Effect of slip boundary condition on flow and heat transfer of a double fractional Maxwell fluid

The purpose of the present paper is to investigate the flow and heat transfer of a double fractional Maxwell fluid with a second order slip model. The fractional governing equations are solved numerically by using the finite difference method. By comparing the analytical solutions of special boundary conditions, the validity of the present numerical method is examined. The effects of the two slip parameters and the fractional parameters on the velocity and temperature distribution are presented graphically and discussed. The results reveal that the fractional Maxwell fluid exhibits a stronger viscosity or elasticity for different fractional parameters, and the oscillation phenomenon will gradually decrease as expected with an increase in slip parameters.     

Unsteady MHD flow and heat transfer of fractional Maxwell viscoelastic nanofluid with Cattaneo heat flux and different particle shapes

The paper aims to investigate the unsteady natural convection flow and heat transfer of fractional Maxwell viscoelastic nanofluid in magnetic field over a vertical plate. The effect of nanoparticle shape is first introduced to the study of fractional Maxwell viscoelastic nanofluid. Fractional shear stress and Cattaneo heat flux model are applied to construct the governing boundary layer equations of momentum and energy, which are solved numerically. The quantities of physical interest are graphically presented and discussed in detail. It is found that particle shape and fractional derivative parameters have profound influence on the flow and heat transfer.     

MHD Maxwell flow modeled by fractional derivatives with chemical reaction and thermal radiation

Heat transfer in a time-dependent flow of incompressible viscoelastic Maxwell fluid induced by a stretching surface has been investigated under the effects of heat radiation and chemical reaction. The magnetic field is applied perpendicular to the direction of flow. Velocity, temperature, and concentration are functions of z and t for the modeled boundary-layer flow problem. To have a hereditary effect, the time-fractional Caputo derivative is incorporated. The pressure gradient is assumed to be zero. The governing equations are non-linear, coupled and Boussinesq approximation is assumed for the formulation of the momentum equation. To solve the derived model numerically, the spatial variables are discretized by employing the finite element method and the Caputo-time derivatives are approximated using finite difference approximations. It reveals that the fractional derivative strengthens the flow field. We also observe that the magnetic field and relaxation time suppress the velocity. The lower Reynolds number enhances the viscosity and thus motion weakens slowly. The velocity initially decreases with increasing unsteadiness parameter δ. Temperature is an increasing function of heat radiation parameter but a decreasing one for the volumetric heat absorption parameter. The increasing value of the chemical reaction parameter decreases concentration. The Prandtl and Schmidt numbers adversely affect the temperature and concentration profiles respectively. The fractional parameter changes completely the velocity profiles. The Maxwell fluids modeled by the fractional differential equations flow faster than the ordinary fluid at small values of the time t but become slower for large values of the time t.     

Heat Transfer in an Upper Convected Maxwell Fluid with Fluid Particle Suspension

An analysis is carried out to study the magnetohydrodynamic (MHD) flow and heat transfer characteristics of an electrically conducting dusty non-Newtonian fluid, namely, the upper convected Maxwell (UCM) fluid over a stretching sheet. The stretching velocity and the temperature at the surface are assumed to vary linearly with the distance from the origin. Using a similarity transformation, the governing nonlinear partial differential equations of the model problem are transformed into coupled non-linear ordinary differential equations and the equations are solved numerically by a second order finite difference implicit method known as the Keller-box method. Comparisons with the available results in the literature are presented as a special case. The effects of the physical parameters on the fluid velocity, the velocity of the dust particle, the density of the dust particle, the fluid temperature, the dust-phase temperature, the skin friction, and the wall-temperature gradient are presented through tables and graphs. It is observed that, Maxwell fluid reduces the wall-shear stress. Also, the fluid particle interaction reduces the fluid temperature in the boundary layer. Furthermore, the results obtained for the flow and heat transfer characteristics reveal many interesting behaviors that warrant further study on the non-Newtonian fluid flow phenomena, especially the dusty UCM fluid flow phenomena.     

