comparison is made with the available results for some particular cases of this study. Dual solutions exist when the surface and the fluid move in the opposite directions.     

Three-Dimensional Linear Instability Analysis of Thermocapillary Return Flow on a Porous Plane

A three-dimensional linear instability analysis of thermocapillary convection in a fluid-porous double layer system, imposed by a horizontal temperature gradient, is performed. The basic motion of fluid is the surface-tension-driven return flow, and the movement of fluid in the porous layer is governed by Darcy＇s law. The slippery effect of velocity at the fluid-porous interface has been taken into account, and the influence of this velocity slippage on the instability characteristic of the system is emphasized. The new behavior of the thermocapillary convection instability has been found and discussed through the figures of the spectrum.     

Stretching a Curved Surface in a Viscous Fluid

This work is concerned with the viscous flow due to a curved stretching sheet. The similarity solution of the problem is obtained numerically by a shooting method using the Runge-Kutta algorithm. The physical quantities of interest like the fluid velocity and skin friction coefficient are obtained and discussed under the influence of dimensionless curvature. It is evident from the results that dimensionless curvature causes an increase in boundary layer thickness and a decrease in the skin friction coefficient.     

Elastic Deformation Analysis on MHD Viscous Dissipative Flow of Viscoelastic Fluid:An Exact Approach

This communication is devoted to analyze elastic deformation on electrically conducted viscoelastic fluid in the presence of viscous dissipation effects. Non-linear analysis is computed through exact solutions for velocity,temperature and concentration profiles. Special emphasis is provided for elastic deformation in the presence of magnetohydrodynamics effects. Concentration profile is discussed significantly in the presence constructive and destructive chemical reaction. Results are displayed through graphs and discussed for physical parameters that are used in present analysis. Notable findings include that temperature and thermal boundary layer thickness is an increasing function of Prandtl number and a decreasing function of elastic deformation. In addition, heat transfer rate is enhanced by increasing the conjugate parameter(γ) which measures the strength of surface heating.     

Effects of transpiration on unsteady MHD flow of an upper convected Maxwell （UCM） fluid passing through a stretching surface in the presence of a first order chemical reaction

The aim of this article is to present the effects of transpiration on the unsteady two-dimensional boundary layer flow of non-Newtonian fluid passing through a stretching sheet in the presence of a first order constructive/destructive chemical reaction. The upper-convected Maxwell （UCM） model is used here to characterize the non-Newtonian behavior of the fluid. Using similarity solutions, the governing nonlinear partial differential equations are transformed into ordinary ones and are then solved numerically by the shooting method. The flow fields and mass transfer are significantly influenced by the governing parameters. The fluid velocity initially decreases as the unsteadiness parameter increases and the concentration decreases significantly due to the increase in the unsteadiness. The effect of increasing values of transpiration （suction） and the Maxwell parameter is to suppress the velocity field; however, the concentration is enhanced as transpiration （suction） and the Maxwell parameter increase. Also, it is found that the fluid velocity decreases as the magnetic parameter increases; however, the concentration increases in this case.     

Three Kinds of Velocity Structure Function in Turbulent Flows

Based on the local multi-scale eddy structures in turbulent flows, we elucidate the essential difference between the real turbulent field with a finite Reynolds number and the Kolmogorov fully developed random field. The motion of fluid particles in the real turbulent field is not fully random. There exist multi-scale structures due to the effect of viscosity. Actually the movements of fluid particles in the turbulent field are restricted by such eddy structures. Furthermore, concept of the locally averaged velocity structure function is put forward to describe the relative strain distortion of two adjacent turbulent eddy structures at a certain scale. The time sequence of the longitudinal velocity component at different vertical locations in turbulent boundary layer has been elaborately measured by the constant temperature anemometry of model 1FA-300 in a wind tunnel. The experiment proves that the locally averaged velocity structure function is in agreement with the wavelet-coefficient structure function.     

MHD Flow and Heat Transfer of a Generalized Burgers' Fluid due to a Periodic Oscillating and Periodic Heating Plate

This paper investigates the MHD flow and heat transfer of the incompressible generalized Burgers' fluid due to a periodic oscillating plate with the effects of the second order slip and periodic heating plate. The momentum equation is formulated with multi-term fractional derivatives, and by means of viscous dissipation, the fractional derivative is considered in the energy equation. A finite difference scheme is established based on the G1-algorithm, whose convergence is confirmed by the comparison with the analytical solution in an example. Meanwhile the numerical solutions of velocity, temperature and shear stress are obtained. The effects of involved parameters on velocity and temperature fields are presented graphically and analyzed in detail. Increasing the fractional derivative parameter α, the velocity and temperature have a decreasing trend, while the influences of fractional derivative parameter β on the velocity and temperature behave conversely. Increasing the absolute value of the first order slip parameter and the second order slip parameter both cause a decrease of velocity. Furthermore, with the decreasing of the magnetic parameter, the shear stress decreases.     

Effects of variable properties on MHD heat and mass transfer flow near a stagnation point towards a stretching sheet in a porous medium with thermal radiation

The effect of variable viscosity and thermal conductivity on steady magnetohydrodynamic(MHD) heat and mass transfer flow of viscous and incompressible fluid near a stagnation point towards a permeable stretching sheet embedded in a porous medium are presented,taking into account thermal radiation and internal heat genberation/absorbtion.The stretching velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation point.The Rosseland approximation is used to describe the radiative heat flux in the energy equation.The governing fundamental equations are first transformed into a system of ordinary differential equations using a scaling group of transformations and are solved numerically by using the fourth-order Rung-Kutta method with the shooting technique.A comparison with previously published work has been carried out and the results are found to be in good agreement.The results are analyzed for the effect of different physical parameters,such as the variable viscosity and thermal conductivity,the ratio of free stream velocity to stretching velocity,the magnetic field,the porosity,the radiation and suction/injection on the flow,and the heat and mass transfer characteristics.The results indicate that the inclusion of variable viscosity and thermal conductivity into the fluids of light and medium molecular weight is able to change the boundary-layer behavior for all values of the velocity ratio parameter λ except for λ = 1.In addition,the imposition of fluid suction increases both the rate of heat and mass transfer,whereas fluid injection shows the opposite effect.     

Drag Force of Non—newtonian Fluid on a Continuous Moving Surface with Strong Suction／Blowing

A theoretical analysis for the laminar boundary layer flow of a non-Newtonian fluid on a continuous moving flat plate with surface strong suction/blowing is made. The types of potential flows necessary for similar solutions to the boundary layer are determined and both analytical and numerical solutions are presented. It is shown that the solution of the boundary layer problem depends not only on the ratio of the velocity of the plate to the velocity of the free stream, but also on the suction/blowing parameter. The skin friction decreases with increasing the parameters of power law and blowing. In the case of existing suction, the shear force decreases with the increases of tangential velocity, the largest shear force occurs at wall and the smallest shear force occurs at the edge of the boundary laryer. However, in the case of existing surface blowing, the shear force initially increases with tangential velocity and the biggest shear force occurs at the interior of the boundary layer, the skin friction approaches to zero as the blowing rate approaches the critical value.