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.     

Non-local continuum mechanics and fractional calculus

The present work introduces fractional calculus into the continuum mechanics area describing non-local constitutive relations. Considering a one-dimensional body and assuming total stored energy depending not only upon the local strain but also upon a fractional derivative of the stain, an elastic model with non-local stress–strain behavior is introduced. Fractional calculus provides a natural framework for describing non-local constitutive relations and requires no assumptions for the interval of non-local influence. Furthermore, the proposed method works in finite intervals contrary to the existing theories requiring infinite domains.     

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     

Discrete fractional diffusion equation

Nonlinear Dynamics - The tool of the discrete fractional calculus is introduced to discrete modeling of diffusion problem. A fractional time discretization diffusion model is presented in the...     

Chaotic oscillation of a nonlinear power system

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Analysis of laminar thermal entrance region of elliptical and rectangular channels with Kantorowich method

The Kantorowich method of variational calculus is introduced to solve the problem of laminar forced heat convection in a channel of an arbitrary cross section. Employing this procedure, the development of the temperature field and the heat transfer data in the thermal entrance region of elliptical and rectangular channels are determined, assuming a linear variation of wall temperature in the direction of flow. The uniform and the fully developed velocity profiles are considered. The local Nusselt numbers are tabulated for different aspect ratios.     

Integral Representations of Increments of Stochastic Processes

A formulation is presented in which the increment of a stochastic process is represented as an integral of the derivative of the process. It is shown that this representation is an alternative to the more common approach of writing equations for the differentials of stochastic processes. A possible advantage of the integral formulation is that its reliance on derivatives, rather than differentials, makes the operations of stochastic calculus more closely resemble those of ordinary deterministic calculus. This similarity to well-known mathematics may serve to make stochastic calculus accessible to a broader audience than in the past. The integral formulation is herein shown to be compatible with the Itô differential rule for non-Gaussian processes and is used to describe the increment of the nonstationary response of a system governed by a vector stochastic equation with parametric delta-correlated excitation.     

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.     

A non-iterative transformation method for Newton's free boundary problem

In book II of Newton's Principia Mathematica of 1687 several applicative problems are introduced and solved. There, we can find the formulation of the first calculus of variations problem that leads to the first free boundary problem of history. The general calculus of variations problem is concerned with the optimal shape design for the motion of projectiles subject to air resistance. Here, for Newton's optimal nose cone free boundary problem, we define a non-iterative initial value method which is referred in the literature as a transformation method. To define this method we apply invariance properties of Newton's free boundary problem under a scaling group of point transformations. Finally, we compare our non-iterative numerical results with those available in the literature and obtained via an iterative shooting method. We emphasize that our non-iterative method is faster than shooting or collocation methods and does not need any preliminary computation to test the target function as the iterative method or even provide any initial iterate. Moreover, applying Buckingham Pi-Theorem we get the functional relation between the unknown free boundary and the nose cone radius and height.     

Treadmilling swimmers near a no-slip wall at low Reynolds number

The motion of a circular treadmilling low Reynolds number swimmer near a no-slip wall is studied analytically. First, the exact solution of Jeffrey and Onishi [Q. J. Mech. Appl. Math., 34 (1981)] for a translating and rotating solid cylinder near a no-slip wall is rederived using a novel conformal mapping approach that differs from the original derivation which employed bipolar coordinates. Then it is shown that this solution can be combined with the reciprocal theorem, and the calculus of residues, to produce an explicit non-linear dynamical system for the treadmilling swimmer's velocity and angular velocity. The resulting non-linear dynamical system governing the swimmer motion is used to corroborate the qualitative results obtained by an approximate model of the same swimmer recently presented in Crowdy and Or [Phys. Rev. E., 81 (2010)].     

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.     

刚架结构变更定理与极限状态函数的连续生成

本文基于计算结构力学概念提出了刚架结构连续变更定理,用数值分析与递推公式相结合的方法连续生成结构极限状态函数,给出了结构退化为机构的新判据。由于引用了计算结构力学方法,重分析不必重新形成结构刚度阵,不必反复求逆矩阵,大大提高了计算效率。     

有限元秩亏方程的最小范数问题

为避免因确定边界约束条件时的不同而引起的应力，位移解不一致问题，作者认为在无明显稳定边界情况下应用整体最小范数法解算有限元方程，如存在相对稳定边界，则可采用部分最小范数解，在一定条件下，最小范数解更能反映位移场的真实状态，并使计算误差分布均匀。     

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.     

Exact solutions for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit

The exact solutions are obtained for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit. The fractional calculus in the constitutive relationship of a non-Newtonian fluid is introduced. We construct the solutions by means of Fourier transform and the discrete Laplace transform of the sequential derivatives and the double finite Fourier transform. The solutions for Newtonian fluid between two infinite parallel plates appear as limiting cases of our solutions.     

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.     

A new approach to model strain change of gelled waxy crude oil under constant stress

Deformation of gelled waxy crude oil with loaded stress is worthy of research for the flow assurance of pipelining system. A dispersion parameter was introduced to characterize the disruption degree of wax crystal structure in crude oil with shear action. Based on fractional calculus theory, a rheological model incorporating dispersion parameter was proposed to describe creep of gelled waxy crude. A discrete and numerical algorithm was proposed to solve the model. Combining with the experimental results of five kinds of waxy crude oil, the model parameters were regressed and found to change monotonously with test temperature. Multiple creep curves of gelled waxy crude oil at a certain temperature can be described with this model.     

Forward and inverse modeling of near-well flow using discrete edge-based vector potentials

Homogeneous effective permeabilities in the near-well region are generally obtained using analytical solutions for transient flow. In contrast, this paper focuses on heterogeneous permeability obtained from steady flow solutions, although extensions to unsteady flow are introduced too. Exterior calculus and its discretized form have been used as a guide to derive the system of algebraic equations. Edge-based vector potentials describing 3-D steady and unsteady flow without mass balance error stabilize the solution. After a physics-oriented introduction, relatively simple analytic examples of forward and inverse discrete modeling demonstrate the applicability.     

Applying the calculus of variations to circular motion

We demonstrate that having found a condition for the stationary points in multivariable calculus, that condition may be substituted back into the original equation and still yield the correct stationary points. With that, we emphasise the conditions that must be met in solving multivariable stationary point problems. We further use the analogy of the stationary points problem with finding stationary paths in calculus of variations to apply the latter to circular paths in an axisymmetric potential. Surprisingly, we find that this classical problem does not meet the required conditions. We subsequently derive new conditions that must be met and suggest a possible application.     

Theory and application of numerical simulation method of capillary force enhanced oil production

A kind of second-order implicit upwind fractional step finite difference methods are presented for the numerical simulation of coupled systems for enhanced(chemical)oil production with capillary force in the porous media.Some techniques,e.g.,the calculus of variations,the energy analysis method,the commutativity of the products of difference operators,the decomposition of high-order difference operators,and the theory of a priori estimate,are introduced.An optimal order error estimate in the l~2 norm is derived.The method is successfully used in the numerical simulation of the enhanced oil production in actual oilfields.The simulation results are satisfactory and interesting.