ExactSolutiononUnsteadyCouetteFlowofGeneralizedMaxwellFluidwithFractionalDerivative资源-CSDN文库

首发论文

76 浏览量 2019-12-30 16:23:14 上传评论收藏 412KB PDF 举报

资源推荐

资源详情

资源评论

http://www.paper.edu.cn

Exact Solution on Unsteady Couette Flow of

Generalized Maxwell Fluid with Fractional

Derivative

Shaowei Wang Mingyu Xu

nstitute of Applied Math, School of Math. and System Science

Shandong University, Jinan, PRC 250100

wangshaowei@math.sdu.edu.cn xumingyu@163169.net

Abstract

In this paper the unsteady Couette flow of a generalized Maxwell fluid with fractional

derivative (GMF) is studied. The exact solution is obtained with the help of integral

transforms (Laplace transform and Weber transform) and generalized Mittag-Leffler

function. It was shown that the distribution and establishment of the velocity is

governed by two non-dimensional parameters

, b and fractional derivative

the model. The result of classical (Newtonian fluid) Couette flow can be contained as

a special case of the result given by this paper, and the decaying of unsteady part of

GMF displays power law behavior, which has scale invariance.

Keywords: unsteady Couette flow; generalized Maxwell fluid; fractional calculus;

exact solution

1 Introduction

The non-Newtonian fluids are increasing being considered more important and appropriate in

technological applications than the Newtonian fluids. Strictly speaking, the linear relation between

stress and the rate of strain does not exist for a lot of real fluids, such as blood, oils, paints and

polymeric solution. In general, the analysis of the behavior of the fluid motion of the non-Newtonian

fluids tends to be much more complicated and subtle in comparison with that of the Newtonian fluids.

There has been fairly large number of flows of Newtonian fluids for which a closed form analytical

This work was supported by the National Natural Science Foundation of China (Grant No. 10272067) and the

Doctoral Program Foundation of the Education Ministry of P.R. China (Grant No.20030422046).

http://www.paper.edu.cn

solution is possible. However, for non-Newtonian fluids such exact solutions are rare. In order to

describe the rheological properties of wide classes of materials, the rheological constitutive equations

with fractional derivatives have been introduced for a long time, which are discussed in the papers

given by Friedrich[1], Bagley[2], Glöckle and Nonnenmacher[3], Rossikhin and Shitikova[4,5],

Mainardi[6], Mainardi and Gorenflo[7], Makris and Coustantinous[8] and the references therein. And

they obtained the ideal results which are good agreement with the experimental data. The starting

point of the fractional derivative model of non-Newtonian fluid is usually a classical differential

equation which is modified by replacing the time derivative of an integer order by the so-called

Riemann-Liouville fractional differential operator.

The fluid considered in this paper is a generalized Maxwell fluid with fractional derivative (GMF).

In one dimension its constitutive equation may be expressed in terms of scalar form[1,9,10]

γκτκτ

ββαα

DGD

000

, (1)

where

is the shear stress,

is the shear strain,

is the relaxation time and is a

shear modulus,

is a viscosity constant (

and

are fractional parameters

such that

10 ≤≤≤

. And is the Riemann-Liouville (R-L) differential operator defined

as:

⎭

⎬

⎫

⎩

⎨

⎧

−

−Γ

∫

tfD

)(

)1(

)]([

. (

) (2)

Furthermore, Friedrich[1] proved that this kind of rheological constitutive equation shows

fluid-like behaviors only in the case that the derivative of stress is fractional and one of strain is first

order in time. Therefore, the following constitutive equation of GMF is used in this paper:

γµτκτ

αα

, (3)

where

dtd /

is the rate of shear strain.

The unsteady Couette flow problem has been considered in several works for a long time containing

various effects as in the book and paper given by Joseph[11], Bernardin[12] and Demirel[13]. In this

paper we use the constitutive equation (3) to study the unsteady Couette flow of GMF. By using the

Laplace transform, Weber transform and generalized Mittag-Leffler function, we get the exact solution

of the problem. It was shown that the result of classical Couette flow can be contained as a special

case of the result given by this paper and the decaying of unsteady part of GMF displays power law

behavior, which has its scale invariance.

剩余8页未读，继续阅读

评论收藏

内容反馈

weixin_38723559

粉丝: 1
资源: 961

Exact Solution on Unsteady Couette Flow of Generalized Maxwell F...

最新资源

Exact Solution on Unsteady Couette Flow of Generalized Maxwell F...

Unsteady shear flow of a viscoelastic fluid with the fractional Maxwell model in rectangular tube

2-2017整数规划模型On the exact solution of the no-wait flow shop problem with due date constraints.pdf

CONSERVATION LAWS, EXACT SOLUTIONS OF TIME-SPACE FRACTIONAL GENERALIZED GINZBURG-(1).pdf

Derivation of an exact solution on a principal component analysis for a set of root visual patterns

New exact solutions of a stratified fluid model

convex quadratic programming for exact solution of 0 1 quadratic programs

An Experimental Comparison of Min-Cut/Max-Flow Algorithms

The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs.pdf

论文研究-On stabilizability and exact observability of complex stochastic systems.pdf

A first course in turbulence

Exact Design of Digital Microfluidic Biochips 2019.pdf

Fast exact shortest-path distance queries on large networks

Handbook of Research on Soft Computing and Nature-Inspired Algorithms

Exact Solutions of a Class of Partial Differential Equations

eXact说明书eXact使用手册eXact操作说明下载

GENERALIZATION OF COMPUTER-ASSISTED PROSODY TRAINING

node-exact-online:Exact Online API 的 Node.js 包装器

Fast Exact Search in Hamming Space with Multi-Index Hashing

Advanced Engineering Mathematics

A review of content-based image retrieval in medical applications.pdf

EXACT系统解决方案

On robustness of exact controllability and exact observability under cross perturbations of the generator in Banach spaces

Excel中的EXACT函数.pdf

np难问题近似算法(绝版好书)

最新资源