Some duct flows of a fractional Maxwell fluid

The aim of this paper is to establish the analytical solutions corresponding to two types of unsteady flows of fractional Maxwell fluid in a duct of rectangular cross-section. The fractional calculus approach is used in solving the problems. With the help of the methods of separation of variables and Laplace transforms, the expressions for the velocity field and the volume flux are presented under series forms in terms of the generalized G functions. Similar solutions for Newtonian and ordinary Maxwell fluids, performing the same motions, are also obtained as the limiting cases of our solutions. Furthermore, the influence of pertinent parameters on the flows is delineated and appropriate conclusions are drawn.     

Maxwell fluid flow between vertical plates with damped shear and thermal flux: Free convection

In this article, we studied free convection flow of Maxwell fluid between two parallel plates a distance d apart from each other. The Caputo time-fractional derivative is used in model and the model is fractionalized through mechanical laws (generalized shear stress constitutive equation and generalized Fourier's law). Closed form solutions are found by means of Laplace and sine-Fourier transforms which are suitable for our boundary conditions. The solutions are expressed in the form of Mittag–Leffler function and generalized G–function of Lorenzo and Hartley. The viscous fractional and ordinary Maxwell and fractional model are presented as special cases. The effects of fractional and physical parameters are graphically illustrated.     

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.     

Axisymmetric convection flow of fractional Maxwell fluid past a vertical cylinder with velocity slip and temperature jump

A novel finite volume method is developed to investigate the axisymmetric convection flow and heat transfer of fractional viscoelastic fluid past a vertical cylinder. Fractional cylindrical governing equations are formulated by fractional Maxwell model and generalized Fourier's law. The velocity slip and temperature jump boundary conditions are considered across the fluid-solid interface. Numerical results are validated by exact solutions of special case with source terms. The effects of fractional derivative parameter and boundary condition parameters on flow and heat transfer characteristics are discussed. The viscoelastic fluid performs evident shear thickening property in the fractional Maxwell constitutive relation. Moreover, the boundary condition parameters have remarkable influence on velocity and temperature distributions.     

Flows of a generalized second grade fluid in a cylinder due to a velocity shock

Unsteady axial flows of second grade fluids with generalized fractional constitutive equation in a circular cylinder are studied. Flows are generated by a time-dependent pressure gradient in the axial direction, an external magnetic field perpendicular on the flow direction and by the cylinder motion. Two different problems are analyzed; one in which the cylinder velocity supports a shock at the instant t = 0 and another in which the cylinder motion is a translation with time-dependent velocity along the axis of cylinder. The generalized fractional constitutive equation of second grade fluid is described by the Caputo time-fractional derivative. Analytical solutions for the velocity field are obtained by using the Laplace transform with respect to time variable and the finite Hankel transform of order zero with respect to the radial coordinate. The influence of the fractional parameter of Caputo derivative on the fluid velocity has been studied by numerical simulations and graphical illustrations. It is found that the fractional fluid flows are faster than the ordinary second grade fluid.     

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.     

Multi-layer flows of immiscible fractional Maxwell fluids in a cylindrical domain

Laminar unsteady multilayer axial flows of fractional immiscible Maxwell fluids in a circular cylinder are investigated. The flow of fluids is generated by a time-dependent pressure gradient in the axial direction and by the translational motion of a cylinder along his axis. The considered mathematical model is based on the fractional constitutive equation of Maxwell fluids with Caputo time-fractional derivatives. Analytical solutions for the fractional differential equations of the velocity fields with boundary and interfaces conditions have been determined by using the Laplace transform coupled with the Hankel transform of order zero and the Weber transform of order zero. The influence of the memory effects on the motion of the fluid has been investigated for the particular case of three fractional Maxwell fluids. It is found that for increasing values of the fractional parameter the fluid velocity is decreasing. The memory effects have a stronger influence on the velocity of the second layer.     

