Hall current,radiation absorption,and diffusion thermoeffects on unsteady MHD rotating chemically reactive second-grade fluid flow past a porous inclined plate in the presence of an aligned magnetic field,thermal radiation,and chemical reaction

In this paper, we analyze the effects of Hall current, radiation absorption and diffusion thermo on unsteady magnetohydromagnetic free convection flow of a viscous incompressible electrically conducting and chemically reacting second-grade fluid past an inclined porous plates in the presence of an aligned magnetic field, thermal radiation, and chemical reaction. An exact analytical solution of the governing equations for fluid velocity, fluid temperature, and species concentration subject to appropriate initial and boundary conditions is obtained using the perturbation technique. Expressions for shear stress, rate of heat transfer, and rate of mass transfer at the plate are derived. The numerical values of primary and secondary fluid velocities, fluid temperature and species concentration are displayed graphically, whereas those of shear stress and rate of mass transfer at the plate are presented in tabular form for various values of pertinent flow parameters. In addition, the skin friction on the boundary, the heat flux expressed in terms of the Nusselt number, and the rate of mass transfer described in the Sherwood number are all derived, and their behavior is studied computationally. It can be deduced that an increase in radiation absorption and hall current parameters over the fluid region increases the velocity produced. The resulting velocity continually increases to a very high level, with contributions coming from thermal and solutal buoyancy forces. Skin friction may decrease by manipulating the rotation parameter, but the Hall effect can worsen it. When the parameter for the chemical reaction increases, there is a concomitant rise in the mass transfer rate.     

Integral Analysis of Laminar Indirect Free Convection Boundary Layers with Weak Blowing for Schmidt No. 1

Laminar natural convection at unity Schmidt number over a horizontal surface with a weak normal velocity at the wall is studied using an integral analysis. To characterise the strength of the blowing, we define a non-dimensional parameter called the blowing parameter. After benchmarking with the no blowing case, the effect of the blowing parameter on boundary layer thickness, velocity and concentration profiles is obtained. Weak blowing is seen to increase the wall shear stress. For blowing parameters greater than unity, the diffusional flux at the wall becomes negligible and the flux is almost entirely due to the blowing.     

Investigation of combined heat and mass transfer by Lie group analysis with variable diffusivity taking into account hydrodynamic slip and thermal convective boundary conditions

The present paper investigates heat and mass transfer over a moving porous plate with hydrodynamic slip and thermal convective boundary conditions and concentration dependent diffusivity. The similarity representation of the system of partial differential equations of the problem is obtained through Lie group analysis. The resulting equations are solved numerically by Maple with Runge–Kutta–Fehlberg fourth–fifth order method. A representative set of results for the physical problem is displayed to illustrate the influence of parameters (velocity slip parameter, convective heat transfer parameter, concentration diffusivity parameter, Prandtl number and Schmidt number) on the dimensionless axial velocity, temperature and concentration field as well as the wall shear stress, the rate of heat transfer and the rate of mass transfer. The analytical solutions for velocity and temperature are obtained. Very good agreements are found between the analytical and numerical results of the present paper with published results.     

Thermal characteristics of forced convection in combined pressure and shear-driven flow of a non-Newtonian third-grade fluid through parallel plates

Heat transfer in a non-Newtonian third-grade fluid, flowing under the action of pressure gradient and shear, through two parallel plates, is considered. The upper plate moves with a constant velocity. Constant wall heat fluxes are applied to the plates. Effect of viscous dissipation is included, which has a major role in heat transfer of non-Newtonian fluids. The governing equations are nonlinear and are solved semi-analytically by using the least-square method (LSM). Then, using the solution for velocity in the energy equation, the solution is obtained by a direct integration process. Further, approximate analytical solutions are obtained by the perturbation method, which validates the results generated by the LSM. The effects of the third-grade fluid parameter on the velocity and temperature and also on the physical quantity, such as Nusselt's number, are discussed. Further, viscous dissipation effects on the temperature distribution have been analyzed. Observations show that the movement of the upper plate results in a significant decrease in temperature near the upper plate. For the unit heat flux ratio, the temperature difference between the surface and fluid is more at the upper surface due to the enhanced convective heat transfer caused by the moving upper plate. Nusselt's number increases significantly with an increase in the heat flux ratio.     

