Exact analytical solution of a convective diffusion from a wedge to a flow with a first order chemical reaction at the surface

The problem of mass transfer in a fluid flow past a wedge with a chemical reaction of the first order at the surface is investigated. The approach suggested previously by Apelblat [1] for the solution of a mass transfer problem with a first order chemical reaction at the interface in boundary layer flow is generalized for the case of flow past a wedge. The solution of the problem is derived in a closed analytical form.     

Mass transfer on a continuous flat plate moving in parallel or reversely to a free stream in the presence of a chemical reaction

An analysis is made to investigate the mass transfer in two-dimensional boundary layer flow past a flat plate moving in parallel or reversely to a free stream with first order chemical reaction. The similarity transformations are used to transform the governing equations and the reduced nonlinear self-similar ordinary differential equations are solved numerically using a shooting method. The dual solutions of the velocity and concentration distributions are obtained. With increase of the Schmidt number the mass transfer is found to be enhanced for the upper solution branch and reduced for the lower solution branch. The concentration overshoot for constructive chemical reaction is observed, i.e., for constructive reaction mass absorption occurs. Moreover, for destructive chemical reaction the mass transfer increases.     

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.     

Effect of Prandtl number on leading edge accretion and ablation: A numerical study of unsteady boundary layer flow over a flat plate

An efficient numerical method, namely, the Runge‐Kutta fourth order integration scheme with shooting technique is employed to give a suitable solution for the unsteady magnetohydrodynamic boundary layer flow of viscous incompressible fluid with accretion or ablation effects over a flat plate under the influence of homogenous first order chemical reaction. When compared to the other numerical techniques such as perturbation methods, this approach provides the accurate numerical results valid uniformly for all nondimensional time. The unsteady behavior of chemically reacting magnetohydrodynamic boundary layer flow is investigated by analyzing the nature of buoyancy and magnetic parameters in the momentum equation. Also, results are extended to the energy and concentration equations by considering the viscous dissipation, Joule heating and chemical reaction effects. With the help of suitable similarity transformations, the highly nonlinear, coupled, time‐dependent partial differential equations are reduced to ordinary differential equations. Furthermore, the numerical solutions in terms of velocity, temperature and concentration profiles within the boundary layer are presented for the various values of control parameters. Also, the impact of physical parameters on the flow, heat and mass transfer characteristics are examined thoroughly. The present investigation reports that, the increasing magnetic parameter increases the temperature field and decreases the velocity field. Also, Eckert number enhance the thermal field whereas, the chemical reaction parameter decays the concentration field. Before concluding the considered problem, present results are validated with the previous results and are found to be in good agreement.     

Effect of chemical reaction and thermal radiation on heat and mass transfer flow of MHD micropolar fluid in a rotating frame of reference

This paper studies the effect of first order chemical reaction and thermal radiation on hydromagnetic free convection heat and mass transfer flow of a micropolar fluid via a porous medium bounded by a semi-infinite porous plate with constant heat source in a rotating frame of reference. The plate is assumed to oscillate in time with constant frequency so that the solutions of the boundary layer are the same oscillatory type. The dimensionless governing equations for this investigation are solved analytically using small perturbation approximation. The effect of the various dimensionless parameters entering into the problem on the velocity, temperature and concentration profiles across the boundary layer are investigated through graphs. Also the results of the skin friction coefficient, couple stress coefficient, the rate of heat and mass transfer at the wall are prepared with various values of the parameters.     

Dual solutions in boundary layer stagnation-point flow and mass transfer with chemical reaction past a stretching/shrinking sheet

In this paper, an analysis is presented to study dual nature of solution of mass transfer with first order chemical reaction in boundary layer stagnation-point flow over a stretching/shrinking sheet. The governing equations are transformed into a set of self-similar ordinary differential equations by similarity transformations. The transformed equations are solved numerically using very efficient shooting method. The study reveals that the dual solutions of velocity and concentration exist for certain values of velocity ratio parameter (the ratio of stretching/shrinking rate and straining rate). The concentration boundary layer thickness decreases with increasing values of Schmidt number and reaction-rate parameter for both solutions.     

