Thermophoresis and MHD mixed convection three-dimensional flow of viscoelastic fluid with Soret and Dufour effects

Heat and mass transfer effects in three-dimensional mixed convection flow of viscoelastic fluid over a stretching surface with convective boundary conditions are investigated. The fluid is electrically conducting in the presence of constant applied magnetic field. Conservation laws of energy and concentration are based upon the Soret and Dufour effects. First order chemical reaction effects are also taken into account. By using the similarity transformations, the governing boundary layer equations are reduced into the ordinary differential equations. The transformed boundary layer equations are computed for the series solutions. Dimensionless velocity, temperature, and concentration distributions are shown graphically for different values of involved parameters. Numerical values of local Nusselt and Sherwood numbers are computed and analyzed. It is found that the behaviors of viscoelastic, mixed convection, and concentration buoyancy parameters on the Nusselt and Sherwood numbers are similar. However, the Nusselt and Sherwood numbers have qualitative opposite effects for Biot number, thermophoretic parameter, and Soret-Dufour parameters.

     

Convection from an inverted cone in a porous medium with cross-diffusion effects

In this study convection from an inverted cone in a porous medium with cross-diffusion is studied numerically. Diffusion-thermo and thermo-diffusion effects are assumed to be significant. The governing equations are transformed into nonlinear ordinary differential equations and then solved numerically using a shooting method together with a sixth order Runge–Kutta method. Verification of the accuracy and correctness of the results is achieved by solving the equations using an independent linearisation method. The effects of the Dufour and the Soret parameters are investigated. The results for the skin friction, Nusselt number and the Sherwood number are presented graphically and in tabular form.     

Velocity and temperature slip effects on squeezing flow of nanofluid between parallel disks in the presence of mixed convection

Velocity and temperature slip effects on squeezing flow of nanofluid between parallel disks in the presence of mixed convection is considered. Equations that govern the flow are transformed into a set of differential equations with the help of transformations. For the purpose of solution, homotopy analysis method is used. The BVPh2.0 package is utilized for the said purpose. Deviations in the velocity, temperature and the concentration profiles are depicted graphically. Mathematical expressions for skin friction coefficient, Nusselt and the Sherwood numbers are derived and the variations in these numbers are portrayed graphically. From the results obtained, we observed that the coefficient of skin friction increases with increase in Hartmann number M for the suction flow (A > 0), while in the blowing flow (A < 0) a fall is seen with increasing M. However, for rising values of velocity parameter β the effect of skin friction coefficient is opposite to that accounted for M. Variations in thermophoresis parameter N _T and thermal slip parameter γ give rise in Nusselt number for both the suction and injection at wall. For both the suction and injection at wall, Sherwood number gets a rise with growing values of Brownian motion parameter N _B, while a drop is seen in Sherwood number for increasing values of thermophoresis parameter N _T. For the sake of comparison, the same problem is also solved by employing a numerical scheme called Runge–Kutta–Fehlberg (RKF) method. Results thus obtained are compared with existing ones and are found to be in agreement.

     

Thermal radiation and slip effects on MHD stagnation point flow of non-Newtonian nanofluid over a convective stretching surface

The present analysis examines the combine effects of thermal radiation and velocity slip along a convectively nonlinear stretching surface. Moreover, MHD effects are also considered near the stagnation point flow of Casson nanofluid. Slipped effects are considered with the porous medium to reduce the drag reduction at the surface of the sheet. Main structure of the system is based upon the system of partial differential equations attained in the form of momentum, energy, and concentration equations. To determine the similar solution system of PDEs is rehabilitated into the set of nonlinear ordinary differential equations (ODEs) by employing compatible similarity transformation. Important physical parameters are acquired through obtained differential equations. To determine the influence of emerging parameters, resulting set of ODE’s in term of unknown function of velocity, temperature, and concentration are successfully solved via Keller’s box-scheme. All the obtained unknown functions are discussed in detail after plotting the results against each physical parameter. To analyze the behavior at the surface: skin friction, local Nusselt and Sherwood numbers are also illustrated against the velocity ratio parameter A, Brownian motion Nb, thermophoresis Nt, and thermal radiation parameters R. Results obtained from the set of equations described that skin friction is decreasing function of A, and local Nusselt and Sherwood number demonstrate the significant influenced by Brownian motion Nb, thermophoresis Nt, and radiation parameters R.

