Coupled heat and mass transfer by MHD free convection flow along a vertical plate with streamwise temperature and species concentration variations

The problem of steady, laminar, coupled heat and mass transfer by MHD free convective boundary‐layer flow along a vertical flat plate with the combined effects of streamwise sinusoidal variations of both the surface temperature and the species concentration in the presence of Soret and Dufour effects is considered. A suitable set of dimensionless variables is used to transform the governing equations of the problem into a non‐similar form. The resulting non‐similar equations have the property that they reduce to various special cases previously considered in the literature. An adequate and efficient implicit, tri‐diagonal finite difference scheme is employed for the numerical solution of the obtained equations. Various comparisons with previously published work are performed and the results are found to be in excellent agreement. A representative set of numerical results for the velocity, temperature, and concentration profiles as well as the surface shear stress, rate of heat transfer, and the rate of mass transfer is presented graphically for various parametric conditions and is discussed. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21033     

Natural convection boundary layer flow over a truncated cone in a porous medium saturated by a nanofluid

This work studies the natural convection boundary layer flow over a truncated cone embedded in a porous medium saturated by a nanofluid with constant wall temperature and constant wall nanoparticle volume fraction. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. A suitable coordinate transformation is performed, and the obtained nonsimilar equations are solved by the cubic spline collocation method. The effect of the Brownian motion parameter and thermophoresis parameter on the temperature, nanoparticle volume fraction and velocity profiles are discussed. The effects of the thermophoresis parameter, Brownian parameter, Lewis number, and buoyancy ratio on the local Nusselt number have been studied. Results show that an increase in the thermophoresis parameter or the Brownian parameter tends to decrease the local Nusselt number. Moreover, the local Nusselt number increases as the buoyancy ratio or the Lewis number is decreased.     

Free convection of non-Newtonian nanofluids about a vertical truncated cone in a porous medium

This work studies the free convection heat transfer over a truncated cone embedded in a porous medium saturated by a non-Newtonian power-law nanofluid with constant wall temperature and constant wall nanoparticle volume fraction. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. A coordinate transformation is performed, and the obtained nonsimilar equations are solved by the cubic spline collocation method. The effects of the power-law index, Brownian motion parameter, thermophoresis parameter and buoyancy ratio on the temperature, nanoparticle volume fraction and velocity profiles are discussed. The reduced Nusselt numbers are plotted as functions of the power-law index, thermophoresis parameter, Brownian parameter, Lewis number, and buoyancy ratio. Results show that increasing the thermophoresis parameter or the Brownian parameter tends to decrease the reduced Nusselt number. Moreover, the reduced Nusselt number increases as the power-law index is increased.     

Thermal performance of Ag–water nanofluid flow over a curved surface due to chemical reaction using Buongiorno's model

An analysis of heat and mass transfer is carried out under the influence of chemical reaction, friction heating, and heat generation/absorption over a curved surface. The impacts of random motion attributes of nanoparticles and thermophoresis are also applied in the expressions of energy and concentration. With the help of assigned transformations, the nonlinear partial differential equations are changed to dimensionless nonlinear ordinary differential equations. Then, the numerical solution is obtained using fourth‐fifth order Runge‐Kutta‐Fehlberg method via the shooting technique. The impacts of relevant parameters on velocity, temperature, and concentration are depicted through graphs and tables. The results illustrate that the lowest concentration distribution of nanofluid is related to the higher value of chemical reaction parameter. Moreover, it is found that thermophoresis and Brownian motion parameters have a propensity to increase the temperature profile while curvature parameter decreases the velocity profile. Also, velocity and temperature fields show a similar behavior for the increasing values of volume fraction of the nanoparticles, while a reverse trend is observed in the concentration profile under the same condition. To authenticate the results of the current study, the obtained data were compared with previously published data.     

Non-Darcy mixed convection from a horizontal plate embedded in a nanofluid saturated porous media

The problem of steady mixed convection boundary-layer flow over an impermeable horizontal flat plate embedded in a porous medium saturated by a nanofluid is numerically studied. The model used for the nanofluid incorporates only the effect of the volume fraction parameter. The surface of the plate is maintained at a constant temperature and a constant nano-particle volume fraction. The resulting governing partial differential equations are transformed into a set of two ordinary (similar) equations, which are solved using the bvp4c function from Matlab. A comparison is made with the available results in the literature, and the present results are in very good agreement with the known results. A representative set of numerical results for the reduced heat transfer from the plate, dimensionless velocity and temperature profiles is graphically and tabularly presented. Also, the salient features of the results are analyzed and discussed.     

