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.     

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.     

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.     

带有分数阶热流条件的时间分数阶热波方程及其参数估计问题

研究了Caputo导数定义下带有分数阶热流条件的一维时间分数阶热波方程及其参数估计问题.首先,对正问题给出了解析解;其次,基于参数敏感性分析,利用最小二乘算法同时对分数阶阶数α和热松弛时间τ进行参数估计;最后对不同的热流分布函数所构成的两个初边值问题,分别进行参数估计仿真实验,分析温度真实值和估计值的拟合程度.实验结果表明,最小二乘算法在求解时间分数阶热波方程的两参数估计问题中是有效的.本文为分数阶热波模型的参数估计提供了一种有效的方法.     

平行板微管道间Maxwell流体的高Zeta势周期电渗流动

本文研究了两平行板微管道中线性黏弹性流体的周期电渗流动, 其中线性黏弹性流体的本构关系是由广义Maxwell模型描述的. 将电渗力作为体力, 解析求解了非线性的Poisson-Boltzmann (P-B)方程, 柯西动量方程和广义Maxwell本构方程. 通过数值计算, 分析了无量纲壁面Zeta势ψ₀ 、 周期电渗流 (electroosmotic flow, EOF) 振荡雷诺数Re和无量纲弛豫时间λ ₁ω 对速度剖面的影响. 结果表明: 对给定的电动宽度K(表示微管道的特征尺度与双电层厚度的比值)、 弛豫时间λ ₁ω 和振荡雷诺数Re, 高Zeta势ψ₀ 产生较大的EOF速度振幅, 并且速度剖面的变化主要集中在双电层 (electric double-layer, EDL) 的狭窄的区域. 此外, 随着弛豫时间的增长流体的弹性显著增加, 速度的变化可以延伸到整个流动的区域中. 对给定的雷诺数Re, 较长的弛豫时间λ₁ω 导致EOF速度剖面较快的变化, 且速度剖面的振幅逐渐增大.     

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.     

Transient electro-osmotic flow of generalized Maxwell fluids in a straight pipe of circular cross section

The transient electro-osmotic flow of a generalized Maxwell fluid with fractional derivative in a narrow capillary tube is examined. With the help of an integral transform method, analytical expressions are derived for the electric potential and transient velocity profile by solving the linearized Poisson-Boltzmann equation and the Navier-Stokes equation. It was shown that the distribution and establishment of the velocity consists of two parts, the steady part and the unsteady one. The effects of relaxation time, fractional derivative parameter, and the Debye-Hückel parameter on the generation of flow are shown graphically and analyzed numerically. The velocity overshoot and oscillation are observed and discussed.     

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.     

The variant of post-Newtonian mechanics with generalized fractional derivatives

In this article, we investigate mathematically the variant of post-Newtonian mechanics using generalized fractional derivatives. The relativistic-covariant generalization of the classical equations for gravitational field is studied. The equations (i) match the weak Newtonian limit on the moderate scales and (ii) deliver a potential higher than Newtonian on certain large-distance characteristic scales. The perturbation of the gravitational field results in the tiny secular perihelion shift and exhibits some unusual effects on large scales. The general representation of the solution for the fractional wave equation is given in the form of retarded potentials. The solutions for the Riesz wave equation and classical wave equation are clearly distinctive in an important sense. The hypothetical gravitational Riesz wave demonstrates the space diffusion of the wave at the scales of metric constant. The diffusion leads to the blur of the peak and disruption of the sharp wave front. This contrasts with the solution of the D'Alembert classical wave equation, which obeys the Huygens principle and does not diffuse.     

Numerical simulation for separation of multi-phase immiscible fluids in porous media

Based on a lattice Boltzmann method and general principles of porous flow, a numerical technique is presented for analysing the separation of multi-phase immiscible fluids in porous media. The total body force acting on fluid particles is modified by axiding relative permeability in Nithiarasu＇s expression with an axiditional surface tension term. As a test of this model, we simulate the phase separation for the case of two immiscible fluids. The numerical results show that the two coupling relative permeability coefficients K12 and K21 have the same magnitude, so the linear flux-forcing relationships satisfy Onsager reciprocity. Phase separation phenomenon is shown with the time evolution of density distribution and bears a strong similarity to the results obtained from other numerical models and the flows in sands. At the same time, the dynamical rules in this model are local, therefore it can be run on massively parallel computers with well computational efficiency.     