     

Numerical study of radiative Maxwell viscoelastic magnetized flow from a stretching permeable sheet with the Cattaneo–Christov heat flux model

In this article, the Cattaneo–Christov heat flux model is implemented to study non-Fourier heat and mass transfer in the magnetohydrodynamic flow of an upper-convected Maxwell fluid over a permeable stretching sheet under a transverse constant magnetic field. Thermal radiation and chemical reaction effects are also considered. The nonlinear partial differential conservation equations for mass, momentum, energy and species conservation are transformed with appropriate similarity variables into a system of coupled, highly nonlinear ordinary differential equations with appropriate boundary conditions. Numerical solutions have been presented for the influence of elasticity parameter (α), magnetic parameter (M ²), suction/injection parameter \((\lambda ),\) Prandtl number (Pr), conduction–radiation parameter (R _d ), sheet stretching parameter (A), Schmidt number (Sc), chemical reaction parameter \(\left( {\gamma_{c} } \right)\), modified Deborah number with respect to relaxation time of heat flux (i.e., non-Fourier Deborah number) on velocity components, temperature and concentration profiles using the successive Taylor series linearization method (STSLM) utilizing Chebyshev interpolating polynomials and Gauss–Lobatto collocation. The effects of selected parameters on skin friction coefficient, Nusselt number and Sherwood number are also presented with the help of tables. Verification of the STSLM solutions is achieved with existing published results demonstrating close agreement. Further validation of skin friction coefficient, Nusselt number and Sherwood number values computed with STSLM is included using Mathematica software shooting quadrature.

     

Magnetohydrodynamic three-dimensional nonlinear convective flow of viscoelastic nanofluid with heat and mass flux conditions

The present research focuses on three-dimensional nonlinear convective flow of viscoelastic nanofluid. Here, the flow is generated due to stretching of a impermeable surface. The phenomenon of heat transport is analyzed by considering thermal radiation and prescribed heat flux condition. Nanofluid model comprises of Brownian motion and thermophoresis. An electrically conducting fluid is accounted due to consideration of an applied magnetic field. The dimensionless variables are introduced for the conversion of partial differential equations into sets of ordinary differential systems. The transformed expressions are explored through homotopic algorithm. Behavior of different dimensionless parameters on the non-dimensional velocities, temperature and concentration are scrutinized graphically. The values of skin friction coefficients, Nusselt and Sherwood numbers are also calculated and elaborated. It is visualized that the heat transfer rate increases with Prandtl number and radiation parameter is higher.

     

MHD flow of radiative micropolar nanofluid in a porous channel: optimal and numerical solutions

The flow of a radiative and electrically conducting micropolar nanofluid inside a porous channel is investigated. After implementing the similarity transformations, the partial differential equations representing the radiative flow are reduced to a system of ordinary differential equations. The subsequent equations are solved by making use of a well-known analytical method called homotopy analysis method (HAM). The expressions concerning the velocity, microrotation, temperature, and nanoparticle concentration profiles are obtained. The radiation tends to drop the temperature profile for the fluid. The formulation for local Nusselt and Sherwood numbers is also presented. Tabular and graphical results highlighting the effects of different physical parameters are presented. Rate of heat transfer at the lower wall is seen to be increasing with higher values of the radiation parameter while a drop in heat transfer rate at the upper wall is observed. Same problem has been solved by implementing the numerical procedure called the Runge–Kutta method. A comparison between the HAM, numerical and already existing results has also been made.