Nonsimilarity boundary layer flow of copper–water nanofluid near stagnation point in a porous medium

The effects of nanoparticle shape are first introduced to study the nonsimilar solutions of stagnation point boundary layer flow of water–copper nanofluid saturated in a porous medium. Two cases of solid matrix of porous medium, including glass balls and aluminum foam, are considered. By using a new empirical correlation for the heat capacitance, thermal conductivity, and thermal diffusivity of the nanofluid saturated in a porous medium, the governing equations of the problem are constructed and reduced by dimensionless variables and nonsimilar transformations, and the homotopy analysis method is adopted to solve the partial differential equations. The results indicate that the heat transfer is significantly enhanced with the increase of permeability of the porous medium on the surface of the stagnation point boundary layer flow. In addition, it is found that the empirical shape of the nanoparticle has an impact on the heat transfer.     

Free convection boundary layer flow over a horizontal cylinder of elliptic cross section in porous media saturated by a nanofluid

This work studies the free convection boundary layer flow over a horizontal cylinder of elliptic cross section in porous media saturated by a nanofluid with constant wall temperature and constant wall nanoparticle volume fraction. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. A coordinate transformation is performed, and the obtained nonsimilar governing equations are then solved by the cubic spline collocation method. The effects of the Brownian motion parameter and thermophoresis parameter on the profiles of the temperature, nanoparticle volume fraction and velocity profiles are presented. The local Nusselt number is presented as a function of the thermophoresis parameter, Brownian parameter, Lewis number and the aspect ratio when the major axis of the elliptical cylinder is vertical (slender orientation) and horizontal (blunt orientation). Results show that the local Nusselt number is increased as the thermophoresis parameter or the Brownian parameter is decreased. The local Nusselt number increases as the buoyancy ratio or the Lewis number is decreased. Moreover, the local Nusselt number of the elliptical cylinder with slender orientation is higher than those of the elliptical cylinder with blunt orientation over the lower half cylinder.     

HEAT AND MASS TRANSFER FROM A PERMEABLE CYLINDER IN A POROUS MEDIUM WITH MAGNETIC FIELD AND HEAT GENERATION/ABSORPTION EFFECTS

We formulate the problem of coupled heat and mass transfer by natural convection from a horizontal cylinder embedded in a uniform porous medium in the presence of an external magnetic field and internal heat generation or absorption effects. The cylinder surface is maintained at a constant temperature and a constant concentration and is permeable to allow for possible fluid wall suction or blowing. The resulting governing equations are nondimensionalized and transformed into a nonsimilar form and then solved numerically by an implicit, iterative, finite-difference method. Comparisons with previously published work are performed and excellent agreement is obtained. A parametric study of the physical parameters is conducted and a representative set of numerical results for the stream function, temperature, concentration profiles, and the Nusselt and Sherwood numbers is illustrated graphically to show interesting features of the solutions.     

Numerical study of developing laminar forced convection of a nanofluid in an annulus

Laminar forced convection of a nanofluid consisting of Al₂O₃ and water has been studied numerically. Two dimensional elliptical governing equations have been solved to investigate the hydrodynamics and thermal behaviors of the fluid flow throughout an annulus. Single phase approach is used for the nanofluid modeling. The velocity and temperature profiles are presented in the fully developed region. The axial evolution of temperature, convective heat transfer coefficient and the friction coefficient at the inner and outer walls' region are shown and discussed. It is shown that the dimensionless axial velocity profile does not significantly change with the nanoparticle volume fraction. But, the temperature profiles are affected by the nanoparticle concentration. In general convective heat transfer coefficient increases with nanoparticle concentration. However, when the order of magnitude of heating energy is much higher than the momentum energy the friction coefficient depends on the nanoparticle concentration. At higher Reynolds numbers for which the momentum energy increases, this dependency on the nanoparticle volume fraction decreases.     

Entropy analysis of nanofluid magnetohydrodynamic convection flow past an inclined surface: A numerical review

A numerical investigation is conducted to review the entropy study of magnetohydrodynamic (MHD) convection nanofluid flow from an inclined surface. In evaluating the thermophoresis and Brownian motion impacts, Buongiorno's model is applied to nanofluid transfer. Using Keller's implicit box technique, the governing partial differential conservation equations and wall and free stream boundary conditions are made into the dimensionless form and solved computationally. For different thermos physical parameter values, the numerical results are discussed both graphically and numerically. Verification of the present code with previous Newtonian responses is also included. To analyze the variability in fluid velocity, temperature, nanoparticle volume fraction, entropy, Bejan number, shear stress rate, wall heat, and mass transfer rates, graphical and tabulated results are reported. The study suggests applications in the manufacturing of nanomaterial fabrication, and so on.     

