Unsteady Free Convection Research Articles

Unsteady MHD free convective flow past a vertical plate: An automated solution approach

The case of unsteady two-dimensional laminar free convection flow over a vertical plate by an incompressible viscous fluid is analysed in the presence of uniform magnetic field perpendicular to the flow. The governing equations in a vector form are transformed into non- dimensional form. Then, the dimensionless equations are solved using automated solution technique which is FEniCS. The effects of magnetic parameter on the velocity and temperature profiles are obtained and discussed in this paper

Free convection flow with chemical reaction effect between oscillating parallel plates

This research purpose is to investigate the exact solutions for unsteady free convection flow between oscillating parallel plates with mass diffusion and chemical reaction. The governing equations are modelled and reduced using non-dimensional variables. The method used is Laplace transform method. Solutions for velocity, temperature, and concentration fields as well as skin friction, Nusselt and Sherwood number are obtained. For physical understanding, analytical results for velocity, temperature and concentration profile are plotted graphically with respect to the Schmidt number, Prandtl number, oscillating parameter, Grashof number, mass Grashof number and chemical reaction parameter. Increasing Prandtl number and Schmidt number decreases the concentration, velocity, temperature, and skin friction but increases the Sherwood and Nusselt numbers.

Numerical Solutions of a Heat Transfer for Fractional Maxwell Fluid Flow with Water Based Clay Nanoparticles; A Finite Difference Approach

Fractional-order mathematical modelling of physical phenomena is a hot topic among various researchers due to its many advantages over positive integer mathematical modelling. In this context, the appropriate solutions of such fractional-order physical modelling become a challenging task among scientists. This paper presents a study of unsteady free convection fluid flow and heat transfer of Maxwell fluids with the presence of Clay nanoparticle modelling using fractional calculus. The obtained model was transformed into a set of linear nondimensional, partial differential equations (PDEs). The finite difference scheme is proposed to discretize the obtained set of nondimensional PDEs. The Maple code was developed and executed against the physical parameters and fractional-order parameter to explain the behavior of the velocity and temperature profiles. Some limiting solutions were obtained and compared with the latest existing ones in literature. The comparative study witnesses that the proposed scheme is a very efficient tool to handle such a physical model and can be extended to other diversified problems of a complex nature.

Effects of Dufour and thermal diffusion on unsteady MHD free convection and mass transfer flow through an infinite vertical permeable sheet

In this paper, the effects of Dufour and thermal diffusion and on unsteady MHD (magnetohydrodynamic) free convection and mass transfer flow through an infinite vertical permeable sheet have been investigated numerically. The non-dimensional governing equations are solved numerically by using the superposition method with the help of “Tec plot” software. The numerical solution regarding the non-dimensional velocity, temperature, and concentration variables against the non-dimensional coordinate variable has been carried out for various values of pertinent numbers and parameters like the suction parameter left( {v_{0} } right), Prandtl number left( {P_{r} } right), magnetic parameter left( M right), Dufour number left( {D_{f} } right), Soret number left( {S_{0} } right), Schmidt number left( {S_{c} } right), and for constant values of modified local Grashof number left( {G_{{text{m}}} } right) and local Grashof number left( {G_{r} } right).The velocity field decreases for increasing the suction parameter which is focusing on the common fact that the usual suction parameter stabilizing the effect on the boundary layer growth. The thermal boundary layer thickness becomes thinner for rising values of the Dufour and Soret numbers. The skin friction enhances for uplifting values of Soret number and Dufour number but reduces for moving suction parameter, Magnetic force number, Prandtl number, and Schmidt number. The heat transfer rate increases for increasing the suction parameter, Dufour number, Prandtl number, and Soret number. The mass transfer rate increases for enhancing the values of suction parameter, Magnetic force number, Soret number, and Prandtl number but decreases for Dufour number and Schmidt number.

State Space Approach to Unsteady FreeConvection Effects on Perfectly Conducting Viscoelastic Fluid

State Space Approach to Unsteady FreeConvection Effects on Perfectly Conducting Viscoelastic Fluid

Effects of fractional derivative and heat source/sink on MHD free convection flow of nanofluids in a vertical cylinder: A generalized Fourier's law model