     

Double-diffusive convection with variable viscosity from a vertical truncated cone in porous media in the presence of magnetic field and radiation effects

This work is focused on the study of combined heat and mass transfer or double-diffusive convection near a vertical truncated cone embedded in a fluid-saturated porous medium in the presence of thermal radiation, magnetic field and variable viscosity effects. The viscosity of the fluid is assumed to be an inverse linear function of the fluid temperature. A boundary-layer analysis is employed to derive the non-dimensional governing equations. The governing equations for this investigation are transformed into a set of non-similar equations and solved numerically using the fourth-order Runge–Kutta integration scheme with the Newton–Raphson shooting technique. Comparisons with previously published work on special cases of the problem are performed and the results are found to be in excellent agreement. A parametric study illustrating the influence of the radiation parameter, magnetic field parameter, viscosity-variation parameter, buoyancy ratio and the Lewis number on the fluid velocity, temperature and solute concentration profiles as well as the Nusselt number and Sherwood number is conducted. The results of this parametric study are shown graphically and the physical aspects of the problem are highlighted and discussed.     

Onset of double-diffusive convection of a sparsely packed micropolar fluid in a porous medium layer saturated with a nanofluid

The onset of convection of a sparsely packed micropolar fluid in a porous medium layer saturated by a nanofluid is examined by using a linear and nonlinear stability analyses. The Darcy–Brinkman–Forchheimer model is employed for the porous medium layer. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The critical Rayleigh number, wave number for stationary and oscillatory modes and frequency of oscillations are obtained analytically using linear theory, and the nonlinear analysis is made with minimal representation of the truncated Fourier series analysis involving only two terms. The effect of various parameters on the stationary and oscillatory convections is shown pictorially. The dependence of stationary or oscillatory convection on the porous parameter and parameters involved in micropolar fluids is also discussed. We also study the effect of time on transient Nusselt number and Sherwood number which are found to be oscillatory when time is small. However, when time becomes very large, both the transient Nusselt value and Sherwood value approach to their steady-state values.     

降膜蒸发过程的数值模拟和传热传质分析

利用CFD软件对逆流降膜蒸发过程进行了实验模拟研究,研究了速度边界层、热边界层和浓度边界层的变化对降膜蒸发传热传质特性的影响规律;通过建立一维逆流降膜蒸发的数学方程编程求解出了对流传热传质的Nu数和Sh数,利用Fluent软件模拟出的实验结果采用回归分析得出了气液流量比Raw、流道的长宽比αL、空气进口无量纲温度θai以及空气进口Re数与Nu数、Sh数之间的无量纲关系式,可为降膜蒸发换热器的设计提供参考。     

Magnetohydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet: a numerical study

A numerical simulation for mixed convective three-dimensional slip flow of water-based nanofluids with temperature jump boundary condition is presented. The flow is caused by nonlinear stretching surface. Conservation of energy equation involves the radiation heat flux term. Applied transverse magnetic effect of variable kind is also incorporated. Suitable nonlinear similarity transformations are used to reduce the governing equations into a set of self-similar equations. The subsequent equations are solved numerically by using shooting method. The solutions for the velocity and temperature distributions are computed for several values of flow pertinent parameters. Further, the numerical values for skin-friction coefficients and Nusselt number in respect of different nanoparticles are tabulated. A comparison between our numerical and already existing results has also been made. It is found that the velocity and thermal slip boundary condition showed a significant effect on momentum and thermal boundary layer thickness at the wall. The presence of nanoparticles stabilizes the thermal boundary layer growth.

     

Non-coaxial rotating flow of viscous fluid with heat and mass transfer

Heat and mass transfer in unsteady non-coaxial rotating flow of viscous fluid over an infinite vertical disk is investigated. The motion in the fluid is induced due to two sources. Firstly, due to the buoyancy force which is caused because of temperature and concentration gradients. Secondly, because of non-coaxial rotation of a disk such that the disk executes cosine or since oscillation in its plane and the fluid is at infinity. The problem is modeled in terms of coupled partial differential equations with some physical boundary and initial conditions. The dimensionless form of the problem is solved via Laplace transform method for exact solutions. Expressions for velocity field, temperature and concentration distributions are obtained, satisfying all the initial and boundary conditions. Skin friction, Nusselt number and Sherwood number are also evaluated. The physical significance of the mathematical results is shown in various plots and is discussed for several embedded parameters. It is found that magnitude of primary velocity is less than secondary velocity. In limiting sense, the present solutions are found identical with published results.

