Analysis of the flow of non-Newtonian visco-elastic fluids in fractal reservoir with the fractional derivative

In both the oil reservoir engineering and seepage flow mechanics, heavy oil with relaxation property shows non-Newtonian rheological characteristics. The relationship between shear rate g& and shear stress t is nonlinear. Because of the relaxation phenomena of heavy oil flow in porous media, the equation of motion can be written as[1] 2,rrvpqkppqtrrtll秏骣+=-+琪抖桫 (1) where lv and lp are velocity relaxation and pressure retardation times. For most porous media, the above motion equation (1)…     

微平行管道内Jeffrey流体的非定常电渗流动

研究了微平行管道内线性黏弹性流体的非定常电渗流动, 其中线性黏弹性流体的本构关系是由Jeffrey流体模型来描述的. 利用Laplace变换法, 求解了线性化的Poisson-Boltzmann方程、 非定常的柯西动量方程和Jeffrey流体本构方程, 给出了黏弹性Jeffrey流体电渗速度的解析表达式, 分析了无量纲弛豫时间λ₁和滞后时间λ₂对速度剖面的影响. 发现滞后时间为零时, 弛豫时间越小, 速度剖面图越接近牛顿流体的速度剖面图; 随着弛豫时间和滞后时间的增加, 速度振幅也变得越来越大, 随着时间的增加, 速度逐渐趋于恒定. 关键词： 双电层 微平行管道 Jeffrey流体 非定常电渗流动     

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.     

Fundamental solutions to time-fractional heat conduction equations in two joint half-lines

Heat conduction in two joint half-lines is considered under the condition of perfect contact, i.e. when the temperatures at the contact point and the heat fluxes through the contact point are the same for both regions. The heat conduction in one half-line is described by the equation with the Caputo time-fractional derivative of order α, whereas heat conduction in another half-line is described by the equation with the time derivative of order β. The fundamental solutions to the first and second Cauchy problems as well as to the source problem are obtained using the Laplace transform with respect to time and the cos-Fourier transform with respect to the spatial coordinate. The fundamental solutions are expressed in terms of the Mittag-Leffler function and the Mainardi function.     

Fractional Cattaneo heat equation in a semi-infinite medium

To better describe the phenomenon of non-Fourier heat conduction, the fractional Cattaneo heat equation is introduced from the generalized Cattaneo model with two fractional derivatives of different orders. The anomalous heat conduction under the Neumann boundary condition in a semi-infinity medium is investigated. Exact solutions are obtained in series form of the H-function by using the Laplace transform method. Finally, numerical examples are presented graphically when different kinds of surface temperature gradient are given. The effects of fractional parameters are also discussed.     

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamental solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy-to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the efficiency of the present schemes.     

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。     

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.     

Transient response in flexure to general uni-directional loads of variable cross-section beam with concentrated tip inertias immersed in a fluid

This paper presents a method for solving problems of transient response in flexure due to general unidirectional dynamic loads of beams of variable cross section with tip inertias. An elastodynamic theory which includes effects of continuous mass and rigidity of the beam has been applied. In the analysis the general dynamic load is expanded into a Fourier series and the beam is divided into many small uniform thickness segments. The equation of motion of each segment is mapped onto the complex domain by use of the Laplace transform method. The solutions of each set of adjoining segments are related to each other at the boundaries by the use of the transfer matrix method. The displacement, the bending slope, the bending moment and the shearing force at each boundary and at arbitrary time are obtained from the Laplace transform inversion integral by using the residue theorem. The theoretical results given in this paper are applicable to problems of dynamic response due to arbitrary loads varying with time of beams of arbitrary shape with concentrated tip inertias. As applications of the present theoretical results, numerical calculations have been carried out for two cases: a uniform beam with a tip inertia and a non-uniform beam (a truncated cone) with a tip inertia. Both are immersed in a fluid and subjected to large waves such as cnoidal waves.     

Comparative study on electroosmosis modulated flow of MHD viscoelastic fluid in the presence of modified Darcy’s law

Magnetic field plays an important role in numerous fields such as biological, chemical, mechanical and medical research. In clinical and medical research the high field magnets are extremely important to create 3D images of anatomical and diagnostic importance from nuclear magnetic resonance signals. In view of these applications, the purpose of present work is to explore the impact of an external magnetic field on the viscoelastic fluid flow in the existence of electroosmosis, porous medium and slip boundary conditions. The governing equation is modified under the suitable dimensionless quantities. The resulting non-dimensional differential equation is evaluated by analytical as well as numerical (finite difference and cubic B-spline) methods. The convergence analysis is also presented for the numerical methods. The variations of sundry parameters on velocity, volume flow rate and skin friction are presented through graphical representations. The current analysis depicts that, the higher velocities are noticed in viscoelastic fluid as compared with Newtonian fluid. The velocity enhances with rising of slip and Darcy parameters. Volume flow rate rises with the slip and viscoelastic parameters. Skin friction is a decreasing function of zeta potential, Darcy number and Hall current parameter. The limiting solutions can be captured for the Newtonian fluid model by setting the viscoelastic parameter to zero.     

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.     

Meshless analysis of three-dimensional steady-state heat conduction problems

Steady-state heat conduction problems arisen in connection with various physical and engineering problems where the functions satisfy a given partial differential equation and particular boundary conditions, have attracted much attention and research recently. These problems are independent of time and involve only space coordinates, as in Poisson’s equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, it is difficult to find an analytical solution, the only choice left is an approximate numerical solution. This paper deals with the numerical solution of three-dimensional steady-state heat conduction problems using the meshless reproducing kernel particle method (RKPM). A variational method is used to obtain the discrete equations. The essential boundary conditions are enforced by the penalty method. The effectiveness of RKPM for three-dimensional steady-state heat conduction problems is investigated by two numerical examples.     

Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition

The axisymmetric time-fractional diffusion-wave equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in a cylinder under the prescribed linear combination of the values of the sought function and the values of its normal derivative at the boundary. The fundamental solutions to the Cauchy, source, and boundary problems are investigated. The Laplace transform with respect to time and finite Hankel transform with respect to the radial coordinate are used. The solutions are obtained in terms of Mittag-Leffler functions. The numerical results are illustrated graphically.     

Initialized fractional differential equations with Riemann-Liouville fractional-order derivative

The initial value problem of fractional differential equations and its solving method are studied in this paper. Firstly, for easy understanding, a different version of the initialized operator theory is presented for Riemann-Liouville’s fractional-order derivative, addressing the initial history in a straightforward form. Then, the initial value problem of a single-term fractional differential equation is converted to an equivalent integral equation, a form that is easy for both theoretical and numerical analysis, and two illustrative examples are given for checking the correctness of the integral equation. Finally, the counter-example proposed in a recent paper, which claims that the initialized operator theory results in wrong solution of a fractional differential equation, is checked again carefully. It is found that solving the equivalent integral equation gives the exact solution, and the reason behind the result of the counter-example is that the calculation therein is based on the conventional Laplace transform for fractional-order derivative, not on the initialized operator theory. The counter-example can be served as a physical model of creep phenomena for some viscoelastic materials, and it is found that it fits experimental curves well.     

On conformable double Laplace transform

In this study authors introduce the conformable double Laplace transform which can be used to solve fractional partial differential equations that represents many physical and engineering models. In these models the derivatives and integrals are in the sense of newly defined conformable type. Then some properties of conformable double Laplace transform are expressed. Finally fractional heat equation and fractional telegraph equation which is used in various applications in science and engineering investigated as an application of this new transform.     

Pseudospin Symmetry in Position-Dependent Mass Dirac-Coulomb Problem by Using Laplace Transform and Convolution Integral

The exact pseudospin symmetry solutions of Dirac equation with position-dependent mass (PDM) Coulomb potential in the presence of Colulomb-like tensor potential are obtained by using Laplace transform (LT) approach. The energy eigenvalue equation of the Dirac particles is found and some numerical results are given. By using Laplace convolution integral, the corresponding radial wave functions are presented in terms of confluent hypergeometric functions.     

Fractional Green–Naghdi theory for thermoelectric MHD

ABSTRACT

In this work, we use the Green–Naghdi theory of thermomechanics of continua to derive a linear theory of MHD thermoelectric fluid with fractional order of heat transfer. This theory permits propagation of thermal waves at finite speed. The one-dimensional model of the theory is applied to Stokes’ flow of unsteady incompressible fluid due to a moving flat plate in the presence of both heat sources and a transverse magnetic field. The problem was solved using the Laplace transform technique. The solution in the transformed domain is obtained by a direct approach. A numerical method based on a Fourier-series expansion is used for the inversion process. The thermoelectric effects with fractional parameter on the temperature and velocity fields are analyzed and discussed in detail with the aid of graphical illustrations.     

Methods of investigation of boundary value problems in viscoelasticity theory

The present paper is devoted to some classical and modern methods of solving boundary value problems of viscoelasticity theory, including the Volterra principle, Il’yushin’s approximation method, Pobedrya’s method of numerical realization of an elastic solution, the method of Laplace (Laplace-Carson) transform and Z-transform, the method of time steps, the usage of viscoelastic models with fractional time derivatives (fractal models), and methods using a new representation of constitutive relations of nonlinear viscoelasticity To Boris Efimovich Pobedria on the occasion of his 70th birthday     

Analysis of Laser-generated Rayleigh wave on viscoelastic materials

In order to understand the viscoelasticity of material, this research has been conducted to study the propagation characteristics of viscoelastic Rayleigh wave theoretically. A model is presented for the pulsed laser generation of ultrasound on viscoelastic medium surface. Referred to the Kelvin model, the frequency equation and the normal displacement of viscoelastic Rayleigh wave were derived, the influence of the viscoelastic modulus on dispersion and attenuation was discussed. From the theoretical calculation, it is shown that the effect of viscoelasticity on the attenuation of Rayleigh wave is more than that on its dispersion. In the case of a weak viscosity, the attenuation of viscoelastic Rayleigh wave is directly proportional to viscosity modulus; the effect of shear viscosity on the attenuation is much more than that of bulk viscosity. The transient response of viscoelastic Rayleigh wave was also simulated using Laplace and Hankel inversion transform, which are showed in good agreement with the theoretic predictions. The model provides a useful tool for the determination of viscoelastic parameters of medium.     

Electro–magneto interaction in fractional Green-Naghdi thermoelastic solid with a cylindrical cavity

A unified mathematical model of Green–Naghdi’s thermoelasticty theories (GN), based on fractional time-derivative of heat transfer is constructed. The model is applied to solve a one-dimensional problem of a perfect conducting unbounded body with a cylindrical cavity subjected to sinusoidal pulse heating in the presence of an axial uniform magnetic field. Laplace transform techniques are used to get the general analytical solutions in Laplace domain, and the inverse Laplace transforms based on Fourier expansion techniques are numerically implemented to obtain the numerical solutions in time domain. Comparisons are made with the results predicted by the two theories. The effects of the fractional derivative parameter on thermoelastic fields for different theories are discussed.