This research looked at the unsteady free convection flows of an incompressible viscous fluid with heat/sink in a vertical cylinder containing a mixture of 47 nm alumina nanoparticles in water. The flow direction is subjected to a perpendicular magnetic field. The generalization entails taking into account a new version of the constitutive equation for thermal flux, known as the generalized Atangana-Baleanu derivative, which is based on the generalized time-fractional derivative with Mittag-Leffler kernel. Using the Laplace transform and the finite Hankel transform, closed forms of analytical solutions for temperature and velocity fields, represented with Bessel and generalized G–function of Lorenzo and Hartley functions, are determined. The generalized solutions are appropriate for particularizations to yield solutions corresponding to fractional derivatives with power-law kernel and exponential kernel. The Mittag-Leffler function is a one-parametric function. It is also possible to acquire the usual situation, which corresponds to classical Fourier's law. To compare models based on generalized Atangana-Baleanu, Atangana-Baleanu, Caputo, and Caputo-Fabrizio time fractional derivatives, numerical simulations produced with the program Mathcad are carried out and visually depicted.

Heat and mass transfer for Soret and Dufour's consequences on unsteady MHD free convection flow over a porous media with heat absorption

AbstractThe significance of this article lies in explaining the influence of Soret and Dufour numbers on an unsteady MHD free convection of flow of heat and mass transfer through porous media. The substances and radiation along the viscous, incompressible, and conductive compounds respond to the unstable convection of the liquid. Using physical quantities, the dimensional governing equations are converted to non‐dimensional equations. Finite element Galerkin method is applied to numerically solve the resulting partial differential equations. Flow parameters on velocity, temperature, and concentration are studied and explained graphically to reflect their effects. Similarly, the skin friction number and Nusselt number are also observed and recorded in tables.

Application of water based drilling clay-nanoparticles in heat transfer of fractional Maxwell fluid over an infinite flat surface

In the present paper, unsteady free convection flow of Maxwell fluid containing clay-nanoparticles is investigated. These particles are hanging in water, engine oil and kerosene. The values for nanofluids based on the Maxwell-Garnett and Brinkman models for effective thermal conductivity and viscosity are calculated numerically. The integer order governing equations are being extended to the novel non-integer order fractional derivative. Analytical solutions of temperature and velocity for Maxwell fluid are build using Laplace transform technique and expressed in such a way that they clearly satisfied the boundary conditions. To see the impact of different flow parameters on the velocity, we have drawn some graphs. As a result, we have seen that the fractional model is superior in narrate the decay property of field variables. Some limiting solutions are obtained and compared with the latest existing literature. Moreover, significant results can be observed for clay nanoparticles with different base fluids.

Characteristics of MHD Casson fluid past an inclined vertical porous plate

This paper addresses the effects of Soret on unsteady free convection flow of viscous incompressible fluid through a porous medium with high porosity bounded by a vertical infinite moving plate under the influence of thermal diffusion, chemical reaction, and heat source. The fluid is considered to be gray, absorbing, and emitting but non-scattering medium. Rosseland approximation is considered to describe the radiative heat flux in the energy equation. The dimensionless governing equations for this investigation are solved analytically using perturbation technique. The effects of various governing parameters on the velocity, temperature, concentration, skin-friction coefficient, Nusselt number and Sherwood number are shown in figures and tables and analyzed in detail. It was noticed that velocity distribution increased with increasing buoyancy parameters; temperature decreased with increasing Prandtl number, and concentration decreased with increasing the Schmidt number and chemical reaction parameter.

Convergence, stability, and numerical solution of unsteady free convection magnetohydrodynamical flow between two slipping plates

In this study, the unsteady free convection magnetohydrodynamical flow of a viscous, incompressible, and electrically conducting fluid between two horizontally directed slipping plates is considered. The external magnetic filed is applied uniformly in the y‐direction and the fluid is assumed to be of low conductivity so that the induced magnetic field is negligible. So the relevant variables, that is, the velocity and the temperature, depend only on one coordinate, the y‐axis. The governing equations of velocity and temperature fields are obtained from the continuity, momentum, and energy equations. The boundary conditions for the velocity are taken in the most general form as Robins type which contain slipping parameter. Moreover, the upper plate is heated exponentially and the lower plate is adiabatic. Finite difference method (FDM) is used to simulate the numerical solutions of the problem in which the explicit forward difference in time variable t and central difference in space variable y is used. Hartmann number, Prandtl number, decay factor, and slipping parameter influences on the flow and temperature are shown graphically. It is seen that as the Hartmann number increases, the velocity magnitude drops, which is the well‐known flattening tendency of the MHD flow. Also, the increase in decay factor causes an increase in both the velocity and temperature magnitudes at increasing time levels, but it does not change further close to the steady‐state. Furthermore, the convergence and stability conditions of the considered scheme are obtained in terms of Hartmann number, Prandtl number, and the slip length.