Variable fluid properties and variable heat flux effects on the flow and heat transfer in a non-Newtonian Maxwell fluid over an unsteady stretching sheet with slip velocity

The effects of variable fluid properties and variable heat flux on the flow and heat transfer of a non-Newtonian Maxwell fluid over an unsteady stretching sheet in the presence of slip velocity have been studied. The governing differential equations are transformed into a set of coupled non-linear ordinary differential equations and then solved with a numerical technique using appropriate boundary conditions for various physical parameters. The numerical solution for the governing non-linear boundary value problem is based on applying the fourth-order Runge-Kutta method coupled with the shooting technique over the entire range of physical parameters. The effects of various parameters like the viscosity parameter, thermal conductivity parameter, unsteadiness parameter, slip velocity parameter, the Deborah number, and the Prandtl number on the flow and temperature profiles as well as on the local skin-friction coefficient and the local Nusselt number are presented and discussed. Comparison of numerical results is made with the earlier published results under limiting cases.     

Generalized Unsteady MHD Natural Convective Flow of Jeffery Model with ramped wall velocity and Newtonian heating; A Caputo-Fabrizio Approach

This theoretical investigation aims to highlight the unsteady freely convective fractional motion of a Jeffery fluid near an infinite vertical plate. The additional effects of ramped velocity condition, Newtonian heating, magnetohydrodynamics (MHD), and nonlinear radiative heat flux are also examined. A system of fractional order partial differential equations is established by choosing Caputo-Fabrizio fractional derivative as a foundation. Laplace transformation followed by an adequate choice of unit-less parameters is executed to solve the subsequent ordinary differential equations. Stehfest’s and Zakian’s numerical algorithms are invoked to find and justify the inverse Laplace transform of velocity and shear stress. Temperature and velocity gradients are evaluated at the wall to effectively probe the rate of heat transfer and shear stress. In this regard, numerical computations of Nusselt number and shear stress for several inputs of connected parameters are tabulated. Furthermore, graphical elucidations of velocity and temperature profiles are provided to observe the rise and fall subjected to variation in several parameters. Additionally, the velocity profile for both ramped boundary condition and constant boundary condition is analyzed to get a deep insight into the physical phenomenon of the considered problem. Finally, a comparative analysis between Jeffery fluid and second grade fluid is carried out for both factional and ordinary cases, and it is determined that Jeffery fluids exhibit rapid motion in both cases.     

MHD Three-Dimensional Flow of Viscoelastic Fluid with Convective Surface Boundary Condition

This article reports the magnetohydrodynamic (MHD) three-dimensional flow of viscoelastic fluid over a stretching surface with heat transfer. Mathematical analysis is formulated using convective boundary conditions. Computations of dimensionless velocity and temperature fields are presented. The tabulated values show excellent agreement between present and previous limiting analysis. Graphical results show the impact of embedded parameters entering into the problem.     

Influence of Non-linear Radiation Heat Flux on Rotating Maxwell Fluid over a Deformable Surface: A Numerical Study

Mathematical model for Maxwell fluid flow in rotating frame induced by an isothermal stretching wall is explored numerically. Scale analysis based boundary layer approximations are applied to simplify the conservation relations which are later converted to similar forms via appropriate substitutions. A numerical approach is utilized to derive similarity solutions for broad range of Deborah number. The results predict that velocity distributions are inversely proportional to the stress relaxation time. This outcome is different from that observed for the elastic parameter of second grade fluid. Unlike non-rotating frame, the solution curves are oscillatory decaying functions of similarity variable. As angular velocity enlarges, temperature rises and significant drop in the heat transfer coefficient occurs. We note that the wall slope of temperature has an asymptotically decaying profile against the wall to ambient ratio parameter. From the qualitative view point, temperature ratio parameter and radiation parameter have similar effect on the thermal boundary layer. Furthermore, radiation parameter has a definite role in improving the cooling process of the stretching boundary.A comparative study of current numerical computations and those from the existing studies is also presented in a limiting case. To our knowledge, the phenomenon of non-linear radiation in rotating viscoelastic flow due to linearly stretched plate is just modeled here.     