Boundary Layer Flow of Rotating Two Phase Nanofluid Over a Stretching Surface

An analysis is carried out to discover the influence of a rotating nanofluid over a stretching surface. The two phase nanofluid model is used for this study. Two types of nanoparticles, namely copper and titanium oxide are used in our analysis with water as the base fluid. The governing system of partial differential equations along with the corresponding boundary conditions are presented and then transformed into a set of nonlinear ordinary differential equations using suitable similarity transformations. These equations are solved numerically by means of an iterative procedure called the midpoint integration scheme along with Richardson extrapolation. The results for flow and heat transfer characteristics are presented through graphs against nanoparticle volume fraction and rotation parameter for both types of nanoparticles. Quantities of physical interest such as local skin friction coefficients and local heat flux rate at the stretching surface are computed and analyzed. Numerical values for skin frictions and local heat flux rate are computed in the absence of nanoparticle volume fraction and rotation and they are found to be in very good agreement with the existing published literature.     

Nanofluid Flow in a Porous Channel with Suction and Chemical Reaction using Tiwari and Das's Nanofluid Model

In this paper, an analysis is made for a nanofluid flow in a porous channel by introducing the conservation equation of nanoparticle volume fraction into Tiwari and Das's nanofluid model. The suction and chemical reaction are also considered in this work. The governing partial differential equations are simplified by employing a new variable and transformed into a system of high‐order nonlinear ordinary differential equations by similarity transformations. The Keller box method is used to solve this problem numerically. In addition, the influences of significant physical parameters on the distributions of the velocity and temperature as well as nanoparticle concentration are graphically presented and discussed in detail. It is found that there exists a critical value of the permeable parameter which determines the influence law of nanoparticle volume fraction parameter on skin friction coefficient and local Sherwood number. The results also indicate that the concentration increases sharply with the Schmidt number and chemical reaction parameter.     

Effect of nanoparticle shapes on irreversibility analysis of nanofluid in a microchannel with individual effects of radiative heat flux,velocity slip and convective heating

In this study, the flow of a nanoliquid in a microchannel is examined. Two distinct metallic nanoparticles, titanium and silver, are used in this study. The slip regime and convective boundary are considered to compute the momentum and energy balance equation. The mathematical expressions are made dimensionless by using nondimensional quantities. A numerical approach called Runge‐Kutta‐Fehlberg scheme is employed to obtain the solution. Effects of the internal heat source and radiative flux on fluid model are examined. The upshots of the pertinent flow parameter and the physical features are visualized through graphical elucidations. The effect of flow constraints on the second law analysis for the described physical phenomenon is predicted. Conclusion indicates that lowering of temperature of the nanofluid is obtained by higher values of nanoparticle volume fraction. The causes of irreversibility in a thermal system is explored in this investigation. The results indicate that nonspherical nanoparticles has higher thermal conductivity ratio as compared with spherical nanoparticles. Minimization of entropy can be attained through increasing volume fraction of titanium and silver nanoparticles. Besides, it is emphasized that entropy generation is high in case of disc‐shaped nanoparticles, followed by needle and sphere shapes.     

Nonsimilar boundary layer flow of Cross fluid induced by a heated stretched sheet

This paper deals with the nonisothermal boundary layer flow of Cross fluid due to a stretching sheet. Unlike previous studies on boundary layer flow of Cross fluid, a nonsimilar formulation is adopted to transform the boundary layer equations into nondimensional form. The problem is characterized by three dimensionless parameters, namely, the Deborah number, the Prandtl number, and dimensionless distance along the sheet. The transformed equations are simulated by a numerical scheme with the help of MAPLE software. The velocity and temperature profiles inside the boundary layer are calculated and shown graphically. The skin friction coefficient and Nusselt number at various axial stations are also tabulated for several values of Deborah number and Prandtl number.     

Insight into the dynamics of micropolar fluid about a vertical cone when nonlinear thermal radiation is significant: The case of triple mixed convection

This novel research investigates the nonlinear triple diffusive combined convective micropolar liquid flow past a vertical cone in the presence of nonlinear thermal radiation, cross-diffusion, and a convective boundary condition. We aim to analyze this present problem using nonsimilar transformations. This report presents the significance of nonlinear mixed convection, energy flux due to the concentration gradient, and mass flux due to the temperature gradient and nonlinear thermal radiation in the dynamics of the fluid subject with micropolar fluid is presented. The differential equations defining the boundary-layer parameters are then transformed into dimensionless view, taking into account the nonsimilar transformation. Furthermore, the method of quasilinearization and implicit finite difference approximation is used to work out the nondimensional governing equations for the solution. The velocity pattern diminishes, while dimensionless temperature and concentration distributions enhance with growing values of microrotation parameter. Furthermore, species concentrations of the fluid increase with increasing Soret effect values, while opposite results appear for mass transfer rates. Also, drag coefficient enhances for assisting buoyancy flow whilst diminishes for opposing buoyancy flow with increasing values of the microrotation parameter. The microrotation pattern reduces with growing values of the nonsimilarity characteristics. Furthermore, the Prandtl number is displayed on a comparison graph, and the results are very similar.     