Effects of Magnetohydrodynamics and Heat Transfer in Casson Fluid Through a Channel

Unsteady flow of Casson fluid past through a vertical channel has been studied by some researchers due to its importance of applications in science and technology. Therefore, the main purpose of this paper is to obtain exact solutions for unsteady free convection flows of Casson fluid with effects of magnetohydrodynamics (MHD) past through vertical channel. This paper is continued study from published article [18] with additional effects of magnetohydrodynamics (MHD). Dimensional governing equations are converted into dimensionless forms by using appropriate dimensionless variables. Dimensionless parameters are obtained through dimensionless process such as Casson fluid, time, Prandtl number, Grashof number and magnetic field. Laplace transform method is used to solve the dimensionless equations with associated initial and boundary conditions. Solutions for velocity and temperature profiles are obtained. Skin friction and Nusselt number are also calculated. The obtained analytical results for velocity and temperature are plotted graphically to discuss the influence of dimensionless parameters on profiles. It is observed that fluid velocity increases with increases of Grashof number, Gr and time, t whereas it decreases with increases of Casson parameter, γ, magnetic field parameter, M and Prandtl number, Pr. Besides that, it is found that temperature profiles decrease with high value of Prandtl number, Pr while increases with high value of time, t. In order to validate the results, the obtained results in limiting cases are compared with the published results and it is found to be in a mutual agreement.

Unsteady MHD free convection flow past a vertical permeable flat plate in a rotating frame of reference with constant heat source and variable thermal boundary condition in a nanofluid

Abstract The problem of unsteady magnetohydrodynamics (MHD) free convection flow of nano-fluid past an oscillation moving vertical semi-infinite permeable flat plate with constant heat source and vertical variable thermal boundary condition in a rotating frame of reference is studied theoretically. The boundary layer has the oscillatory type of solution because slip velocity is assumed to oscillate in time with constant frequency. The governing equations are solved using small perturbation approximation. For this investigation, we use two types of nano-particles in base fluid, namely Cu (copper) and Al2O3 (aluminium oxide). The effect of various types of suspended metallic nano-particles and non-dimensional physical parameters on the fluid flow and heat transfer rate are presented and discussed through the graphs.

Influence of radiative heat flux in nonlinear convection of quartic order in Casson fluid past a vertical permeable plate with variable suction and Hall current

The current study centralizes on unsteady free convection slip flow of Casson fluid past a vertical permeable plate with Hall current, radiative heat flux, and variable suction. The nonlinear convection is subjected to quartic order. Perturbation method is used to convert the non-linear coupled partial differential equation of the momentum and energy to a system of ordinary differential equations. The dimensionless governing equations are solved analytically for velocity and temperature profiles. The graphs are plotted for sundry parameters for variations in the distinct flow fields w.r.t distance from the plate. Variation in the skin friction for the axial and transverse cases are presented in the form of graphs for various parameters. It is observed that with the increase in the order of non-linear convection and value of radiation parameter, the velocity field increases in Casson fluid. The increase in heat absorption parameter and Prandtl number decreases the temperature profile and increase in radiative heat flux parameter increases the temperature profile.

Heat Transfer Analysis for Viscous Fluid Flow with the Newtonian Heating and Effect of Magnetic Force in a Rotating Regime

In this article, an unsteady free convection flow of MHD viscous fluid over a vertical rotating plate with Newtonian heating and heat generation is analyzed. The dimensionless governing equations for temperature and velocity fields are solved using the Laplace transform technique. Analytical solutions are obtained for the temperature and components of velocity fields. The obtained solutions satisfy the initial and boundary conditions. Some physical aspects of flow parameters on the fluid motion are presented graphically.

Radiation effect on free convection flow between oscillating parallel plates with mass diffusion

This study investigates radiation effect on unsteady oscillatory free convection between two parallel plates with mass diffusion. Appropriate non-dimensional variables are used to reduce the dimensional governing equations to dimensionless equation along with imposed initial and boundary conditions. The solutions for velocity, temperature, and concentration profiles are obtained by using Laplace transform method. To illustrate the behavior of the fluid flow, graphical results are presented with influenced of Schmidt number, Prandtl number, radiation parameter, oscillating parameter, Grashof, and mass Grashof number. The corresponding expressions for skin friction, Nusselt number, and Sherwood number are also calculated. It is observed that increasing radiation parameter, Prandtl, and Schmidt number will increase the Nusselt number but the skin friction will be reduced.

