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.     

Further results on the variable viscosity with magnetic field on flow and heat transfer to a continuous moving flat plate

The steady, laminar-boundary-layer flow along an isothermal, continuously moving, flat plate is studied taking into account the variation of viscosity with temperature in the presence of a magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. The resulting, governing equations are non-dimensionalized and are transformed using a similarity transformation and then solved numerically by the shooting method. Comparison with previously published work is performed and full agreement is obtained. A parametric study of all parameters involved is conducted, and a representative set of numerical results for the velocity and temperature profiles as well as the skin-friction parameter and the Nusselt number is illustrated graphically to show typical trends of the solutions. It is worth pointing out that, when the variation of viscosity with temperature is strong in the presence of the effect of a magnetic field, the results of the present work are completely different from those that studied the same problem in the absence of magnetic field.     

Temperature-Dependent Viscosity Effects on Non-Darcy Hydrodynamic Free Convection Heat Transfer from a Vertical Wedge in Porous Media

An analysis is presented to investigate the effect of temperature-dependent viscosity on free convection flow along a vertical wedge adjacent to a porous medium in the presence of heat generation or absorption. The governing fundamental equations are transformed into the system of ordinary differential equations using scaling group of transformations and are solved numerically by using the fifth-order Rung-Kutta method with shooting technique for various values of the physical parameters. The effects of variable viscosity parameter on the velocity, temperature and concentration are discussed. Numerical results for the problem considered are given and illustrated graphically.     

Heat transfer effects on Hiemenz flow of nanofluid over a porous wedge sheet in the presence of suction/injection due to solar energy: Lie group transformation

The objective of the present work is to investigate theoretically the Hiemenz flow and heat transfer of an incompressible viscous nanofluid past a porous wedge sheet in the presence of suction/injection due to solar energy (incident radiation). The wall of the wedge is embedded in a uniform Darcian porous medium in order to allow for possible fluid wall suction or injection and has a power-law variation of the wall temperature. The partial differential equations governing the problem under consideration are transformed by a special form of Lie symmetry group transformations viz. one-parameter group of transformation into a system of ordinary differential equations, which are solved numerically using Runge-Kutta-Gill based shooting method. The conclusion is drawn that the flow field and temperature are significantly influenced by thermal radiation, nanoparticle volume fraction, and porosity of the sheet.     

Variable Fluid Property Effect on Heat Transfer and Frictional Flow Characteristics of Water Flowing through Microchannel

The effects of temperature-dependent viscosity and thermal conductivity on heat transfer and frictional flow characteristics of water flowing through a microchannel are numerically investigated in this work. The hydrodynamically and thermally developing flow with no-slip, notemperature jump, and constant wall heat flux boundary condition is numerically studied using 2D continuum-based conservation equations. A significant deviation in Nusselt number from conventional theory is observed due to flattening of axial velocity profile due to temperaturedependent viscosity variation. The Nusselt number shows a significant deviation from conventional theory due to flattening of the radial temperature profile due to temperature-dependent thermal conductivity variation. It is noted that the deviation in Nusselt number from conventional theory is maximum for combined temperature-dependent viscosity and thermal conductivity variations. The effects of temperature-dependent viscosity and thermal conductivity on the Fanning friction factor are also investigated. Additionally, the effects of variable fluid properties on Poiseuille number, Prandtl number, and Peclet number are also investigated.     

MHD free convection flow over an isothermal vertical cone with temperature dependent viscosity

Free convection flow over an isothermal vertical cone immersed in a fluid with variable viscosity and MHD is studied in this paper. Using appropriate variables, the basic equations are transformed into the non-dimensional boundary-layer equations. These equations are then solved numerically using a very efficient implicit finite-difference method known as Crankl-Nicolson scheme. Detailed results for the velocity, temperature, skin friction, and heat transfer rates for a selection of parameter sets consisting of the viscosity parameter, magnetic field parameter, and Prandtl number are discussed. In order to validate our numerical results, the present results are compared with the available work in the literature and are found to be in an excellent agreement.     

Lattice Boltzmann Simulation of Free-Surface Temperature Dispersion in Shallow Water Flows

We develop a lattice Boltzmann method for modeling free-surface temperature dispersion in the shallow water flows. The governing equations are derived from the incompressible Navier-Stokes equations with assumptions of shallow water flows including bed frictions, eddy viscosity, wind shear stresses and Coriolis forces. The thermal effects are incorporated in the momentum equation by using a Boussinesq approximation. The dispersion of free-surface temperature is modelled by an advection-diffusion equation. Two distribution functions are used in the lattice Boltzmann method to recover the flow and temperature variables using the same lattice structure. Neither upwind discretization procedures nor Riemann problem solvers are needed in discretizing the shallow water equations. In addition, the source terms are straightforwardly included in the model without relying on well-balanced techniques to treat flux gradients and source terms. We validate the model for a class of problems with known analytical solutions and we also present numerical results for sea-surface temperature distribution in the Strait of Gibraltar.     

