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     

Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel

The unsteady flow of viscoelastic fluid with the fractional derivative Maxwell model (FDMM) in a channel is studied in this note. The exact solutions are obtained for an arbitrary pressure gradient by means of the finite Fourier cosine transform and the Laplace transform. Two special cases of pressure gradient are discussed. Some results given by the classical models with integer-order are included in this note.     

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.     

Unsteady incompressible Couette flow in a uniform transverse magnetic field

Summary The unsteady plane Couette flow of an incompressible, viscous and infinitely conducting fluid in a uniformly imposed transverse magnetic field is studied. The problem is solved in general in a series form by means of a finite Fourier transform, and explicit solutions for two special cases are worked out.     

Plane strain consolidation of a multi-layered soil with anisotrpic permeability and compressible constituents

对多层地基的平面应变固结问题进行了研究,并同时考虑了土体的渗透各向异性和孔隙 流体的可压缩性. 从平面应变Biot固结的控制方程出发,对时间t, 坐标z和x进行 Laplace和Fourier变换,建立了地基表面(z=0)和任意深度z处的基本量 在Laplace-Fourier变换域内的传递矩阵关系. 利用传递矩阵 法,结合土层连续条件和边界条件,并应用Laplace-Fourier逆变换技术,推导出渗透各向 异性可压缩多层地基平面应变固结的理论解. 基于该解,编制了计算程序,并进行了 数值计算. 讨论了土体的渗透各向异性、孔隙流体的可压缩性以及地基的分层特性对地基固 结的影响,分析结果表明：土体的渗透各向异性、孔隙流体的可压缩性,以及地基的分层特 性对地基的固结行为有着重要的影响.     

Dynamic Green’s Functions for an Anisotropic Multilayered Poroelastic Half-Space

The dynamic responses of an anisotropic multilayered poroelastic half-space to a point load or a fluid source are studied based on Stroh formalism and Fourier transforms. Taking the boundary conditions and the continuity of the materials into consideration, the three-dimensional Green’s functions of generalized concentrated forces (force and fluid source) applied at the free surface, interface and in the interior of a layer are derived in the Fourier transformed domain, respectively. The actual solutions in the frequency domain can further be acquired by inverting the Fourier transform. Finally, numerical examples are carried out to verify the presented theory and discuss the Green’s fields due to three cases of a concentrated force or a fluid source applied at three different locations for an anisotropic multilayered poroelastic half-space.

     

Spectral element modelling and analysis of a pipeline conveying internal unsteady fluid

In this paper, a spectral element model is developed for the uniform straight pipelines conveying internal unsteady fluid. Four coupled pipe-dynamics equations are derived first by using the Hamilton's principle and the principles of fluid mechanics. The transverse displacement, the axial displacement, the fluid pressure and the fluid velocity are all considered as the dependent variables. The coupled pipe-dynamics equations are then linearized about the steady-state values of the fluid pressure and velocity. As the final step, the spectral element model represented by the exact dynamic stiffness matrix, which is often called spectral element matrix, is formulated by using the frequency-domain solutions of the linearized pipe-dynamics equations. The fast Fourier transform (FFT)-based spectral dynamic analyses are conducted to evaluate the accuracy of the present spectral element model and also to investigate the structural dynamic characteristics and the internal fluid transients of an example pipeline system.     

The exact solution of Stokes' second problem including start-up process with fractional element

The start-up process of Stokes' second problem ofa viscoelastic material with fractional element is studied. Thefluid above an infinite flat plane is set in motion by a suddenacceleration of the plate to steady oscillation. Exact solutionsare obtained by using Laplace transform and Fourier transform.It is found that the relationship between the first peakvalue and the one of equal-amplitude oscillations dependson the distance from the plate. The amplitude decreases forincreasing frequency and increasing...     

The exact solution of Stokes＇ second problem including start-up process with fractional element

The start-up process of Stokes＇ second problem of a viscoelastic material with fractional element is studied. The fluid above an infinite flat plane is set in motion by a sudden acceleration of the plate to steady oscillation. Exact solutions are obtained by using Laplace transform and Fourier transform. It is found that the relationship between the first peak value and the one of equal-amplitude oscillations depends on the distance from the plate. The amplitude decreases for increasing frequency and increasing distance.     

Decay of a potential vortex in a Maxwell fluid

The velocity field and the associated tangential tension corresponding to a potential vortex in a Maxwell fluid are determined by means of the Hankel transform. The similar solutions for a Newtonian fluid appear as a limiting case of our solutions.     

