Solving generalized fractional problem on a funnel-shaped domain depicting viscoelastic fluid in porous medium

Abstract

In this paper, a series of generalized-formed equations describing the fluid flow, combined with the heat and mass transfer of viscoelastic fluid under the impacts of Dufour and Soret effects are presented. The constitutive relation of viscoelastic fluid, modified Fick’s model and Fourier’s law are all considered as fractional types. Assume that the ultrafiltration process happens in a funnel, and this particular convex domain renders the problem real and more difficult. The highlight of this work is to deal with a series of complex and coupled controlling equations on a funnel-shaped domain. A finite volume method with unstructured grid mesh is adopted, and presented in details. The verified numerical results show that the algorithm is effective, and the numerical method has a good accuracy even with a coarse mesh.

Introduction

Filtration of viscoelastic fluid is widely observed in industrial applications, such as in food processing and so on, and the permeability behavior of viscoelastic fluid has roused the attentions of many scholars. Sajid et al. [1] studied hybrid convection of viscoelastic fluid fully developed between two parallel vertical permeable plates. Naganthran et al. [2] investigated the stability of viscoelastic fluid flowing around the stagnation point on a permeable shrinkage plane. Aziz et al. [3] investigated the unsteady flow of a third-order fluid permeating a panel in a porous medium. For viscoelastic fluids which possess non-local and non-instantaneous dynamic characteristics, the fractional derivative can better characterize the fluid behaviors. However, due to the definition of fractional model, it brings great resource consumption in calculation than in the cases with integer-order models. As widely used numerical methods, finite element method, spectral element method, and meshless method have their own advantages and can solve problems from different engineering applications as well as some analytical methods [4], [5]. In this paper, we provide effective numerical schemes to solve the generalized fractional differential equation depicting the viscoelastic fluid in the porous medium. By using the finite volume method with unstructured meshes [6], difficulties in the calculation caused by the trapezoidal domain, the mutually coupled heat and mass transfer equations, and convection terms in the momentum equation are resolved tactically. The present method can be extended to similar problems on any convex domain and a good accuracy can be obtained even with a coarse mesh.

Assume an incompressible viscoelastic fluid flows through a conical filter containing porous media with an initial velocity and in the gravitational field (Fig. 1). Consider the cross section of this filter, i.e., a two-dimensional problem. Buoyancy and infiltration due to porous media are taken into account. Assume the $x$-axis lies along the center line of the cone, and the $y$-axis is perpendicular to it. For viscoelastic fluid, where the shear stress at a certain point is affected by the velocity gradient both nearby and far-away, the fractional derivative model is used to characterize the fluid behavior better compared with the integer derivative model [7], [8]:

$${\tau }_{xy}={\tilde{\mu }}_{\alpha }\frac{{\partial }^{\alpha }u}{\partial y^{\alpha }},$$

where ${\tilde{\mu }}_{\alpha }$ and $\alpha \left(0<\alpha <1\right)$ are the generalized dynamic viscosity and space-fractional derivative, respectively. The Riemann–Liouville fractional derivative is used [9]:

$$\frac{{\partial }^{\alpha }u}{\partial y^{\alpha }}=\frac{1}{\Gamma (1-\alpha )}\frac{\partial }{\partial y}{\int }_0^y{(y-\xi )}^{-\alpha }u(x,\xi ,t)d\xi ,$$

where $\Gamma \left(\cdot \right)$ denotes the Gamma function.

Furthermore, with the impact of thermal inhomogeneity combined with mass inhomogeneity, the density gradient generated in the saturated media leads to the buoyancy drive while the flow, heat and mass transfer are mutually coupled [10]. Thus, the modified Fourier’s law [11] and a similar fractional Fick’s model [12] are adopted as:

$$J_q=-{\tilde{\lambda }}_{\gamma }\frac{{\partial }^{\gamma }T}{\partial y^{\gamma }}-\frac{\rho {\tilde{D}}_{\gamma }{K}_T}{C_s}\frac{{\partial }^{\gamma }C}{\partial y^{\gamma }},J_m=-\rho {\tilde{D}}_{\gamma }\frac{{\partial }^{\gamma }C}{\partial y^{\gamma }}-\frac{\rho {\tilde{D}}_{\gamma }{K}_T}{T_m}\frac{{\partial }^{\gamma }T}{\partial y^{\gamma }},$$