Radiative heat transfer analysis of non-Newtonian dusty Casson fluid flow along a complex wavy surface

In this article, numerical solutions are obtained to observe the influence of thermal radiation on Casson particulate suspension flow past a complex isothermal wavy surface. Rosseland diffusion approximation is employed to express the contribution of radiative heat flux over the Casson fluid model. Using coordinate transformations, the two-phase model is converted into a suitable form and then integrated numerically by employing implicit finite-difference method. The numerical results are discussed in detail in terms of shear stress, rate of heat transfer, streamlines, and isotherms. It is found that the rate of heat transfer increases extensively when radiation parameter and mass concentration parameter are penetrated into the mechanism.     

Mass transfer and free convection flow through A porous medium

A theoretical analysis of the steady free convective and mass transfer flow is presented, when a viscous and incompressible fluid flows through a porous medium occupying a semi-infinite region of the space bounded by an infinite vertical porous plate. The fluid is subjected to a normal suction velocity, and the heat flux at the plate is constant. The free-stream velocity is assumed constant.     

Forced convection heat transfer of Williamson fluid flow in porous media over horizontal plate with constant heat flux

Forced convection of Williamson fluid flow in porous media under constant surface heat flux conditions is investigated numerically. A model of Darcy–Forchheimer–Brinkman is used and the corresponding governing equations are expressed in dimensionless forms and solved numerically using bvp4c with MATLAB package. Boundary layer velocity, shear stress, and temperature profiles, in addition to the local Nusselt number parameter over a horizontal plate, are found. The effects of the Forchheimer parameter, Nusselt number, Darcy parameter, porous inertia, and Williamson parameter on the velocity profiles, temperature profiles, coefficient of friction, and coefficient of heat transfer are investigated. The results showed that as the Darcy parameter increases, boundary layer velocity and shear stress increase, while the temperature and Nusselt number decrease. In addition, as Williamson's parameter increases, velocity within the boundary layer, shear stress, and Nusselt number decrease while the temperature profile increases. Also, with larger values of the Forchheimer parameter, the velocity of the boundary layer, shear stress, temperature, and Nusselt number increase. Furthermore, the Nusselt number and the coefficient of friction are obtained on the surface of the horizontal plate.     

MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition

A numerical approach has been used to study the heat and mass transfer from a vertical plate embedded in a porous medium experiencing a first-order chemical reaction and exposed to a transverse magnetic field. Instead of the commonly used conditions of constant surface temperature or constant heat flux, a convective boundary condition is employed which makes this study unique and the results more realistic and practically useful. The momentum, energy, and concentration equations derived as coupled second-order, ordinary differential equations are solved numerically using a highly accurate and thoroughly tested finite difference algorithm. The effects of Biot number, thermal Grashof number, mass transfer Grashof number, permeability parameter, Hartmann number, Eckert number, Sherwood number and Schmidt number on the velocity, temperature, and concentration profiles are illustrated graphically. A table containing the numerical data for the plate surface temperature, the wall shear stress, and the local Nusselt and Sherwood numbers is also provided. The discussion focuses on the physical interpretation of the results as well their comparison with the results of previous studies.     