Dynamic Analysis of an Infinite Cylindrical Hole in a Saturated Poroelastic Medium

In this paper, the dynamic response of an infinite cylindrical hole embedded in a porous medium and subjected to an axisymmetric ring load is investigated. Two scalar potentials and two vector potentials are introduced to decouple the governing equations of Biot’s theory. By taking a Fourier transform with respect to time and the axial coordinate, we derive general solutions for the potentials, displacements, stresses and pore pressures in the frequency-wave-number domain. Using the general solutions and a set of boundary conditions applied at the hole surface, the frequency-wave-number domain solutions for the proposed problem are determined. Numerical inversion of the Fourier transform with respect to the axial wave number yields the frequency domain solutions, while a double inverse Fourier transform with respect to frequency as well as the axial wave number generates the time-space domain solution. The numerical results of this paper indicate that the dynamic response of a porous medium surrounding an infinite hole is dependant upon many factors including the parameters of the porous media, the location of receivers, the boundary conditions along the hole surface as well as the load characteristics.     

The Rayleigh-Stokes-Problem for a fluid of Maxwellian type

The velocity fields corresponding to an incompressible fluid of Maxwellian type subjected to a linear flow on an infinite flat plate and within an infinite edge are determined by means of the Fourier sine transforms. They are in close proximity of those of a second grade fluid. The well known solutions for a Navier-Stokes fluid appear as a limiting case of our solutions.     

渗透各向异性可压缩地基固结的平面应变分析

对多层地基的平面应变固结问题进行了研究,并同时考虑了土体的渗透各向异性和孔隙流体的可压缩性.从平而应变Biot固结的控制方程出发,对时间t,坐标z和x进行Laplace和Fourier变换,建立了地基表面(z=O)和任意深度z处的基本量在Laplauce-Fourier变换域内的传递矩阵关系.利用传递矩阵法,结合土层连续条件和边界条件,并应用Laplace-Fourier逆变换技术,推导出渗透各向异性可压缩多层地基平面应变固结的理论解.基于该解,编制了计算程序,并进行了数值计算.讨论了土体的渗透各向异性、孔隙流体的可压缩性以及地基的分层特性对地基同结的影响,分析结果表明:土体的渗透各向异性、孔隙流体的可压缩性,以及地基的分层特性对地基的固结行为有着重要的影响.     

Flexural-gravity wave resistances due to a surface-moving line source on a fluid covered by a thin elastic plate

Analytical solutions for the flexural-gravity wave resistances due to a line source steadily moving on the surface of an infinitely deep fluid are investigated within the framework of the linear potential theory. The homogenous fluid, covered by a thin elastic plate, is assumed to be incompressible and inviscid, and the motion to be irrotational. The solution in integral form for the wave resistance is obtained by means of the Fourier transform and the explicitly analytical solutions are derived with the aid of the residue theorem. The dispersion relation shows that there is a minimal phase speed c_min, a threshold for the existence of the wave resistance. No wave is generated when the moving speed of the source V is less than c_min while the wave resistances firstly increase to their peak values and then decrease when V ? c_min. The effects of the flexural rigidity and the inertia of the plate are studied.     

饱和土地基在简谐荷载作用下的解析分析

基于多孔介质混合物理论,用解析的方法研究了不可压饱和土地基受到简谐荷载作用下的动力响应问题。利用Fourier积分变换求解耦合方程组,得到了二维饱和土介质在简谐荷载作用下的通解。针对表面透水的具有下卧基岩的饱和土层以及半无限饱和土地基的边界条件,获得了固体骨架位移、孔隙流体位移、固体骨架有效应力以及孔隙流体压力的积分形式解答,并通过数值算例分析了饱和土地基在简谐荷载作用下的响应。     

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.     

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.     

MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium

Exact analytical solutions for magnetohydrodynamic (MHD) flows of an incompressible second grade fluid in a porous medium are developed. The modified Darcy's law for second grade fluid has been used in the flow modelling. The Hall effect is taken into account. The exact solutions for the unsteady flow induced by the time-dependent motion of a plane wall between two side walls perpendicular to the plane has been constructed by means of Fourier sine transforms. The similar solutions for a Newtonian fluid, performing the same motion, appear as limiting cases of the solutions obtained here. The influence of various parameters of interest on the velocity and shear stress at the bottom wall has been shown and discussed through several graphs. A comparison between a Newtonian and a second grade fluids is also made.     

受移动简谐力作用的多孔弹性半平面问题

研究了匀速移动的振动荷载作用下半无限多孔饱和固体中产生的应力和孔隙水压力．应用Fourier变换求解该问题的控制偏微分方程,考虑了荷载的移动速度及振动频率对多孔饱和固体中应力与孔隙水压力的影响,并与相应的弹性介质的解答进行了比较．结果显示多孔饱和半平面中应力和孔隙水压力随荷载的移动速度与振动频率的增加而增大,多孔饱和固体在移动荷载下的动力响应与相应的单相弹性固体的动力响应有较大的差别。     

Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary

Based on a modified Darcy's law, Stokes’ first problem was investigated for a second grade fluid in a porous half-space with a heated flat plate. Exact solutions of the velocity and temperature fields were obtained using Fourier sine transforms. In contrast to the classical Stokes’ first problem, there is a steady-state solution for the second grade fluid in the porous half-space, which is a damping exponential function with respect to the distance from the flat plate. The well-known solutions for Newtonian fluids in non-porous or porous half-space appear in limiting cases of our solutions.