Modern development on the features of magnetic field and heat sink/source in Maxwell nanofluid subject to convective heat transport

Nanofluids are forthcoming new generation heat transfer fluids, which have been scrutinized precisely, in current years. Thermophysical assets of these fluids have noteworthy impact on their heat transfer features. In this current investigation a mathematical relation for two dimensional (2D) flow of magnetite Maxwell nanofluid influenced by a stretched cylinder is established. To visualize the stimulus of Brownian moment and thermophoresis phenomena on Maxwell fluid Buongiorno's relation has been considered. Moreover, heat sink/source and convective condition are also presented for heat transport mechanism. The homotopic scheme has been developed for the solutions of nonlinear ordinary differential equations (ODEs). The achieved outcomes are planned and consulted in aspects for somatic parameters. It is noteworthy that the velocity of Maxwell fluid display conflicting performance for curvature parameter and Deborah number. It is also reported that the liquid velocity decays for magnetic parameter, whereas the nanoliquid temperature and concentration field enhance for magnetic parameter. Furthermore, the liquid temperature intensifies for the progressive values of thermophoresis parameter and Brownian motion. Additionally, endorsement of current significances is organized via benchmarking with earlier famous limiting situations and we pledge a marvelous communication with these outcomes.     

Multi-layer flows of immiscible fractional Maxwell fluids with generalized thermal flux

Unsteady laminar flows and heat transfer of n-immiscible fractional Maxwell fluids in a channel are investigated under influence of time-dependent pressure gradient. The isothermal channel walls have translational motions in their planes with time-dependent velocities. Governing equations of the mathematical model are based on the generalized constitutive equations for shear stress and thermal flux described by the time-fractional Caputo derivative. Analytical and semi-analytical solutions for velocity, shear stress, and temperature fields are obtained by using finite sine-Fourier and Laplace transforms. In the case of semi-analytical solutions, the inverse Laplace transforms are obtained numerically by employing the Talbots algorithms. Using the software Mathcad, numerical calculations have carried out and results are presented in graphical illustrations in order to analyze the memory effects on the fluid temperature and motion. It is found that in fluids with thermal memory the heat transfer is slower compared with the ordinary fluid, while the fractional velocity parameters act as braking/accelerating factors of the fluids.     

Heat transfer in MHD flow of dusty viscoelastic (Walters’ liquid model-B) stratified fluid in porous medium under variable viscosity

The paper investigates the effects of heat transfer in MHD flow of viscoelastic stratified fluid in porous medium on a parallel plate channel inclined at an angle θ. A laminar convection flow for incompressible conducting fluid is considered. It is assumed that the plates are kept at different temperatures which decay with time. The partial differential equations governing the flow are solved by perturbation technique. Expressions for the velocity of fluid and particle phases, temperature field, Nusselt number, skin friction and flow flux are obtained within the channel. The effects of various parameters like stratification factor, magnetic field parameter, Prandtl number on temperature field, heat transfer, skin friction, flow flux, velocity for both the fluid and particle phases are displayed through graphs and discussed numerically.     

Advanced Study of Unsteady Heat and Chemical Reaction with Ramped Wall and Slip Effect on a Viscous Fluid

This paper investigate the effect of slip boundary condition, thermal radiation, heat source, Dufour number,chemical reaction and viscous dissipation on heat and mass transfer of unsteady free convective MHD flow of a viscous fluid past through a vertical plate embedded in a porous media. Numerical results are obtained for solving the nonlinear governing momentum, energy and concentration equations with slip boundary condition, ramped wall temperature and ramped wall concentration on the surface of the vertical plate. The influence of emerging parameters on velocity,temperature and concentration fields are shown graphically.     

Heat transfer analysis in the flow of Walters' B fluid with a convective boundary condition

Radiative heat transfer in the steady two-dimensional flow of Walters' B fluid with a non-uniform heat source/sink is investigated. An incompressible fluid is bounded by a stretching porous surface. The convective boundary condition is used for the thermal boundary layer problem. The relevant equations are first simplified under usual boundary layer assumptions and then transformed into a similar form by suitable transformations. Explicit series solutions of velocity and temperature are derived by the homotopy analysis method(HAM). The dimensionless velocity and temperature gradients at the wall are calculated and discussed.     

