Lie group analysis and similarity solution for fractional Blasius flow

https://doi.org/10.1016/j.cnsns.2016.01.010Get rights and content

Highlights

  • We present an investigation for Blasius flow of fractional viscoelastic fluids.

  • Fractional similarity equation is derived firstly by using Lie group transformation.

  • Numerical solution is obtained for specified parameters by generalized shooting method.

  • The effects of fractional derivative and Reynolds number on the velocity field is analyzed graphically.

Abstract

This paper presents an investigation for boundary layer flow of viscoelastic fluids past a flat plate. Fractional-order Blasius equation with spatial fractional Riemann–Liouville derivative is derived firstly by using Lie group transformation. The solution is obtained numerically by the generalized shooting method, employing the shifted Grünwald formula and classical fourth order Runge–Kutta method as the iterative scheme. The effects of the order of fractional derivative and the generalized Reynolds number on the velocity profiles are analyzed and discussed. Numerical results show that the smaller the value of the fractional order derivative leads to the faster velocity of viscoelastic fluids near the plate but not to hold near the outer flow. As the Reynolds number increases, the fluid is moving faster in the whole boundary layer consistently.

Introduction

Fluids encountered in natural science and industry usually manifest viscoelastic behavior, namely the materials exhibit both viscous and elastic properties. The viscoelastic characteristics of the liquids reported by recent measurements [1], [2] result in distinct differences between the numerical predictions based on the Navier–Stokes equations and experimental data. Consequently, a modified linear Maxwell constitutive equation was introduced and the ability was verified by investigating two common problems [2]. In the fluid-structure interaction problems, such as blood flow, fluids show non-Newtonian properties and viscoelastic behavior often occurs [3]. To numerically simulating these complex flow, the rheological model of Oldroyd-B was employed [3] and further Oldroyd-B, FENE-P, and Owens model are utilized respectively in [4]. These constitutive equations mentioned above, linear or nonlinear type, are all described with integer order derivatives.

The fractional calculus has become an increasingly attractive alternative for describing viscoelastic materials. The non-integer order derivatives to viscoelasticity problems had emerged initially as an empirical tool. The physical origin of this new approach may come from the observations that the stress relaxation phenomenon could be modeled by fractional powers of time [5]. Then a theoretical explanation of the suitability of time-fractional order derivative was established with molecular theories [5] and the effects of fractional derivative on the spectrum of relaxation modes of viscoelastic material were further investigated with Rouse model [6]. In the study of relaxation and creep properties of soft biological tissues, the experiments shown fractional order Voigt model performed better compared to the integer order models [7]. Mainardi’s latest book [8] gives detailed interpretation for the application of fractional calculus to model linear viscoelasticity behavior. Due to the complex properties demonstrated by viscoelastic fluids, indeed, it seems that certain fractional viscoelastic model has its merits in particular applications. Therefore, many integer order viscoelastic models had been generalized to its fractional order formula, replaced the integer order operator by fractional order. Second-grade fluid constitutive equation with time fractional derivative was applied to the Rayleigh–Stokes problem and the exact solutions of the velocity and temperature were obtained using the Fourier sine transform and the fractional Laplace transform [9]. Time fractional derivative Maxwell rheological equation and Oldroyd-B type were investigated for flows in oscillating rectangular duct or on an accelerating plate respectively [10], [11]. While, the exact analytical solutions depend a lot on the assumption of particular form of velocity field or stress field to simplify the flow governing equations.

More recently, spatial-fractional derivatives have been applied to the solid mechanics for describing non-local constitutive relations [12], which suit problems of small length scales or long-range cohesive interactions between material particles [13]. Although time-fractional derivatives have draw many researcher attention as mentioned above, spatial-fractional viscoelastic rheological models are still an almost uncharted territory [14]. Hence, incorporating the characteristics of laminar boundary layer flow, this paper presents a spatial-fractional viscoelastic constitutive model and shows its application in the classical Blasius flow model.

