Exact solution for the rotational flow of a generalized second grade fluid in a circular cylinder

This paper deals with the rotational flow of a generalized second grade fluid, within a circular cylinder, due to a torsional shear stress. The fractional calculus approach in the constitutive relationship model of a second grade fluid is introduced. The velocity field and the resulting shear stress are determined by means of the Laplace and finite Hankel transforms to satisfy all imposed initial and boundary conditions. The solutions corresponding to second grade fluids as well as those for Newtonian fluids are obtained as limiting cases of our general solutions. The influence of the fractional coefficient on the velocity of the fluid is also analyzed by graphical illustrations.     

On the MHD flow of fractional generalized Burgers＇ fluid with modified Darcy＇s law

This work is concerned with applying the fractional calculus approach to the magnetohydrodynamic (MHD) pipe flow of a fractional generalized Burgers’ fluid in a porous space by using modified Darcy’s relationship. The fluid is electrically conducting in the presence of a constant applied magnetic field in the transverse direction. Exact solution for the velocity distribution is developed with the help of Fourier transform for fractional calculus. The solutions for a Navier–Stokes, second grade, Maxwell, Oldroyd-B and Burgers’ fluids appear as the limiting cases of the present analysis.     

Decay of vortex velocity and diffusion of temperature in a generalized second grade fluid

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid was introduced. The velocity and temperature fields of the vortex flow of a generalized second fluid with fractional derivative model were described by fractional partial differential equations. Exact analytical solutions of these differential equations were obtained by using the discrete Laplace transform of the sequential fractional derivatives and generalized Mittag-Leffier function. The influence of fractional coefficient on the decay of vortex velocity and diffusion of temperature was also analyzed.     

Unsteady flows of a generalized second grade fluid with the fractional derivative model between two parallel plates

The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced. Exact analytical solutions are obtained for a class of unsteady flows for the generalized second grade fluid with the fractional derivative model between two parallel plates by using the Laplace transform and Fourier transform for fractional calculus. The unsteady flows are generated by the impulsive motion or periodic oscillation of one of the plates. In addition, the solutions of the shear stresses at the plates are also determined. The project supported by the National Natural Science Foundation of China (10372007, 10002003) and CNPC Innovation Fund     

广义Maxwell黏弹性流体在两平板间的非定常流动

将分数阶微积分运算引入Maxwell黏弹性流体的本构方程，研究了黏弹性流体在两平板问的非定常流动．对于广义Maxwell黏弹性流体的分数阶导数模型，导出了对时间具有分数阶导数的特殊运动方程，利用分数阶微积分的Laplace变换理论，得到了流动的解析解．     

Coupling model for unsteady MHD flow of generalized Maxwell fluid with radiation thermal transform

This paper introduces a new model for the Fourier law of heat conduction with the time-fractional order to the generalized Maxwell fluid. The flow is influenced by magnetic field, radiation heat, and heat source. A fractional calculus approach is used to establish the constitutive relationship coupling model of a viscoelastic fluid. We use the Laplace transform and solve ordinary differential equations with a matrix form to obtain the velocity and temperature in the Laplace domain. To obtain solutions from the Laplace space back to the original space, the numerical inversion of the Laplace transform is used. According to the results and graphs, a new theory can be constructed. Comparisons of the associated parameters and the corresponding flow and heat transfer characteristics are presented and analyzed in detail.     

An exact solution for flow past an accelerated horizontal plate in a rotating fluid with the generalized Oldroyd-B model

In this paper, the generalized Oldroyd-B with fractional calculus approach is used. An exact solution in terms of Fox-H function for flow past an accelerated horizontal plate in a rotating fluid is obtained by using discrete Laplace transform method. A comparison among the influence of various parameters in the Oldroyd-B model and the angular velocity of the fluid on the velocity profiles is made through numerical method in graphic form.     