Unsteady mixed convection on a wedge in a porous medium

A numerical solution has been presented for the unsteady mixed convection on a wedge which is embedded in a fluid-saturated porous medium. The flow is impulsively sets into motion from rest and the temperature of the wedge is also suddenly changed from that of the ambient fluid. A second-order upwind finite-difference scheme has been used to solve the governing nonsimilar equations. The results are found to be in excellent agreement with some previously published data.     

Natural Convection Flow with Surface Radiation Along a Vertical Wavy Surface

In this study, natural convection boundary layer flow of thermally radiating fluid along a heated vertical wavy surface is analyzed. Here, the radiative component of heat flux emulates the surface temperature. Governing equations are reduced to dimensionless form, subject to the appropriate transformation. Resulting dimensionless equations are transformed to a set of parabolic partial differential equations by using primitive variable formulation, which are then integrated numerically via iterative finite difference scheme. Emphasis has been given to low Prandtl number fluid. The numerical results obtained for the physical parameters, such as, surface radiation parameter, R, and radiative length parameter, ξ, are discussed in terms of local skin friction and Nusselt number coefficients. Comprehensive interpretation of velocity distribution is also given in the form of streamlines.     

Convective Heat Transfer in a Vertical Rectangular Duct Filled with a Nanofluid

An analysis is performed to study natural convective heat transfer in a vertical rectangular duct filled with a nanofluid. One of the vertical walls of the duct is cooled by a constant temperature, while the other wall is heated by a constant temperature. The other two sides of the duct are thermally insulated. The transport equations for a Newtonian fluid are solved numerically with a finite volume method of second‐order accuracy. The influence of pertinent parameters such as Grashof number, Brinkman number, aspect ratio and solid volume fraction on the heat transfer characteristics of natural convection is studied. Results for the volumetric flow rate and skin friction for Copper and Diamond nanoparticles are also drawn. The Nusselt number for various types of nanoparticle such as silver, copper, diamond and titanium oxide are also tabulated. The results indicate that inclusion of nanoparticles into pure water improves its heat transfer performance; however, there is an optimum solid volume fraction which maximizes the heat transfer rate.     

Heat Transfer Phenomena of Assemblages of Smooth Slip Spheres in Newtonian Fluids

Combined effects of slip velocity and volume fraction of slip spheres on the heat transfer characteristics of multiple slip spheres are numerically investigated within the framework of a free surface cell model combined with a linear slip velocity along the surface of the slip spheres. The governing conservation equations of the mass, momentum, and energy are solved by a segregated approach using a simplified marker and cell algorithm implemented on a staggered grid arrangement in spherical coordinates. The convection and diffusion terms of conservation equations are discretized using quadratic upstream interpolation for convective kinematics and second‐order central differencing schemes, respectively. Prior to obtaining new results, this numerical solver is validated by comparison of present results with the existing literature values. Further new results are obtained for a range of conditions as; Reynolds number, Re: 0.1–200; Prandtl number, Pr: 1–100; volume fraction of slip spheres, Φ: 0.1–0.5 and slip parameter, λ: 0.01–100. The effects of these dimensionless parameters on isotherm contours and local and average Nusselt numbers are thoroughly delineated. Finally, a new empirical correlation for the average Nusselt number of multiple smooth slip spheres is proposed on the basis of present numerical results.     

Heat and Mass Transfer by Mixed Convection from a Vertical Slender Cylinder with Chemical Reaction and Soret and Dufour Effects

This work is focused on the study of heat and mass transfer by mixed convection over a vertical slender cylinder in the presence of chemical reaction and thermal‐diffusion and diffusion‐thermo effects. The resulting equations have the property whereby they reduce to various special cases previously considered in the literature. An adequate implicit, tri‐diagonal finite‐difference scheme is employed for the numerical solution of the obtained equations. Various comparisons with previously published work are performed and the results are found to be in excellent agreement. Representative results for the local skin‐friction coefficient, local Nusselt number, and the local Sherwood number illustrating the influence of the surface transverse curvature parameter, Richardson number, concentration to thermal buoyancy ratio, Schmidt number, chemical reaction, and the Dufour and Soret numbers are presented and discussed. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 42(7): 618–629, 2013; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21045