Thermal analysis of buoyancy-motivated Casson fluid flow with time-independent chemical reaction under Lorentz forces

The present research study examines the magneto-hydrodynamic natural convection visco-elastic boundary layer of Casson fluid past a nonlinear stretching sheet with Joule and viscous dissipation effects under the influence of chemical reaction. To differentiate the visco-elastic nature of Casson fluid with Newtonian fluids, an established Casson model is considered. The present physical problem is modeled by utilizing the considered geometry. The resulting system of coupled nonlinear partial differential equations is reduced to a system of nonlinear ordinary differential equations by applying suitable similarity transformations. Numerical solutions of these reduced nondimensional governing flow field equations are obtained by applying the Runge-Kutta integration scheme with the shooting method (RK-4). The physical behavior of different control parameters is described through graphs and tables. The present study describes that the velocity and temperature profiles decreased for increasing values of Casson fluid parameter. Velocity field diminished for the increasing nonlinear parameter whereas velocity profile magnified for increasing free convection parameter. Thermal field enhanced with increasing magnetic parameter in the flow regime. The concentration profile decreased for the rising values of the chemical reaction parameter. The magnitude of the skin-friction coefficient enhanced with increasing magnetic parameter. Increasing Eckert number increases the heat transfer rate and increasing chemical reaction parameter magnifies the mass transfer rate. Finally, the similarity results presented in this article are excellently matched with previously available solutions in the literature.     

Nonlinear radiation on Maxwell fluid in a convective heat transfer with viscous dissipation and activation energy

The present analysis addresses linear and nonlinear radiation effects in hydrodynamic viscous Maxwell fluid flow on a unidirectional stretching surface through viscous dissipation. The relaxation effect is considered in the mathematical model, which elucidates mass transport mechanisms under binary chemical reaction and activation energy. Mathematical modeling contains nonlinear partial differential equations using boundary conditions. Appropriate transformations convert the partial differential equations into ordinary differential equations. Numerical solutions for regular differential equations are brought by Runge–Kutta–Fehlberg numerical quadrature and a shooting method with a tolerance level of 10⁻⁹. The influence of physical variables, such as Deborah relaxation number, rotation parameter, Biot number, activation energy parameter, reaction rate parameter, Eckert number, and Prandtl number are investigated. Increasing the Biot number improves the temperature region in the boundary layer. With high rotation, the increasing Deborah number enhances the fluid temperature substantially throughout the boundary layer.     

Heat and mass transfer of an unsteady MHD natural convection flow of a rotating fluid past a vertical porous flat plate in the presence of radiative heat transfer

Analytical closed-form solution of the unsteady hydro-magnetic natural convection heat and mass transfer flow of a rotating, incompressible, viscous Boussinesq fluid is presented in this study in the presence of radiative heat transfer and a first order chemical reaction between the fluid and the diffusing species. The Rosseland approximation for an optically thick fluid is invoked to describe the radiative flux. Results obtained show that a decrease in the temperature boundary layer occurs when the Prandtl number and the radiation parameter are increased and the flow velocity approaches steady state as the time parameter t, is increased. These findings are in quantitative agreement with earlier reported studies.     

Magnetohydrodynamic chemically reactive viscoelastic fluid flow through an inclined porous layer

Analysis of a time-independent magnetohydrodynamic viscoelastic fluid flow in a deformable inclined porous layer with first-order chemical reaction has been investigated. Walters' fluid model has been used to study viscoelastic fluid. The walls are suctioned/injected at a constant rate. The expression representing the solution for solid displacement, fluid velocity, temperature, and concentration distribution is obtained. The effect of applicable parameters on solid displacement, fluid velocity, temperature, and concentration are discussed graphically, while skin friction, heat transfer, and mass transfer are revealed in a tabular structure. It is noticed that solid displacement, fluid velocity, and temperature profiles decrease when the viscoelastic parameter increase. Solid displacement enhances and the velocity of the fluid reduces owing to the influence of increasing drag parameter, whereas the reverse effect is seen for the volume fraction parameter. Nusselt number at the walls shows the opposite behavior for the viscoelastic parameter and Eckert number. Sherwood number at the walls shows opposite behavior for Reynolds number, Schmidt number, and radiation parameter. Also, the entropy generation number rises as a result of the influence of viscoelasticity and Eckert number.     