Thermal Residual Stresses in One-Directional Functionally Graded Plates Subjected to In-Plane Heat Flux

This study carries out the transient thermal residual stress analyses of functionally graded clamped plates for different in-plane material compositions and in-plane heat fluxes. The heat conduction and Navier equations representing the two-dimensional thermoelastic problem were discretized using the finite-difference method, and the set of linear equations were solved using the pseudo singular value method. Both in-plane temperature distributions and the heat transfer period were affected considerably by the compositional gradient. The type of in-plane heat flux had a minor effect on the temperature profile, but on the heat transfer period. The high stress levels appeared in the ceramic-rich regions. The normal and equivalent stresses exhibited a sharp change in the plates with ceramic-rich as well as metal-rich compositions, and the concentrated on a narrow ceramic layer. A smooth stress variation was achieved through the graded region with a balanced composition of ceramic and metal-phases, and the stress discontinuities disappeared. The in-plane shear stress was negligible. The equivalent stress exhibited a linear temporal variation for both constant and sinusoidal heat fluxes, but a nonlinear variation for the exponential heat flux. In case the heat flux is applied along the metal edge (metal-to-ceramic plate) instead of the ceramic edge, the displacement and stress components exhibited similar distributions to those of a ceramic-to-metal plate but in the opposite direction. As a result, the distribution of in-plane material composition affects only normal stress distributions, whereas the peak stress levels occur in the ceramic-rich regions. Since the normal stresses concentrate along a narrow ceramic layer for ceramic-rich or metal-rich compositions, a balanced in-plane material composition distribution of ceramic and metal would be useful to avoid probable local ceramic fracture or damage.     

MHD mass transfer flow past an inclined plate with variable temperature and plate velocity embedded in a porous medium

An exact solution of unsteady MHD free convective mass transfer flow past an infinite inclined plate embedded in a saturated porous medium with variable plate velocity, temperature, and mass diffusion has been presented. An attempt has been made to analyze the Soret effect and the influence of the angle of inclination on the flow and transport properties, in the presence of thermal radiation, heat source, and chemical reaction. The equations governing the flow, heat, and mass transfer are solved by employing the Laplace transform technique, in closed form. The variations in fluid velocity, temperature, and concentration profiles are shown graphically whereas the numerical values of shear stress, the rate of heat transfer, and the rate of mass transfer from the plate to the fluid are presented in tabular form for various values of the flow parameters. The results show that the flow is accelerated due to the Soret effect while the angle of inclination sustains a retarding effect on fluid velocity. Further it is observed that the viscous drag at the plate and the mass diffusion from the plate to the fluid decrease under the influence of thermal diffusion.     

Integral transform solution of transient forced convection in external flow

The Generalized Integral Transform Technique (GITT), under its partial transformation mode, is employed in the hybrid numerical–analytical solution of transient laminar forced convection over flat plates, subjected to arbitrary time variations of wall heat flux applied from above. From the available Blasius velocity distributions and employing a coordinate transformation to account only for the thermally affected region along the main flow direction, the transient temperature distribution is expanded in terms of eigenfunctions obtained from the diffusion operator in the transversal direction. The resulting coupled system of partial differential equations for the transformed potentials is numerically solved in terms of the dimensionless time variable and longitudinal coordinate, by making use of the method of lines implemented in the Mathematica routine NDSolve. Numerical solutions for the wall temperature and heat transfer coefficient are then readily computed, yielding the time evolution and the longitudinal distribution of these parameters, for any specified wall heat flux time function.     

Convective heat transfer from a partially premixed impinging flame jet. Part II: Time-resolved results

This is the second of a two-part paper on heat transfer from an impinging flame jet reporting time-resolved results. Axial and radial profiles of time-resolved local heat fluxes of methane-air jet flames impinging normal to a cooled plate are reported, including the root mean square (RMS), probability distribution function (PDF), and the power spectral density (PSD) of the heat flux fluctuations as a function of equivalence ratio, Reynolds number, and nozzle-plate spacing. The RMS, PDF, and PSD of the heat flux signal from the stagnation point and along the plate revealed correlation of the local heat flux to the flame structure. Impingement heat flux from premixed nozzle-stabilized flames was characterized by small RMS fluctuations and frequency behavior indicating the formation of weak, buoyancy-driven vortex structures at the shear layer between the hot gases surrounding the flame and the ambient air. Conversely, diffusion flames were characterized by much larger RMS fluctuations and PSD’s indicating the development of much larger vortex structures. Time-resolved heat flux for lifted flames varied according to flame structure and combustion intensity. PSD magnitudes were related to the range of temperatures in the flow; greater temperature ranges produced larger heat flux variations. The contributing frequencies were related to the duration of the heat flux fluctuation; more rapid changes in heat flux produced higher frequency content.     