Analytical and semi-analytical solutions to flows of two immiscible Maxwell fluids between moving plates

Unsteady flows of two immiscible Maxwell fluids in a rectangular channel bounded by two moving parallel plates are studied. The fluid motion is generated by a time-dependent pressure gradient and by the translational motions of the channel walls in their planes. Analytical solutions for velocity and shear stress fields have been obtained by using the Laplace transform coupled with the finite sine-Fourier transform. These analytical solutions are new in the literature and the method developed in this paper can be generalized to unsteady flows of n-layers of immiscible fluids. By using the Laplace transform and classical method for ordinary differential equations, the second form of the Laplace transforms of velocity and shear stress are determined. For the numerical Laplace inversion, two accuracy numerical algorithms, namely the Talbot algorithm and the improved Talbot algorithm are used.     

Creeping flow of a viscous fluid in a uniformly porous slit with porous medium: An application to the diseased renal tubules

In this paper, exact solutions for the creeping flow of Newtonian fluid through a porous slit with uniform reabsorption at the porous walls and a porous medium in between are presented. The momentum equation is converted into the form of stream function and is then solved exactly. The solutions for corresponding problem without porous filling in the channel are also deduced and they match exactly with those present in literature. Expressions for other useful physiological quantities like longitudinal and transverse velocities, pressure difference, mean pressure drop across the slit, volume flow rate, wall shear stress, fractional reabsorption and leakage flux are derived. The absorption velocity for renal tubule in a rat kidney is computed for the relevant fractional reabsorption of 80%. The data are then used to tabulate pressure differences corresponding to various values of medium porosity. The results are also presented graphically and it is shown that there is a possibility of reverse flow, usually farther along the length of the slit, when the values of initial flow rate are not high or when the values of absorption velocity are too high.     

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.     

Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe

Generally speaking, rheological properties of materials are specified by their so-called constitutive equations. The simplest constitutive equation for a fluid is a Newtonian one, on which the classical Navier-Stokes theory is based. The mechanical behavior of many fluids is well described by this theory. However, there are many rheologically compli- cated fluids such as polymer solutions, blood and heavy oils which are inadequately de- scribed by a Newtonian constitutive equation that does …     

Natural convection with damped thermal flux in a vertical circular cylinder

Natural convection flows of an incompressible Newtonian fluid inside a circular cylinder are studied. The heat transfer process is described by a generalized fractional constitutive equation for the thermal flux-temperature gradient. Caputo time-fractional derivative operator, which provides the damping of thermal flux, is considered into the studied model.Analytical solutions to the fluid temperature, thermal flux, fluid velocity and volume flow rate are obtained with the integral transforms method (Laplace transform and finite Hankel transform).Temperature behaviors for small and large values of the time t, as well as the post-transient and transient velocity components are determined. The influence of the memory parameter (the order of the time-fractional derivative) on the temperature, thermal flux, velocity and the volume flow rate is numerically and graphically studied.     

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.     

Exact solutions for the flow of Casson fluid over a stretching surface with transpiration and heat transfer effects

The effects of transpiration on forced convection boundary layer non-Newtonian fluid flow and heat transfer toward a linearly stretching surface are reported.The flow is caused solely by the stretching of the sheet in its own plane with a velocity varying linearly with the distance from a fixed point.The constitutive relationship for the Casson fluid is used.The governing partial differential equations corresponding to the momentum and energy equations are converted into non-linear ordinary differential equations by using similarity transformations.Exact solutions of the resulting ordinary differential equations are obtained.The effect of increasing Casson parameter,i.e.,with decreasing yield stress(the fluid behaves as a Newtonian fluid as the Casson parameter becomes large),is to suppress the velocity field.However,the temperature is enhanced as the Casson parameter increases.It is observed that the effect of transpiration is to decrease the fluid velocity as well as the temperature.The skin-friction coefficient is found to increase as the transpiration parameter increases.     

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.     

An analysis of the semi-analytic solutions of a viscous fluid with old and new definitions of fractional derivatives

In this paper we present the natural convection flow of an incompressible viscous fluid subject to Newtonian heating and constant mass diffusion using a recently developed definition of the Caputo–Fabrizio fractional derivative. Boundary layer equations in dimensionless form are obtained by means of dimensionless variables. The expressions for the temperature, concentration and velocity fields are obtained in the Laplace transformed domain. The inverse Laplace transform for the temperature, concentration and velocity field are found numerically by means of Stehfest's and Tzou's algorithms. A comparative analysis has been carried between the Caputo–Fabrizio and the Caputo fractional model obtained by Vieru (2015) through graphical illustration. At the end, we can see the impact of the flow parameters, including the new fractional parameter, on the flow which is presented graphically. As a result, the fractional viscous fluid model with the Caputo–Fabrizio fractional derivative has a higher velocity than with the Caputo.     