New trends of fractional modeling and heat and mass transfer investigation of (SWCNTs and MWCNTs)-CMC based nanofluids flow over inclined plate with generalized boundary conditions

In this paper, we have considered unsteady MHD viscous fluid flow of Carboxyl methyl cellulose (CMC) as based fluid and Carbon nanotubes CNTs (SWCNTs, MWCNTs) nanoparticles passing through an inclined plate of infinite length. Furthermore, the effects of heat source, chemical reaction, porosity and MHD are considered. Fractional model is developed by Caputo time fractional derivatives though recent trends of fractional modeling. The semi exact solutions are obtained for the governing equations in dimensionless form by Laplace transform method. Influence of fractional and other flow parameters on temperature, concentration and velocity fields are graphically illustrated. The Nusselt number, Sherwood number and skin friction are computed for fixed values of flow parameters and presented in tabular form. As a result, for larger values of fractional parameters temperature, concentration and velocity fields can be enhanced. A comparison has been drawn between SWCTs-CMC and MWCTs-CMC based nanofluids and found that MWCTs based nanofluids are more efficient in heat transfer than SWCTs-CMC based nanofluids. Further, in the absence of nanoparticles the obtained results are reduced to recently published results and this fact have been proved graphically and they are in good agreement.     

Exact Solutions for the Flow of Fractional Maxwell Fluid in Pipe-Like Domains

This paper presents an analysis of unsteady flow of incompressible fractional Maxwell fluid filled in the annular region between two infinite coaxial circular cylinders. The fluid motion is created by the inner cylinder that applies a longitudinal time-dependent shear stress and the outer cylinder that is moving at a constant velocity. The velocity field and shear stress are determined using the Laplace and finite Hankel transforms. Obtained solutions are presented in terms of the generalized G and R functions. We also obtain the solutions for ordinary Maxwell fluid and Newtonian fluid as special cases of generalized solutions. The influence of different parameters on the velocity field and shear stress is also presented using graphical illustration. Finally, a comparison is drawn between motions of fractional Maxwell fluid, ordinary Maxwell fluid and Newtonian fluid.     

Mixed Convection in Viscoelastic Boundary Layer Flow and Heat Transfer over a Stretching Sheet

An investigation is carried out on mixed convection boundary layer flow of an incompressible and electrically conducting viscoelastic fluid over a linearly stretching surface in which the heat transfer includes the effects of viscous dissipation, elastic deformation, thermal radiation, and non-uniform heat source/sink for two general types of non-isothermal boundary conditions. The governing partial differential equations for the fluid flow and temperature are reduced to a nonlinear system of ordinary differential equations which are solved analytically using the homotopy analysis method (HAM). Graphical and numerical demonstrations of the convergence of the HAM solutions are provided, and the effects of various parameters on the skin friction coefficient and wall heat transfer are tabulated. In addition, it is demonstrated that previously reported solutions of the thermal energy equation given in [1] do not converge at the boundary, and therefore, the boundary derivatives reported are not correct.     

振荡流共轭换热格子-Boltzmann模拟

振荡流共轭换热现象广泛存在于热声热机等工程应用中.基于双分布格子-Boltzmann模型,对平行平板间振荡流共轭换热进行了数值模拟.通过假定共轭界面处流体和固体的未知内能分布函数均为对应的平衡态滑移修正格式,提出了一种处理共轭换热边界的新方法.模拟结果表明,该方法可以保证共轭界面上温度连续和热流连续.分析了不同流体与固体导热系数比情况下振荡流共轭换热的速度场、温度场以及热流分布的特点.     

Free convection flow of nanofluid over infinite vertical plate with damped thermal flux

In this article, the unsteady free convection flow and heat transfer of nanofluid past over an infinite vertical plate is considered. The fractional generalized Fourier's law with Caputo time derivatives with power-law model to describe the influence of memory on the nanofluid behavior. The analytical solutions for dimensionless temperature and velocity fields and dimensionless thermal flux are obtained by means of Laplace transformation. The fluid is water based nanofluid containing nanoparticles of CuO or Ag. The effects of fractional and physical parameters are discussed graphically.