Unsteady mixed convection flow in the stagnation region of a heated vertical plate due to impulsive motion

The unsteady mixed convection in the stagnation flow on a heated vertical plate is studied where the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase in the surface temperature (heat flux). The short time as well as the long time solutions are included in the analysis. Both prescribed surface temperature and prescribed surface heat flux conditions are considered. The partial differential equations governing the flow and the heat transfer have been solved numerically using an implicit finite difference scheme. Also, the asymptotic behaviour of the solution for large value of the independent variable is examined when the flow becomes steady. There is a smooth transition from the small-time solution to the large-time solution. The surface shear stress and the heat transfer increase with time and buoyancy parameter. The heat transfer increases with the Prandtl number, but the surface shear stress decreases.     

Incidences of aligned magnetic field on unsteady MHD flow past a parabolic accelerated inclined plate in a porous medium

An exact analysis of a radiative hydromagnetic flow behavior over a tilted parabolic plate through a permeable medium along with variable species concentration and fluid temperature in the presence of a slanted magnetic field parameter, chemical reaction, and heat generation has been carried out in this study. Closed-form analytical benchmark solutions for flow-governing equations are obtained by using the Laplace transform method. Thereafter, the incidences of different important physical entities on the nondimensional velocity field, temperature distribution, and species concentration are presented using graphs, whereas impacts of various physical entities on wall shear stress, heat and mass transfer rates are presented in tables. It is worth noting that an increase in the magnetic field and its inclination angle causes the reduction in the fluid velocity. However, wall shear stress increases with the increase of magnetic field and its inclination angle. The novel results in this article can be used to improve quicker cooling and producing miniaturized heat flow systems with upgraded efficiency and cost-effectiveness.     

Pulsatile flow in heated porous channelEcoulement pulsatoire dans un canal poreux et chauffePulsierende strömung in einem beheizten porösen kanalПyльcaциoннoe тeчeниe в нaгpeвaeмoм пopиcтoм кaнaлe

The combined forced and free convection flow through a porous channel when a pulsatile pressure is applied across its ends is discussed. It is assumed that the ratio of the width of the channel to the length (δ) is small. Even in this physically realistic situation of a finite channel, the transverse velocity remains undisturbed and acquires its suction or blowing value at the channel walls. A salient feature of the investigation is the presence of steady streaming component in the higher approximations due to nonlinearity in the viscous dissipation heat. When the channel is horizontal, the axial velocity distribution obtained is exact and independent of δ. But the pressure and temperature distributions are exact and δ-dependent. On the other hand, if the channel is vertical, only the pressure distribution is obtained exactly. It is independent of the transverse coordinate but varies linearly with the axial coordinate. The velocity and temperature are obtained approximately to order O(δ). Also the shear stress and heat flux at the walls are discussed quantitatively.     

Lie symmetry analysis and numerical solutions for thermosolutal chemically reacting radiative micropolar flow from an inclined porous surface