     

Stability analysis and numerical simulation of gravitactic bioconvection in a rectangular cavity

The bioconvection of gravitactic microorganisms, through linear analysis and numerical simulation is presented. Using the basic state as initial condition for both microorganisms and streamlines, the critical Rayleigh number and the bioconvection are predicted. The dynamic microorganisms’ behavior is influenced by the initial spatial distribution. The stability of the system is dependent on the horizontal wave component that is inversely related to wavelength. The pattern length and the instability of the process are associated with the horizontal component of the wave number and the characteristic wavelength, respectively. Using complex and real eigenvalues, five unstable and three stable rolls are found respectively. The three stable rolls is the dominant pattern when varying the principal variables of this bioconvective process.     

A mathematical model of MHD nanofluid flow having gyrotactic microorganisms with thermal radiation and chemical reaction effects

In this article, we have examined three-dimensional unsteady MHD boundary layer flow of viscous nanofluid having gyrotactic microorganisms through a stretching porous cylinder. Simultaneous effects of nonlinear thermal radiation and chemical reaction are taken into account. Moreover, the effects of velocity slip and thermal slip are also considered. The governing flow problem is modelled by means of similarity transformation variables with their relevant boundary conditions. The obtained reduced highly nonlinear coupled ordinary differential equations are solved numerically by means of nonlinear shooting technique. The effects of all the governing parameters are discussed for velocity profile, temperature profile, nanoparticle concentration profile and motile microorganisms’ density function presented with the help of tables and graphs. The numerical comparison is also presented with the existing published results as a special case of our study. It is found that velocity of the fluid diminishes for large values of magnetic parameter and porosity parameter. Radiation effects show an increment in the temperature profile, whereas thermal slip parameter shows converse effect. Furthermore, it is also observed that chemical reaction parameter significantly enhances the nanoparticle concentration profile. The present study is also applicable in bio-nano-polymer process and in different industrial process.

     

Gaseous slip flow forced convection through ordered microcylinders

This is a theoretical study dealing with longitudinal gaseous slip flow forced convection between a periodic bunch of microcylinders arranged in regular array. The selected geometry has applications in microscale pin fin heat sinks used for cooling of microchips. The flow is considered to be hydrodynamically and thermally fully developed. The two axially constant heat flux boundary conditions of H1 and H2 are considered in the analysis. The velocity and temperature discontinuities at the boundary are incorporated into the solutions using the first order slip boundary conditions. The method considered is mainly analytical in which the governing equations and three of the boundary conditions are exactly satisfied. The remaining symmetry condition on the right-hand boundary of the typical element is applied to the solution through the point matching technique. The results show that both the Poiseuille number and the Nusselt number are decreasing functions of the degree of rarefaction characterized by the Knudsen number. While an increase in the blockage ratio leads to a higher Poiseuille number, the functionality of the Nusselt number on this parameter is not monotonic. At small and moderate values of the blockage ratio, the Nusselt number is higher for a higher blockage ratio, whereas the opposite may be right for higher values of this parameter. It is also observed that the angular variations of the parameters are reduced at smaller blockage ratios. Accordingly, the H1 and H2 Nusselt numbers are the same for small and moderate blockage ratios.     

Natural convection boundary layer of a non-Newtonian fluid about a permeable vertical cone embedded in a porous medium saturated with a nanofluid

An analysis was performed to study the effect of uniform transpiration velocity on free convection boundary-layer flow of a non-Newtonian fluid over a permeable vertical cone embedded in a porous medium saturated with a nanofluid. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing partial differential equations are transformed into a set of non-similar equations and solved numerically by an efficient 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 velocity, temperature, and volume fraction profiles as well as the local Nusselt and Sherwood numbers is illustrated graphically to show interesting features of the solutions.     