Effect of melting heat transfer and thermal radiation on Casson fluid flow in porous medium over moving surface with magnetohydrodynamics

In this study, the effects of variable fluid properties on heat transfer in MHD Casson fluid melts over a moving surface in a porous medium in the presence of the radiation are examined. The relevant similarity transformations are used to reduce the governing equations into a system of highly nonlinear ordinary differential equations and those are then solved numerically using the Runge–Kutta–Fehlbergmethod. The effects of different controlling parameters, namely, the Casson parameter,melting and radiation parameters, Prandtl number,magnetic field, porosity, viscosity and the thermal conductivity parameters on flow and heat transfer are investigated. The numerical results for the dimensionless velocity and temperature as well as friction factor and reducedNusselt number are presented graphically and discussed. It is found that the rate of heat transfer increases as the Casson parameter increases.     

Analysis for flow of Jeffrey fluid with nanoparticles

An analysis of the boundary layer flow and heat transfer in a Jeffrey fluid containing nanoparticles is presented in this paper. Here, fluid motion is due to a stretchable cylinder. The thermal conductivity of the fluid is taken to be temperature-dependent. The partial differential equations of velocity, temperature, and concentration fields are transformed to a dimensionless system of ordinary differential equations. Nonlinear governing analysis is computed for the homotopy solutions. The behaviors of Brownian motion and thermophoresis diffusion of nanoparticles have been examined graphically. Numerical values of the local Nusselt number are computed and analyzed.     

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.     

Convective flow past a vertical plate under the influence of magnetic field and thermal radiation subjected to a variable surface temperature in the presence of heat source/sink

In the present analysis, a numerical study is performed to examine the heat transfer characteristics of a convective flow over a vertical plate under the combined effects of magnetic field and thermal radiation in the presence of heat source/sink. The surface of the plate is subjected to a variable surface temperature. The boundary layer equations governing the flow are reduced to non-dimensional equations valid in the free convection regime using the suitable non-dimensional parameters. The dimensionless governing equations are solved by an implicit finite difference method of Crank—Nicolson type which is fast convergent, more accurate and unconditionally stable. Numerical results are obtained and presented for velocity, temperature, local and average wall shear stress, local and average Nusselt number in air. The present results are compared with the results available in the literature and are found to be in an excellent agreement.     

MHD Stagnation Point Flow of Williamson Fluid over a Stretching Cylinder with Variable Thermal Conductivity and Homogeneous/Heterogeneous Reaction

The present study reveals the effect of homogeneous/hetereogeneous reaction on stagnation point flow of Williamson fluid in the presence of magnetohydrodynamics and heat generation/absorption coefficient over a stretching cylinder. Further the effects of variable thermal conductivity and thermal stratification are also considered. The governing partial differential equations are converted to ordinary differential equations with the help of similarity transformation.The system of coupled non-linear ordinary differential equations is then solved by shooting technique. MATLAB shooting code is validated by comparison with the previously published work in limiting case. Results are further strengthened when the present results are compared with MATLAB built-in function bvp4c. Effects of prominent parameters are deliberated graphically for the velocity, temperature and concentration profiles. Skin-friction coefficient and Nusselt number for the different parameters are investigated with the help of tables.     

Free Convective Nonaligned Non-Newtonian Flow with Non-linear Thermal Radiation

The present study explores the free convective oblique Casson fluid over a stretching surface with non-linear thermal radiation effects. The governing physical problem is modelled and transformed into a set of coupled non-linear ordinary differential equations by suitable similarity transformation, which are solved numerically with the help of shooting method keeping the convergence control of 10^-5 in computations. Influence of pertinent physical parameters on normal, tangential velocity profiles and temperature are expressed through graphs. Physical quantities of interest such as skin friction coefficients and local heat flux are investigated numerically.     

Convection heat transfer in a Maxwell fluid at a non-isothermal surface

Analysis is carried out to study the convection heat transfer in an upper convected Maxwell fluid at a non-isothermal stretching surface. This is a generalization of the paper by Sadeghy et al. [21] to study the effects of free convection currents, variable thermal conductivity and the variable temperature at the stretching surface. Unlike in Sadeghy et al., here the governing nonlinear partial differential equations are coupled. These coupled equations are transformed in to a system of nonlinear ordinary differential equations and are solved numerically by a finite difference scheme (known as the Keller-Box method) and the numerical results are presented through graphs and tables for a wide range of governing parameters. The results obtained for the flow and heat transfer characteristics reveal many interesting behaviors that warrant further study of nonlinear convection heat transfer.     