Steady, laminar, incompressible thermosolutal natural convection flow of micropolar fluid from an inclined perforated surface with convective boundary conditions is studied. Thermal radiative flux and chemical reaction effects are included to represent phenomena encountered in high-temperature materials synthesis operations. Rosseland's diffusion approximation is used to describe the radiative heat flux in the energy equation. A Lie scaling group transformation is implemented to derive a self-similar form of the partial differential conservation equations. The resulting coupled nonlinear boundary value problem is solved with Runge-Kutta fourth order numerical quadrature (shooting technique). Validation of solutions with an optimized Adomian decomposition method algorithm is included. Verification of the accuracy of shooting is also conducted as a particular case of nonreactive micropolar flow from a vertical permeable surface. The evolution of velocity, angular velocity (microrotation component), temperature, and concentration are examined for a variety of parameters including coupling number, plate inclination angle, suction/injection parameter, radiation-conduction parameter, Biot number, and reaction parameter. Numerical results for steady-state skin friction coefficient, couple stress coefficient, Nusselt number, and Sherwood number are tabulated and discussed. Interesting features of the hydrodynamic, heat and mass transfer characteristics are examined.     

Effect of transpiration on combined heat and mass transfer in mixed convection along a vertical plate

The effect of transpiration velocity on the heat and mass transfer characteristics of mixed convection flow along a permeable vertical flat plate under the combined effects of thermal and mass diffusion is analysed. The diffusion-thermo and thermo-diffusion effects as well as the interfacial velocities due to mass diffusion are negligibly small. The plate is maintained at a uniform temperature and species concentration. Numerical results for the local skin-friction, the local Nusselt number and the local Sherwood number, as well as for the velocity, the temperature and the concentration profiles, are presented for diffusion of common species into air only. In general, it has been found for thermally assisted flow that the local surface shear-, heat-, and mass-transfer rates decrease owing to suction of fluid. This trend reversed for blowing of fluid. In addition this trend is higher for species of larger Schmidt number as well as for increasing buoyancy force.     

Direct numerical simulation for a time-developing natural-convection boundary layer along a vertical flat plate

Direct numerical simulations were performed for the transitional and turbulent natural-convection boundary layer for air and water along a hot vertical flat plate. The numerical results for water well reproduce the vortex-like structures as observed experimentally in the thermal field for a high Prandtl-number fluid. When the calculated values are evaluated with the integral thickness of the velocity boundary layer as a length scale, the turbulence statistics such as heat transfer rate, mean velocity, mean temperature, intensity of velocity and temperature fluctuations, Reynolds shear stress and turbulent heat fluxes correspond relatively well with those obtained from the experiments for space-developing flows. This shows that the time-developing direct numerical simulation of the natural-convection boundary layer can provide details that are difficult to obtain in experiments. These data are of great importance to understand the turbulence structures.     

The transient thermo-electro-elastic fields in a piezoelectric plate with a crack

This paper is concerned with a thin plate made by a piezoelectric ceramic material and containing a crack perpendicular to its surfaces. It is assumed that the transient thermal stress is set up by the application of a heat flux as a function of the time and position along the crack edge and the heat flow by convection from the plate surfaces. The plate is also subjected to mechanical and electric loadings. The exact analytical formulae are obtained for transient thermo-electro-elastic fields in the plate. The exact analytical solutions for the stress and electric displacement intensity factors and crack-opening displacement are obtained. Numerical examples show, among others, a dependence of the stress and electric displacement intensity factors on the thermal and elastic, piezoelectric and dielectric constants of the piezoelectric materials.     

Effects of suction and free convection currents on the oscillatory flow past an infinite vertical porous plate with constant heat flux

Two-dimensional flow analysis of an incompressible, viscous fluid past an infinite porous vertical plate, in the presence of constant heat flux, has been presented. the effects of suction, free convection currents and free-stream oscillations have been considered and approximate solutions to the coupled non-linear equations have been derived in case of small amplitude oscillations. the solutions are derived for the mean velocity, the mean plate temperature, the mean skin-friction, the transient velocity, the transient temperature, the amplitude and phase of the skin-friction, and the first and second harmonics of the plate temperature. Velocity and temperature fields have been shown graphically whereas the numerical values of the other physical quantities are given in tabular form. During the course of discussion, the effects of γ (suction parameter), G(the Grashof number), E (Eckert number), P (Prandtl number and ω (frequency) have been discussed.