Stokes＇ first problem for a viscoelastic fluid with the generalized Oldroyd-B model

The flow near a wall suddenly set in motion for a viscoelastic fluid with the generalized Oldroyd-B model is studied. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox H-function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions. The project supported by the National Natural Science Foundation of China (10272067), the Doctoral Program Foundation of the Education Ministry of China (20030422046), the Natural Science Foundation of Shandong Province, China (Y2006A14) and the Research Foundation of Shandong University at Weihai. The English text was polished by Keren Wang.     

Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. The flow near a wall suddenly set in motion is studied for a non-Newtonian viscoelastic fluid with the fractional Maxwell model. Exact solutions of velocity and stress are obtained by using the discrete inverse Laplace transform of the sequential fractional derivatives. It is found that the effect of the fractional orders in the constitutive relationship on the flow field is significant. The results show that for small times there are appreciable viscoelastic effects on the shear stress at the plate, for large times the viscoelastic effects become weak. The project supported by the National Natural Science Foundation of China (10002003), Foundation for University Key Teacher by the Ministry of Education, Research Fund for the Doctoral Program of Higher Education     

A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. A generalized Maxwell model with the fractional calculus was considered. Exact solutions of some unsteady flows of a viscoelastic fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional calculus. The flows generated by impulsively started motions of one of the plates are examined. The flows generated by periodic oscillations of one of the plates are also studied.     

An inverse problem to estimate an unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid

In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid.     

Unsteady Rotating Flows of a Viscoelastic Fluid with the Fractional Maxwell Model Between Coaxial Cylinders

The fractional calculus is used in the constitutive relationship model of viscoelastic fluid. A generalized Maxwell model with fractional calculus is considered. Based on the flow conditions described, two flow cases are solved and the exact solutions are obtained by using the Weber transform and the Laplace transform for fractional calculus.The project supported by the National Natural Science Foundation of China (10272067, 10426024), the Doctoral Program Foundation of the Education Ministry of China (20030422046) and the Natural Science Foundation of Shandong University at Weihai. The English text was polished by Keren Wang.     

A hybrid technique for the solution of unsteady Maxwell fluid with fractional derivatives due to tangential shear stress

The article describes the unsteady motion of viscoelastic fluid for a Maxwell model with fractional derivatives. The flow is produced by cylinder, considering time dependent quadratic shear stress ft² on Maxwell fluid with fractional derivatives. The fractional calculus approach is used in the constitutive relationship of Maxwell model. By applying Laplace transform with respect to time t and modified Bessel functions, semianalytical solutions for velocity function and tangential shear stress are obtained. The obtained semianalytical results are presented in transform domain, satisfy both initial and boundary conditions. Our solutions particularized to Newtonian and Maxwell fluids having typical derivatives. The inverse Laplace transform has been calculated numerically. The numerical results for velocity function are shown in Table by using MATLAB program and compared them with two other algorithms in order to provide validation of obtained results. The influence of fractional parameters and material constants on the velocity field and tangential stress is analyzed by graphs.     

Some helical flows of a Burgers’ fluid with fractional derivative

In this article, fractional calculus approach is used in the constitutive relationship of a Burgers’ fluid model. Integral transforms are used to calculate the velocity and the stress fields for some helical flows of a Burgers’ fluid with fractional derivative. Moreover, the behavior of different physical parameters involve in the Burgers’ fluid model is analyzed through several graphs.     

The Flow Analysis of Viscoelastic Fluid with Fractional Order Derivative in Horizontal Well

The fractional calculus approach is introduced into the seepage mechanics. A three-dimensional relaxation model of viscoelastic fluid is built. The models based on four boundary conditions of exact solution in Laplace space for some unsteady flows in an infinite reservoir is obtained by using the Laplace transform and Fourier sine and cosine integral transform. The pressure transient behavior of non-Newtonian viscoelastic fluid is studied by using Stehfest method of the numerical Laplace transform inversion and Gauss–Laguerre numerical integral formulae. The viscoelastic fluid is very sensitive to the order of the fractional derivative. The change rules of pressure are discussed when the parameters of the models change. The plots of type pressure curves are given, and the results can be provided to theoretical basis and well-test method for oil field.     