Significance of activation energy and Wu's slip features in Cross nanofluid with motile microorganisms

The current article investigates the impact of the bioconvection in an unsteady flow of magnetized Cross nanofluid with gyrotactic microorganisms and activation energy over a linearly stretched configuration. The analysis has been performed by utilizing the realistic Wu's slip boundary and zero mass flux conditions. The effects of nonlinear thermal radiation and the activation energy are also addressed. The governing flow equations are deduced to a dimensionless form by considering suitable transformations which are numerically targeted via a shooting algorithm. The physical visualization of each physical parameter governing the flow problem has been displayed graphically for distribution of velocity, temperature, concentration and motile microorganisms. The numerical treatment for the variation of skin friction coefficient, local Nusselt number, local Sherwood number and motile density number is performed in tabular forms.     

MHD swirling flow and heat transfer in Maxwell fluid driven by two coaxially rotating disks with variable thermal conductivity

We develop a mathematical modeling for an electrically conducting non-Newtonian Maxwell fluid flow occurring between two coaxially parallel stretchable rotating disks at constant distant apart. The pressure and heat transfer analysis is carried out subject to the effects of axial magnetic field and temperature dependent thermal conductivity. The stretching and rotating rates of both disks are assumed different from each other. The two diverse phenomena, such as, when both disks are rotating with different angular velocities in the same as well as in the opposite directions are discussed. The similarity procedure adopted by von Kármán is utilized to reduce the governing momentum and energy equations into nonlinear ordinary differential equations. The solution of the governing problem is obtained numerically using bvp4c scheme in Matlab. The effects of active parameters including stretching rates, Deborah number, magnetic number, Prandtl number, thermal conductivity parameter and Reynolds number are examined for same as well as opposite rotation direction for radial, azimuthal, and axial flows, pressure and temperature fields. The classical flow pattern happening between the disks is significantly altered by the stretching action which is a main physical significances of this study. The azimuthal flow is observed higher for the same direction of disks rotation as compared to opposite disks rotation. The pressure field drops near the lower disk with increasing values of Reynolds number. The role of thermal conductivity parameter is quite useful to enhance the fluid temperature.     

Thermal-diffusion and diffusion-thermo effects on combined heat and mass transfer by hydromagnetic three-dimensional free convection over a permeable stretching surface with radiation

The thermal-diffusion and diffusion-thermo effects on the heat and mass transfer characteristics of free convection past a continuously stretching permeable surface in the presence of magnetic field, blowing/suction and radiation are studied. The fluid viscosity is assumed to vary with temperature. The resulting, governing three-dimensional equations are transformed using a similarity transformation and then solved numerically by the shooting method. Comparison with previously published work is performed and full agreement is obtained. A parametric study showing the effects of variable viscosity parameter β, magnetic parameter M, Dufour number Df, Soret number Sr, radiation parameter R and blowing/suction parameter f₀ on the velocity, temperature, and concentration field of a hydrogen-air mixture as a non-chemical reacting fluid pair, as well as the local skin-friction coefficient, the local Nusselt number, and the local Sherwood number is carried out. These are illustrated graphically and in tabular form to depict special features of the solutions.     

Boundary layer flow and heat transfer of a nanofluid over a permeable unsteady stretching sheet with viscous dissipation

A numerical study of the boundary layer flow past unsteady stretching surface in nanofluid under the effects of suction and viscous dissipation is investigated. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. A similarity solution is presented, which depends on the unsteadiness parameter A, Eckert number Ec, ζ suction or injection parameter, Prandtl number Pr, Lewis number Le, Brownian motion number Nb, and thermophoresis number Nt. The governing partial differential equations were converted to nonlinear ordinary differential equations by using a suitable similarity transformation, which are solved numerically using the Nactsheim-Swigert shooting technique together with Runge-Kutta six-order iteration scheme. The accuracy of the numerical method is tested by performing various comparisons with the previously published work, and the results are found to be in excellent agreement. Numerical results are presented both in tabular and graphical forms illustrating the effects of these parameters on thermal and nanoparticle volume fraction boundary layers. The thermal boundary layer thickens with a rise in the local temperature as the Brownianmotion, thermophoresis, and convective heating each intensify.     

Casson fluid flow: Free convective heat and mass transfer over an unsteady permeable stretching surface considering viscous dissipation

The present study investigates a Casson fluid flow in the presence of free convection of combined heat and mass transfer toward an unsteady permeable stretching sheet with thermal radiation, viscous dissipation and chemical reaction. The governing partial differential equations are reduced to a system of nonlinear ordinary differential equations and then solved by an efficient Runge–Kutta–Fehlberg method. The dimensionless velocity is decreased by increasing values of the chemical reaction and magnetic parameter while fluid temperature is significantly reduced by increasing values of the Prandtl number. The heat transfer rate is reduced with increasing values of thermal radiation and magnetic parameters.