Effect of heat generation on mixed convection in porous cavity with sinusoidal heated moving lid and uniformly heated or cooled bottom walls

Effect of heat generation and absorption on mixed convection flows in a sinusoidal heated lid-driven square cavity filled with a porous medium is investigated numerically. Both the vertical walls of the enclosure are insulated while the bottom wall is uniformly heated or cooled. The top wall is moving at a constant speed and is heated sinusoidally. The governing equations and boundary conditions are non-dimensionalized and solved numerically by using finite volume method approach along with SIMPLE algorithm together with non-uniform grid system. The effect of Darcy and heat generation parameters are investigated in terms of the flow, heat transfer, and Nusselt number. The results for stream function and isotherm are plotted and it is found that there have significant influence with the presence of heat generation and porous medium.

     

Analysis of magnetohydrodynamic flow and heat transfer of Cu–water nanofluid between parallel plates for different shapes of nanoparticles

The present study analyzes the heat transfer in the flow of copper–water nanofluids between parallel plates. For effective thermal conductivity of nanofluids, Hamilton and Crosser's model has been utilized to examine the flow by considering different shape factors. By employing the suitable similarity transformations, the equations governing the flow are transformed into a set of nonlinear ordinary differential equations. The resulting set of equations is solved numerically with the help of Runge–Kutta–Fehlberg numerical scheme. The graphical simulation presents the analysis of variations, in velocity and temperature profiles, for emerging parameters. A comprehensive discussion also accompanies the graphical results. Moreover, the effects of relevant parameters, on skin friction coefficient and Nusselt number, are highlighted graphically. It is noticed that the velocity field is an increasing function of all the parameters involved. Furthermore, the temperature of the fluid is maximum for the platelet-shaped particles followed by the cylinder- and brick-shaped particles.

     

Numerical investigation of heat and mass transfer flow under the influence of silicon carbide by means of plasma-enhanced chemical vapor deposition vertical reactor

The effect of characteristics flow (contour of velocity), mass transfer (Sherwood number) and heat transfer (Nu number) on the growth rate of silicon carbide by means of plasma-enhanced chemical vapor deposition vertical reactor is investigated. The species transport and thermal fluid transport with chemical reaction are taken into account. The steady-state laminar fluid flow and gas flow having ideal behavior are considered. A mixture of silane and propane (2% molar) as main reactant gases and hydrogen (96% molar) as propellant gas are injected into the reactor. Four different diameters of shower head, three different substrate rotation speeds and five different temperatures of the substrate are used. The finite volume method is employed to solve the problem. The governing equations are solved by upwind differencing scheme. The assumption of speed–pressure coupling leads to use of semi-implicit method for pressure-linked equations to solve the governing equation. It is found that the deposition rate reduces with the shower head diameter and value of substrate temperature and enhances with rotational speed of the substrate. Furthermore, the best shower head diameter to achieve maximum rate of deposition is 1 mm. At the end, a comparison as a limiting case of the considered problem with the existing studies is made. Comparing the results of this experiment with prior studies has shown acceptable consistency.

     

Multi-objective optimization of the heat transmission and fluid forces around a rounded cornered square cylinder

The unsteady fluid stream and warmth transmission nearby a square cylinder with sharp and rounded cornered edges are numerically examined, and then the roundness of the corner is predicted and optimized for the minimum fluid forces and maximum heat transmission rate. The roundness of the cylinder corner is changing 0.5D (circle) to 0.71D (square); D is the depth of the cylinder. The fluid flow and the heat transmission features around the sharp and curved cornered square cylinder are evaluated with the streamline, isotherm patterns, pressure coefficient, drag and lift coefficients, local Nusselt number (Nu_local) and average Nusselt number (Nu_avg) at different Re and for several roundness values. These characteristics are predicted by the gene expression programming, and then the multi-objective genetic algorithm is utilized for the optimization. A number of combinations of values of corners have been found in the form of Pareto-optimal solution to compromise the minimum fluid forces with maximum heat transfer rate.