分数阶Oldroyd-B黏弹性Poiseuille流的Laplace数值反演分析

采用Laplace数值反演的Stehfest算法研究了分数阶Oldroyd-B粘弹性流体在两平板间非定常的Poiseuille流动问题.首先,通过数值解与近似解析解的比较验证了Stehfest算法的有效性.其次,运用Stehfest算法对平板Poiseuille流动进行了研究,揭示了分数阶黏弹性平板流的速度过冲和应力过冲现象,指出这些现象对分数导数的阶数存在明显的依赖性.同时,数值结果表明,整数阶本构方程仅仅是分数阶本构方程的特例,分数阶本构方程较整数阶本构方程具有更广泛的适用性。     

Effects of double stratification and heat flux damping on convective flows over a vertical cylinder

Unsteady free convection flows of viscous fluids over a vertical circular cylinder are investigated by taking in consideration thermal and mass stratification and the thermal memory effects. The mathematical model of thermal transport is based on the fractional generalized Fourier's law for thermal flux with the kernel of power-law kind. In this model the histories of the temperature gradient influence the thermal and mass transport process and the fluid motion. On the cylinder's surface the temperature (or the thermal flux) and solute concentration are constant. Solutions in the transformed domain for the perturbation temperature and concentration and fluid velocity are determined using the Laplace transform coupled with the classical method for the ordinary non-homogeneous differential equations. The inverse Laplace transforms are obtained numerically by employing the Stehfest's algorithm. Solutions for the case corresponding to classical Fourier's law are obtained as particular case of general solutions by taking the memory parameter equal to zero. The influence of the thermal memory and of thermal and mass stratifications is numerically and graphically analyzed by using the software Mathcad 15.     

Effects of transpiration on unsteady MHD flow of an upper convected Maxwell （UCM） fluid passing through a stretching surface in the presence of a first order chemical reaction

The aim of this article is to present the effects of transpiration on the unsteady two-dimensional boundary layer flow of non-Newtonian fluid passing through a stretching sheet in the presence of a first order constructive/destructive chemical reaction. The upper-convected Maxwell （UCM） model is used here to characterize the non-Newtonian behavior of the fluid. Using similarity solutions, the governing nonlinear partial differential equations are transformed into ordinary ones and are then solved numerically by the shooting method. The flow fields and mass transfer are significantly influenced by the governing parameters. The fluid velocity initially decreases as the unsteadiness parameter increases and the concentration decreases significantly due to the increase in the unsteadiness. The effect of increasing values of transpiration （suction） and the Maxwell parameter is to suppress the velocity field; however, the concentration is enhanced as transpiration （suction） and the Maxwell parameter increase. Also, it is found that the fluid velocity decreases as the magnetic parameter increases; however, the concentration increases in this case.     

Dynamics of gas bubbles in viscoelastic fluids. I. Linear viscoelasticity

The nonlinear oscillations of spherical gas bubbles in linear viscoelastic fluids are studied. A novel approach is implemented to derive a governing system of nonlinear ordinary differential equations. The linear Maxwell and Jeffreys models are chosen as the fluid constitutive equations. An advantage of this new formulation is that, when compared with previous approaches, it facilitates perturbation methods and numerical investigations. Analytical solutions are obtained using a multiple scale perturbation method and compared with the Newtonian results for various Deborah numbers. Numerical analysis of the full equations supports the perturbation analysis, and further reveals significant differences between the viscoelastic and Newtonian cases. Differences in the oscillation phase and harmonic structure characterize some of the viscoelastic effects. Subharmonic excitations at particular fluid parameters lead to a discrete group modulation of the radial excursions; this appears to be a unique, previously undiscovered phenomenon. Implications for medical ultrasound applications are discussed in light of these current findings.