where ${\tilde{\lambda }}_{\gamma }$ and $C_s$ are the generalized thermal conductivity and the concentration susceptibility. $K_T$ represents the thermal diffusion ratio, while $T_m$ and ${\tilde{D}}_{\gamma }$ are the mean temperature and the generalized diffusion coefficient, respectively. Fractional derivatives of the Riemann–Liouville type are also used in Eq.(3). The value of $\gamma$ ranges from 0 to 1.

By using the fractional constitutive relations of Eqs. (1), (3), the governing equations can be obtained. To further non-dimensionalize the governing equations, we adopt the following dimensionless variables: $x^{\ast }=\frac{x}{L}$, $y^{\ast }=\frac{y}{L}$, $u^{\ast }=\frac{u}{u_{\infty }}$, $v^{\ast }=\frac{v}{v_{\infty }}$, $T^{\ast }=\frac{T-T\infty }{T_w-T_{\infty }}$, $C^{\ast }=\frac{C-C\infty }{C_w-C_{\infty }}$, $t^{\ast }=\frac{t{u}_{\infty }}{L}$, where characteristic length is denoted as $L$. Thus, the dimensionless governing equations become (the ‘*’ in the dimensionless variables is omitted in the followings):

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{1}{Re}\cdot \frac{\partial }{\partial y}\left(\frac{{\partial }^{\alpha }u}{\partial y^{\alpha }}\right)+Ar\cdot T+ArNr\cdot C-\frac{\varepsilon }{Re\cdot Da}u+f(x,y,t),$$

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{1}{Pr}\frac{\partial }{\partial y}\left(\frac{{\partial }^{\gamma }T}{\partial y^{\gamma }}\right)+D_f\frac{\partial }{\partial y}\left(\frac{{\partial }^{\gamma }C}{\partial y^{\gamma }}\right)+g(x,y,t),$$

$$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=\frac{1}{Re}\frac{1}{Sc}\frac{\partial }{\partial y}\left(\frac{{\partial }^{\gamma }C}{\partial y^{\gamma }}\right)+Sr\frac{\partial }{\partial y}\left(\frac{{\partial }^{\gamma }T}{\partial y^{\gamma }}\right)+h(x,y,t),$$

where, $Re=\frac{\rho L^{\alpha }{u}_{\infty }}{{\tilde{\mu }}_{\alpha }}$, $Sc=\frac{{\tilde{\mu }}_{\alpha }}{\rho {\tilde{D}}_{\gamma }}$, $D_f=\frac{{\tilde{D}}_{\gamma }{K}_T(C_w-C_{\infty })}{C_s{C}_p{L}^{\gamma }{u}_{\infty }(T_w-T_{\infty })}$, $Sr=\frac{{\tilde{D}}_{\gamma }{K}_T(T_w-T_{\infty })}{T_m{L}^{\gamma }{u}_{\infty }(C_w-C_{\infty })}$, $Ar=\frac{g{\beta }_{T}L(T_w-T_{\infty })}{{u_{\infty }}^2}$, $Da=\frac{k_0}{L^{1+\alpha }}$, $Nr=\frac{{\beta }_C(C_w-C_{\infty })}{{\beta }_T(T_w-T_{\infty })}$, $Pr=\frac{{\tilde{\lambda }}_{\gamma }}{\rho C_p{L}^{\gamma }{u}_{\infty }}$. $Re$, $Sc$, $D_f$, $Sr$ and $Pr$ are generalized Reynolds number, Schmidt number, Dufour number, Soret number, and Prandtl number, respectively. $Ar$, $Nr$ and $Da$ represent the Archimedes number, the buoyancy ratio number, and the Darcy number. It should be noted that Eqs. (4)–(7) are generalized forms: $f(x,y,t)$, $g(x,y,t)$ and $h(x,y,t)$ are all source terms. Since the pressure can be controlled artificially in actual working conditions [7], a pressure term may be incorporated into $f(x,y,t)$.