Lie group analysis originally advocated by the Sophus Lie has proven to be an efficient approach for the algorithmic determination of exact solutions to partial differential equations [15]. In fluid mechanics, most obtained exact solutions are similarity solutions that the number of independent variables in the governing equations is reduced by one or more [16], [17]. With the increasingly applications of fractional differential equations(FDE), principal procedures of Lie group analysis were extended to FDE for finding the exact solution of the equation[18], [19], [20] in recent years. The results of time fractional KdV-type equation have been obtained by Hu et al. [21]. Chen and Jiang [22] have applied the methods to simplify two classes of fractional partial differential equations successfully. Considering the complicated procedure of numerical computation to fractional nonlinear partial differential equations, it is good enough to solve the symmetry reduced equations by analytical or numerical methods. In particular, differential equations with two independent variables can be transformed to ordinary differential equations by using Lie group analysis.

The paper is organized as follows. In Section 2, the flow governing equations with spatial-fractional derivative are proposed. The detailed symmetry reduction procedures for the governing equations are presented in Section 3. In Section 4, the obtained boundary value fractional-order ordinary equation is further transformed into initial value problem. Then the set of equations are solved by a shooting method with the shifted Grünwald formula and fourth order Runge–Kutta method. Some numerical results and discussions are given in Section 5, followed by conclusions in Section 6.

Section snippets

Mathematical formulation of the model

We consider the steady two-dimensional flow of incompressible viscoelastic fluids past a thick, semi-infinite flat plate. The external stream that is spatially uniform with speed u is taken to be parallel to the plate, which leads to the disappearance of pressure gradient. Such flow scene is fundamental to the boundary layer theory, in which the viscous effects are assumed to be play a significant role in the thin layer near the object surface. The two-dimensional steady Prandtl boundary layer

Lie symmetry analysis and similarity solution

We give detailed derivation procedure of the Lie symmetry analysis next for the boundary value problem (16) and (17). The essence of the Lie symmetry method is to obtain point transformations which keeps Eq. (16) invariant. These transformations can be achieved by solving the determining equations which are linear, homogeneous partial differential equations. According to the Lie theory, let us firstly define a group of one parameter ϵ continuous transforms, x*=x+ϵζ(x,y,ψ)+O(ϵ2),y*=y+ϵη(x,y,ψ)+O(

Numerical approximation for the fractional Blasius equation

To solve the fractional Blasius equation, the boundary value problem (48) is transformed into an initial value problem Dη2+αf(η)+Re˜1+αf(η)f(η)=0,f(0)=0,f(0)=0,f(0)=γ.where the value of γ can be obtained on condition that we have an approximation for f()=1. we introduce the linear transform y1=f,y2=f,y3=f,Then, the (48) can be reduced to a set of differential equations y1=y2,y2=y3,y3α=Re˜1+αy1y3.The boundary conditions are transform into y1(0)=0,y2(0)=0,y3(0)=γ.We note that the

Results and discussion

We first consider the following initial value problem: Dηαy(η)=2η2αΓ(3α)y(η)+η2,y(0)=0,where 0 < α < 1. The analytical solution is known and is given by y(η)=η2. By the Grünwald formula, the numerical solutions are obtained and the comparison with the exact solution are presented at different positions. From the results listed in Table 1, we observed that the Grünwald formula provides a good approximation solution to the exact solution.

An exact analytic solution for boundary value problem

Conclusion

A spatial fractional derivative constitutive equation is proposed for describing the long-range interactions of viscoelastic fluid flowing in boundary layer. By the extended Lie symmetry methods, the resulting two dimensional nonlinear governing equation with the fractional derivative is reduced to a fractional ordinary equation. The shifted Grünwald formula jointed with classical fourth order Runge–Kutta method is used in shooting method to solve the system of transformed equations. The

Acknowledgments

The authors would like to thank A. A. Kasatkin (USAYU, Russia) for kindly suggestions and providing the innovative article [28] and the translated version. This work is supported by the National Natural Science Foundations of China (nos. 51276014 and 51476191).

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