Models for Ground Water Flow: A Numerical Comparison Between Richards' Model and the Fractional Flow Model

In this paper we compare two models for flow in porous media. The first is the well known Richards' equation, which is based on the assumption that the air in the unsaturated zone has infinite mobility. This means that it models a single phase. In the second and more general full two-phase approach, the air is considered as a separate phase. Here, we use the fractional flow equation.We study the difference between the two models numerically by varying the relative contribution of the different physical terms (the gravity and the total velocity) in the fractional flow equation. Richards' equation is considered as the limit of the fractional flow approach when the mobility of the air-phase tends to infinity. In particular, we are interested in determining the parameter intervals where the two models differ significantly, and we will quantify the asymptotic behavior.The equations are studied in the two-dimensional (2D) case. The study is based on a relative permeability depending quadratically on the saturation, and a capillary pressure expressed by a cubic function of the saturation.     

An extended formulation of calculus of variations for incommensurate fractional derivatives with fractional performance index

In this paper, we consider the main problem of variational calculus when the derivatives are Riemann?CLiouville-type fractional with incommensurate orders in general. As the most general form of the performance index, we consider a fractional integral form for the functional that is to be extremized. In the light of fractional calculus and fractional integration by parts, we express a generalized problem of the calculus of variations, in which the classical problem is a special case. Considering five cases of the problem (fixed, free, and dependent final time and states), we derive a necessary condition which is an extended version of the classical Euler?CLagrange equation. As another important result, we derive the necessary conditions for an optimization problem with piecewise smooth extremals where the fractional derivatives are not necessarily continuous. The latter result is valid only for the integer order for performance index. Finally, we provide some examples to clarify the effectiveness of the proposed theorems.     

Stochastic response of an axially moving viscoelastic beam with fractional order constitutive relation and random excitations

A convenient and universal residue calculus method is proposed to study the stochastic response behaviors of an axially moving viscoelastic beam with random noise excitations and fractional order constitutive relationship, where the random excitation can be decomposed as a nonstationary stochastic process, Mittag-Leffler internal noise, and external stationary noise excitation. Then, based on the Laplace transform approach, we derived the mean value function, variance function and covariance function through the Green's function technique and the residue calculus method, and obtained theoretical results. In some special case of fractional order derivative α , the Monte Carlo approach and error function results were applied to check the effectiveness of the analytical results, and good agreement was found. Finally in a general-purpose case, we also confirmed the analytical conclusion via the direct Monte Carlo simulation.     

A Fractional Model of Continuum Mechanics

Although there has been renewed interest in the use of fractional models in many application areas, in reality fractional analysis has a long and distinguished history and can be traced back to the likes of Leibniz (Letter to L’Hospital, 1695), Liouville (J. éc. Polytech. 13:71, 1832), and Riemann (Gesammelte Werke, p. 62, 1876). Recent publications (Podlubny in Math. Sci. Eng. 198, 1999; Sabatier et al. in Advances in fractional calculus: theoretical developments and applications in physics and engineering, Springer, Berlin, 2007; Das in Functional fractional calculus for system identification and controls, Springer, Berlin, 2007) demonstrate that fractional derivative models have found widespread applications in science and engineering. Late fundamental considerations have led to the introduction of fractional calculus in continuum mechanics in an attempt to develop non-local constitutive relations (Lazopoulos in Mech. Res. Commun. 33:753–757, 2006). Attempts have also been made to model microscopic forces using fractional derivatives (Vazquez in Nonlinear waves: classical and quantum aspects, pp. 129–133, 2004). Our approach in this paper differs from previous theoretical work, in that we develop a general framework directly from the classical continuum mechanics, by defining the laws of motion and the stresses using fractional derivatives. The timeliness and relevance of this work is justified by the surge in interest in applications of fractional order models to biological, physical and economic systems. The aim of the present paper is to lay the foundations for a new non-local model of continuum mechanics based on fractional order derivatives which we will refer to as the fractional model of continuum mechanics. Following the theoretical development, we apply this framework to two one-dimensional model problems: the deformation of an infinite bar subjected to a self-equilibrated load distribution, and the propagation of longitudinal waves in a thin finite bar.