On the validity of the quasi-steady state equation for heat or mass transfer problems with an axially moving boundary

The following two questions related to the quasi-steady state equation for heat or mass transfer problems are answered in this paper: (1) “Under what process (moving velocity, time, and length) and material parameters (thermal or mass diffusivity) the quasi-steady state equation is valid ?” and (2) “How large are the errors associated with the quasi-steady state assumption when solving for heat or mass transfer?” To answer these questions, an exact analytical solution for the unidirectional transient (unsteady) heat or mass transfer problems with an axially moving boundary in semi-infinite domains is derived. Then, a comparison between the transient and quasi-steady state solutions is provided for the temperature (or concentration) distribution, temperature (or concentration) gradient at the moving boundary, and thermal (or solutal) boundary layer thickness. It is also demonstrated that the solution developed in this work is the general solution applicable for both the transient state and the quasi-steady state.     

Natural convection boundary layer with suction and mass transfer in a porous medium

The free convection boundary layer flow with simultaneous heat and mass transfer in a porous medium is studied when the boundary wall moves in its own plane with suction. The study also incorporates chemical reaction for the very simple model of a binary reaction with Arrhenius activation energy. For large suction, asymptotic approximate solutions are obtained for the flow variables for various values of the activation energy.     

Entropy generation on MHD flow of a Casson fluid over a curved stretching surface with exponential space-dependent heat source and nonlinear thermal radiation

The present model concentrates on entropy generation on a steady incompressible flow of a Casson liquid past a permeable stretching curve surface through chemical reaction and magnetic field effects. The exponential space-dependent heat source cum heat and mass convective boundary conditions are accounted for. The resulting nonlinear boundary layer model is simplified by the transformation of similarity. Chebyshev spectral technique is involved for obtaining numerical results of the converted system of the mathematical models. Behavior of the determining thermo-physical parameters on the profiles of velocity, temperature, concentration, skin friction, heat, mass transfer rate, rate of entropy generation, and finally the Bejan number are presented. The major point of the present investigation show that the curvature term weakens the mass transfer profile as the fluid temperature reduces all over the diffusion regime. A decrease in heat generation strengthens the species molecular bond, which prevents free Casson particle diffusion. Furthermore, the mass transfer field diminishes in suction and injection flow medium.     

Effect of chemical reaction and heat generation or absorption on double-diffusive convection from a vertical truncated cone in porous media with variable viscosity

This work is focused on the study of combined heat and mass transfer on double-diffusive convection near a vertical truncated cone in a fluid-saturated porous medium in the presence of a first-order chemical reaction and heat generation or absorption with variable viscosity. The viscosity of the fluid is assumed to be an inverse linear function of the temperature. A boundary layer analysis is employed to derive the non-dimensional non-similar governing equations. The governing equations for this investigation are formulated and solved numerically using the fourth-order Runge–Kutta integration scheme with Newton–Raphson shooting technique. Comparisons with previously published work on special cases of the problem are performed and found to be in excellent agreement. A parametric study illustrating the influence of chemical reaction parameter, heat generation or absorption parameter, viscosity-variation parameter, buoyancy ratio and Lewis number on the fluid velocity, temperature, concentration as well as 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.     

Transient natural convective flow of a radiative viscous incompressible fluid past an exponentially accelerated porous plate with chemical reaction species

The present study deals with an unsteady magnetohydrodynamic natural convective flow of a viscous, incompressible fluid past an exponentially accelerated porous plate surrounded by a porous medium with suction or injection. The novelty of the current research is to analyze the behavior of the flow due to mass transfer with first-order chemical reaction in the presence of a heat source in the energy equation. The existence of suction/injection and radiation parameters in the flow enhances the utility of the research as they are an integral part of nuclear reactors, thermal and chemical engineering processes, and many more. The Laplace transform technique (via Bromwich contour) is applied to solve exactly the governing equations. The nature of the flow velocity, temperature, and concentration profiles due to the impact of pertinent flow parameters are presented graphically. The numerical outcomes of coefficient of skin friction, rate of heat transfer, and mass transfer are obtained in tabular form. The results indicate that the skin friction increases slowly with the reaction parameter and largely with the suction parameter, whereas the concentration gradient increases at a much higher rate with the reaction parameter. The fluid injection has a negative impact on the velocity gradient. It is seen that the heat source enhances both velocity and temperature profiles throughout the flow field, whereas the first-order chemical reaction acts reversely on the velocity and mass transfer process. The current research can be applied to identify the cause behind the drag force produced in seepage flow due to the heated or cooled accelerated plate.     