Pressure-driven transient flows of Newtonian fluids through microtubes with slip boundary

Recent advances in microscale experiments and molecular simulations confirm that slip of fluid on solid surface occurs at small scale, and thus the traditional no-slip boundary condition in fluid mechanics cannot be applied to flow in micrometer and nanometer scale tubes and channels. On the other hand, there is an urgent need to understand fluid flow in micrometer scale due to the emergence of biochemical lab-on-the-chip system and micro-electromechanical system fabrication technologies. In this paper, we study the pressure driven transient flow of an incompressible Newtonian fluid in microtubes with a Navier slip boundary condition. An exact solution is derived and is shown to include some existing known results as special cases. Through analysis of the derived solution, it is found that the influences of boundary slip on the flow behaviour are qualitatively different for different types of pressure fields driving the flow. For pressure fields with a constant pressure gradient, the boundary slip does not alter the interior material deformation and stress field; while, for pressure fields with a wave form pressure gradient, the boundary slip causes the change of interior material deformation and consequently the velocity profile and stress field. We also derive asymptotic expressions for the exact solution through which a parameter is identified to dominate the behaviour of the flow driven by the wave form pressure gradient, and an explicit formulae for the critical slip parameter leading to the maximum transient flow rate is established.     

Mixed element FEM level set method for numerical simulation of immiscible fluids

A new realization of a finite element level set method for simulation of immiscible fluid flows is introduced and validated on numerical benchmarks. The new method involves a mixed discretization of the dependent variables, discretizing the flow variables with non-conforming Rannacher–Turek finite elements while using a simple first order conforming discretization of the level set field. A three step segregated solution approach is employed, first a discrete projection method is used to decouple and compute the velocity and pressure separately, after which the level set field can be computed independently.The developed method is tested and validated on a static bubble test case and on a numerical rising bubble test case for which a very accurate benchmark solution has been established. The new approach is also compared against two commercial simulation codes, Ansys Fluent and Comsol Multiphysics, which shows that the developed method is a magnitude or more accurate and at the same time significantly faster than state of the art commercial codes.     

Power-law slip profile of the moving contact line in two-phase immiscible flows

Large-scale molecular dynamics (MD) simulations on two-phase immiscible flows show that, associated with the moving contact line, there is a very large 1/x partial-slip region where x denotes the distance from the contact line. This power-law partial-slip region is verified in large-scale adaptive continuum calculations based on a local, continuum hydrodynamic formulation, which has proved successful in reproducing MD results at the nanoscale. Both MD simulations and numerical solutions of continuum equations indicate the existence of a universal slip profile in the Stokes-flow regime.     

On mechanism of peristaltic flows for power-law fluids

This paper presents a mathematical model for flow induced by peristaltic waves through a deformable tube. An incompressible power-law fluid is considered. The two dimensional model is formulated based upon the fundamental equation of mass conservation and momentum. Exact analytical solutions have been derived for the stream function, axial velocity and pressure gradient which is the main goal of this work. Moreover, pressure rise per wavelength has been evaluated numerically. The present analysis has been performed under long wavelength and low Reynolds number assumptions. The effects of various physical parameters are also discussed through graphs.     

Vortices with fractional flux in two-gap superconductors and in extended faddeev model

We discuss linear topological defects allowed in two-gap superconductors and equivalent extended Faddeev model. We show that, in these systems, there exist vortices which carry an arbitrary fraction of magnetic flux quantum. Besides that, we discuss topological defects which do not carry magnetic flux and describe features of ordinary one-magnetic-flux-quantum vortices in the two-gap system. The results could be relevant for the newly discovered two-band superconductor MgB2.