Flow and heat transfer of viscoelastic fluid with a novel space distributed-order constitution relationship

Abstract

The space distributed-order constitution relationship is firstly formulated to study the flow and heat transfer in the boundary layer. The distributed-order boundary layer equations are derived and approximated by a multi-space fractional order ones, which are then solved by the implicit difference schemes. By comparing the analytical solutions of special boundary conditions, the validity of the present numerical method is examined. Then, the stability and convergence of the implicit finite difference scheme are analysed systematically. Finally, we present two practical examples to illustrate the effectiveness of our numerical method. Furthermore, the numerical technique in this study can be extended to others distributed-order boundary layer models.

Introduction

In recent decades, fractional operators were found to be quite flexible in describing the viscoelastic flow constitution and heat conduction law due to the long-term memory and inheritance properties of different materials [1], [2], [3], [4]. Friendrich [5] introduces firstly Riemann-Liouville fractional operators to establish the Maxwell double fractional model with relaxation parameters. Tan and Xu [6] investigated the non-Newton fluid with time fractional Maxwell model over a suddenly stretching sheet. Khan et al. [7] studied the MHD flow based on modified Darcy's law for the Oldroyd-B fluid. Hayat et al. [8] explored the MHD pipe flow of generalized Burgers' fluid through a permeable space by introducing the fractional calculus. Liu et al. [9] considered the space-time fractional advection diffusion model and discussed the stability and convergence of numerical solution. Liu et al. [10] developed the fractional Cattaneo-christov upper-convective derivative flux model. Zhao et al. [11] established firstly the fractional unsteady nature convection boundary layer equations and obtained the numerical solutions. Chen et al. [12] adopt the double fractional Maxwell model to demonstrate the viscous or elastic behavior of viscoelastic fluids under different fractional parameters.

For the constitution models discussed above, the fractional parameters were usually taken as a constant. However, many physical progresses cannot be accurately described by using classical fractional order models in some complex dynamical system, such as rheological properties of composite materials, decelerating sub-diffusion and decelerating super-diffusion. Recently, the distributed-order constitution models have been applied as a more effective tool than classical fractional order models [13], [14], [15], [16]. For the generalizing the stress-strain relation of inelastic media, Caputo [17], [18] presented the ordinary differential equations with distributed-order derivatives. Diethelm and Ford [19] proposed a numerical algorithm of distributed-order differential equations and the corresponding convergence theory is given. Hu et al. [20] investigated two-sided space fractional and a new time distributed-order diffusion equation. Morgado et al. [21] obtained the numerical solutions of time distributed-order diffusion equations by applying the Chebyshev collocation method. Ren and Chen [22] developed a numerical method for time distributed-order equation with initial singularity. Zhang et al. [23] proposed a Crank-Nicolson alternating direction implicit (ADI) Galerkin-Legendre spectral scheme for the two-dimensional Riesz spatial distributed-order convection diffusion equation, and verified its convergence and stability. Yin et al. [24] extended the convolutional quadrature (CQ) method to distributed-order fractional calculus. In addition, a new structure of ODE solutions with distributed-order is explored and proposed a new correction technique for this new structure to recover the optimal convergence rate. More analyses on the numerical method of distributed-order equations are given in the literature [25], [26], [27], [28].

To the best knowledge of the authors, the space distributed-order model has not been applied to study boundary layer problems. In fact, proposing the distributed-order constitution model is meaningful and worthy to analyses flow and heat transfer in the boundary layer. Inspired by the discussions above, based on the classical fractional order constitution [29], [30], [31], this paper propose the distributed-order law of Newton inner friction and Fourier's law of heat conduction as follows:

$$\tau =\tilde{\mu }\underset{0}{\overset{1}{\int }}\varphi (\alpha )\frac{{\partial }^{\alpha }u}{\partial y^{\alpha }}d\alpha ,\mathrm{q}=\tilde{k}\underset{0}{\overset{1}{\int }}\varpi (\beta )\frac{{\partial }^{\beta }T}{\partial y^{\beta }}d\beta$$

where $\tilde{\mu }$ is viscosity coefficient, $\tilde{k}$ is thermal conductivity, $\varphi (\alpha )$ and $\varpi (\beta )$ are non-negative weight function and satisfies the conditions [32], [33]: $0\le \varphi (\alpha ),\varpi (\beta )\ne 0$, $\alpha ,\beta \in (0,1]$ and $0<{\int }_0^1\varphi (\alpha )d\alpha <\infty ,0<{\int }_0^1\varpi (\beta )d\beta <\infty$. Different from the previous fractional constitutive model, the distributed-order model can describe more accurately flow and heat transfer process due to the continuity of the fractional parameters.

The plan of the manuscript is as follows: In section 2, we establish the boundary layer equations of the distributed-order. In section 3, the implicit difference schemes are designed for solving the boundary layer problems described by Eqs. (2)–(4). Meanwhile, the solvability and validation of the numerical scheme are examined. In section 4, we discuss the stability and convergence of the numerical scheme. In section 5, two practical examples are presented to illustrate the effectiveness of our numerical method. We conclude our paper in section 6.