Radiation and chemical reaction effects on unsteady MHD free convention mass transfer fluid flow in a porous plate

Objective: To study the effects of Radiation and chemical reaction on unsteady MHD free convection flow in a porous plate. Method: The problem is governed by coupled non-linear partial differential equations. The nondimensional equations of the problem have been solved numerically using Finite Element Method. Findings: The effects of governing parameters on the flow variables are discussed quantitatively with the help of graphs for the flow field, temperature field, concentration field, skin friction and Nusselt number. Novelty: The accuracy of the problem has been verified by comparing with the previous published work and the agreement between the results is excellent, which established confidence in the numerical results reported in this study. Keywords: Radiation parameter; Chemical reaction; Eckert number; heat source parameter; micro polar fluid

Effect of Heat Generation (Absorption) on Unsteady MHD flow over a Truncated Cone Embedded in a Stable Thermally Stratified Environment

This paper examines the effect of heat generation (absorption) on unsteady free convection flow of an incompressible electrically conducting fluid over a truncated cone embedded in a thermally stratified medium in the presence of a transverse magnetic field. The governing nonlinear partial differential equations have been solved numerically, using an efficient implicit finite difference scheme along with quasilinearization technique. The numerical results for skin-friction coefficients and velocity components in x- and y- directions, Nusselt number and temperature profiles are obtained and analyzed for different values of governing parameters. It is observed that, heat generation increases skin friction and decreases heat transfer coefficients, whereas opposite trend is observed during heat absorption. Also it is found that heat generation has a significant effect on temperature profile.

Entropy generation and heat transfer analysis in power-law fluid flow: Finite difference method

In this article entropy generation analysis for unsteady laminar free convection flow of power-law fluid is investigated. Fluid flow is in the presence of thermal radiation and magnetic field. The flow and heat transfer are governed by coupled system of PDE's. System of partial differential equation (PDE's) governs the continuity, momentum and energy equations. Using suitable dimensionless parameters, the system of PDE's is transformed into dimensionless form. Numerical computations made employing finite difference method (FDM). This method converts the nonlinear PDE 's to simple algebraic equations and by using variational technique the approximate solution is obtained technique. Impacts of important flow variables on velocity, skin friction and Nusselt number are shown graphically. Skin friction and Nusselt number are discussed for different pertinent flow variables. Velocity, temperature, entropy and Bejan number variations through important variables are discussed in the form of graphs. The obtained results show that velocity decays against Re, Ha and Pr. Temperature is enhanced for higher values of Re, Ec, Tr and Gr. For higher estimation of Ec entropy of the system enhance while show opposite behavior for the Bejan number. Entropy of system boosts for strong magnetic field.

Lie Symmetry Group for Unsteady Free Convection Boundary-Layer Flow over a Vertical Surface

The Lie symmetry group transformation method was used to investigate the partial differential equations that model the motion of a natural convective unsteady flow past to a non-isothermal vertical flat surface. The one-parameter Lie group transformation was applied twice consecutively to convert the motion governing equations into a system of ordinary differential equations. The obtained system of ordinary differential equations was solved numerically using the Lobatto IIIA formula (implicit Runge–Kutta method). The effect of the Prandtl number on the temperature and velocity profiles is illustrated graphically.

The Effect of Variable Viscosity on Unsteady Free Convective Fluid Flow over a Stretching Surface in the Presence of Thermal Radiation and Viscous Dissipation

The unsteady free convective flow of an incompressible, viscous, electrically conducting fluid over a vertical stretching surface with variable viscosity and thermal radiation in the presence of viscous dissipation is considered. The fluid is assumed to be gray, absorbing emitting but non-scattering medium. Thus, the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The viscosity of the fluid is assumed to be an inverse linear function of temperature. Specifically, the study is aimed at determining the effects of variable viscosity parameter ( ), Eckert number ( ), and radiation parameter ( ) on temperature distribution profiles, while holding the Prandtl number constant that is corresponding to air. The specific energy equation governing the fluid flow is solved numerically by the finite difference method. The nonlinear partial differential equation is solved numerically by the finite difference method. MATLAB code is used to solve the Forward Time and Centered Space (FTCS) scheme. Further, stability and consistency analyses are performed on the FTCS scheme to determine the convergence of the scheme. Results for temperature distribution profiles are presented graphically for different values of variable viscosity ( ), Eckert number ( ), radiation parameter ( ). The findings are that an increase in variable viscosity and Eckert number causes an enhancement of temperature distribution profiles whereas an increase in radiation parameter leads to a decrease in temperature distribution profiles. The FTCS scheme was found to be unconditionally stable and consistent. The results obtained in this study follow expected trends and are in good agreement with some previous studies.