Analysis of steady flows in viscous fluid with heat/mass transfer and slip effects

The goal of present article is to discuss the effects of heat and mass transfer with slip on the Couette and generalized Couette flow in a homogeneous and thermodynamically compatible third grade non Newtonian viscous fluid. Exact solutions of velocity and temperature in Couette flow problem are derived. The nonlinear analysis for generalized Couette flow problem is performed by using spectral homotopy analysis method (SHAM). Convergence and residual error are shown explicitly. The obtained solutions satisfy the boundary conditions and the governing equations. The dimensionless Nusselt numbers at the walls for various involved parameters are also given in each case. Graphical illustrations are shown for certain embedded parameters of interest in the derived solutions.     

Mixed free and forced convection in the incompressible boundary layer along a rotating vertical cylinder with fluid injection

We examine the steady incompressible laminar boundary layer flow along a vertical cylinder with isothermal walls. The mixed free and forced convection regime is studied while injection/suction of fluid can take place through the cylinder surface. The two-dimensional boundary layer equations are solved using an efficient finite difference scheme, and velocity and temperature profiles, as well as skin friction, heat transfer and pressure coefficients, are calculated. It is proved that fluid injection can considerably reduce the skin friction and heat transfer at the wall. Also, significant differences are reported when the present results are compared with published results for the zero mass transfer case.     

Effects of thermal radiation on micropolar fluid flow and heat transfer over a porous shrinking sheet

The effects of thermal radiation on the flow of micropolar fluid and heat transfer past a porous shrinking sheet is investigated. The self-similar ODEs are obtained using similarity transformations from the governing PDEs and are then solved numerically by very efficient shooting method. The analysis reveals that for the steady flow of micropolar fluid, the wall mass suction needs to be increased. Dual solutions of velocity and temperature are obtained for several values of the each parameter involved. For increasing values of the material parameter K, the velocity decreases for first solution, whereas, for second solution it increases. Due to increase of thermal radiation, the temperature and thermal boundary layer thickness reduce in both solutions and also the heat transfer from the sheet enhances with thermal radiation.     

Effects of a first-order chemical reaction on turbulent mass transfer

The effects of a first-order chemical reaction on turbulent mass transfer from a wall are investigated using a Lagrangian method that involves the numerical solution for the flow field in conjunction with the tracking of mass markers released from the wall. The markers react according to a Poisson distribution that is correlated to the reaction rate constant. The method allows studying a range of Schmidt number fluids and a range of Damköhler numbers. The behavior of a continuous line source of the reactant, and the behavior of the dissolution of the reactant from the wall are examined. It is found that the mass transfer coefficient increases dramatically when a first-order reaction occurs. A correlation between the mass transfer coefficient and the Damköhler number is proposed for both low and high Schmidt numbers. The fundamental reason for the change in the mass transfer properties is that each marker is affected by different parts of the turbulence velocity field depending on the Schmidt and Damköhler numbers.     

Effect of thermophoresis and Brownian motion on the melting heat transfer of a Jeffrey fluid near a stagnation point towards a stretching surface: Buongiorno's model

This analysis intends to address the coupled effect of phase change heat transfer, thermal radiation, and viscous heating on the MHD flow of an incompressible chemically reactive nanofluid in the vicinity of the stagnation point toward the stretching surface, taking a Jeffrey fluid as the base fluid. Convergent analytical solutions for the nonlinear boundary layer equations are obtained by the successive application of scaling variables and the highly efficacious homotopy analysis method. Error analysis is implemented to endorse the convergence of the solutions. Through parametric examination, influence of various physical parameters occurring in analysis of the profiles of velocity, temperature, and nanoparticle concentration, coefficient of surface drag, rates of mass and heat transfer is explored pictorially. The Deborah number and the melting parameter are found to enhance velocity, and the associated momentum boundary layers are thicker, whereas the magnetic field depreciates the flow rate. Temperature is observed to enhance with the thermophoresis parameter, Prandtl number and Eckert number, whereas a reduction is seen with the thermal radiation parameter and Brownian motion parameter. Nanoparticle concentration is depleted by the chemical reaction parameter, the thermophoresis parameter, and the Lewis number.