- Research
- Open Access
Finite difference-finite element approach for solving fractional Oldroyd-B equation
- Amer Rasheed^{1}Email author,
- Abdul Wahab^{2},
- Shaista Qaim Shah^{3} and
- Rab Nawaz^{4}
https://doi.org/10.1186/s13662-016-0961-9
© Rasheed et al. 2016
- Received: 19 February 2016
- Accepted: 2 September 2016
- Published: 13 September 2016
Abstract
In this article, we study an unsteady flow of an anomalous Oldroyd-B fluid confined between two infinite parallel plates subject to no-slip condition at boundary. The flow is induced by a linear acceleration of the lower plate in its own plane. A standard Galerkin finite element method is adopted to construct an approximate solution blended with a finite difference approximation for Caputo fractional time derivatives. The convergence of the proposed numerical scheme is substantiated, and error estimates are provided in appropriate norms. Some adequate numerical simulations are performed in order to elucidate the dominance of characteristic flow parameters of velocity field in the prescribed configuration.
Keywords
- anomalous fluid
- Oldroyd-B fluid
- finite element method
- stability
- convergence
MSC
- 35A15
- 76A05
- 76A10
- 76D99
- 76M10
1 Introduction
The curiosity to understand mass transfer phenomena in fluids obeying non-Newtonian rheological paradigms is increasing due to broad range of engineering applications and apposite industrial processes. The examples include material plasticizing and solidification processes for manufacturing parts, oil-well drilling, and fossil fuel combustion; see, for instance, [1–3], the review article [4], and references therein. Spirited researchers endeavoring in assorted domains have been experimentally testing, mathematically modeling, establishing numerical approximations, and designing algorithms for analyzing various flow problems in different geometric and flow configurations [3–15]. Mathematicians are particularly exposed to challenging mathematical riddles, for instance, related to solvability, consistency, stability, and thermodynamic compatibility of constitutive flow models, their solutions, and approximations [1, 4, 11, 12, 16–23].
Several approximate and self-consistent non-Newtonian rheological models are proposed over the past decades as no single one can encompass assorted features of all the fluids. These models are classified into differential, rate, and integral types. The interested readers are referred, for instance, to [20, 24, 25] for detailed accounts. In particular, the stress relaxation in polymer processing is usually predicted using rate-type fluid models such as Maxwell, Oldroyd-B, or Burgers fluids [7, 9, 11, 24, 26–29].
In certain non-Newtonian fluids, an anomalous rheological model provides a more realistic fit to the experimental data [13, 30, 31]. For instance, the anomalous Maxwell model yields algebraically decaying stress relaxation modulus resulting in a good agreement to experimental data [31], whereas the Brownian Maxwell fluid fails to do so, at least over complete range of frequencies. Moreover, as indicated by Bagley and Torvik [32, 33], the molecular theory is harmonic to anomalous viscoelastic models. The anomalous nature of the flow is usually modeled with fractional-order time derivatives replacing those of integer order in classical stress-strain relations. The issues concerning well-posedness and thermodynamic stability of the anomalous viscoelastic flow models have been addressed, for instance, in [23, 34, 35].
The constitutive initial-boundary value problems for non-Newtonian fluids rarely have exact and closed-form analytic solutions since these models are strongly nonlinear, whereas sufficient boundary conditions are not often available. Thus, numerical and asymptotic techniques are sought exploiting supplementary information on the flow profile. Unfortunately, the asymptotic solutions are mostly divergent for strongly nonlinear problems and large values of pertinent flow parameters such as Péclet, Reynolds, and Weissenberg numbers [36, 37]. Therefore, great interest in numerical approximation techniques in non-Newtonian fluid mechanics is observed in recent years; see, for instance, [14, 15, 19, 38–40], among many others.
The hot topics in numerical analysis include challenging issues related to instability of approximate solutions due to strong nonlinearity, convection dominance, and parabolic-hyperbolic nature of non-Newtonian flow problems for increasing values of flow parameters. A variety of stabilization and numerical approximation frameworks are consequently introduced and analyzed. More recently, frameworks for approximating solutions to time and/or space fractional differential equations in connection with subdiffusion and superdiffusion, viscoelastic wave propagation, and anomalous flow problems are discussed; see [40–43] and references therein. The so-called L1-finite difference approximation method is invoked together with space approximation schemes to obtain numerical solutions to time-discretized models, such as spatial finite difference, spectral, lumped mass, and Galerkin finite element techniques.
In this article, we provide a numerical exposition of flow phenomena for an incompressible anomalous Oldroyd-B fluid using a standard Galerkin finite element method (FEM) blended with finite difference approximation in time. We consider the fluid confined between two infinite parallel plates, which starts flowing due to a linear acceleration of the lower plate in its own plane, whereas the upper plate is kept rigid and no slip condition at boundaries is imposed. The anomalous behavior of the Oldroyd-B fluid is modeled with left-sided Caputo fractional time derivatives thereby generalizing the canonical Brownian Oldroyd-B fluid model that can be perceived as a limiting case. The objective of the investigation is twofold: (1) understanding the velocity profile in the aforementioned flow and (2) deploying standard Lagrange-Galerkin FEM together with the L1-finite difference scheme and subsequently performing a convergence analysis following the pioneer works in [41–43]. Albeit, the assumptions of a linear plate acceleration and negligible pressure gradient are made for brevity, and the analysis contained herein can be analogously performed otherwise. The results can be extended to the Burgers fluids and will be discussed in a forthcoming investigation.
The rest of this contribution is arranged in the following manner. In Section 2, the flow problem is mathematically formulated. The equations governing the flow are detailed (see Section 2.1), and the associated initial boundary value problem (IBVP) is derived and nondimensionalized (Section 2.2). The finite element approximation to the velocity field is presented in Section 3. First, a few notions and notation are collected (Section 3.1), and a finite-difference-based temporal discretization scheme is presented for the fractional time derivatives (Section 3.2). Then, the spatial discretization of the IBVP is derived using Lagrange interpolation functions (Section 3.3). The convergence analysis of the numerical scheme is performed in Section 4, and the numerical simulations are presented in Section 5. Finally, the findings of the investigation are summarized in Section 6.
2 Formulation of flow problem
We fix the following notation henceforth.
Definition 2.1
Remark 2.2
The fractional derivative \(\partial^{\gamma}_{t} \phi\) converges to the canonical integer-order derivative \({\partial_{t}^{n}\phi}\) as the parameter \(\gamma\in \mathbb {R}\to n\in\mathbb{N}\), where \(n-1<\gamma<n\) (see, e.g., [45], p.92).
The following proposition from [45], Proposition 2.16, will be useful in the sequel.
Proposition 2.3
2.1 Flow configuration and governing equations for an ordinary fluid
Remark 2.4
The thermodynamic stability, necessary condition for well-posedness in the sense of Hadamard, and causality constraints restrict the values of relaxation and retardation times \(\lambda_{1}\) and \(\lambda_{2}\) to be such that \(0<\lambda_{2}<\lambda_{1}\) (refer, e.g., to [20, 46] for further details).
2.2 Flow problem
In this section, the constitutive equations corresponding to a fractional Oldroyd-B fluid are derived together with relevant initial and boundary conditions. Toward this end, we first briefly derive constitutive equations for the flow of a canonical Oldroyd-B fluid and then highlight appropriate changes in order to incorporate anomalous behavior of fluid rheology.
Remark 2.5
As a consequence of Remark 2.2, the anomalous Oldroyd-B model (16) reduces to the canonical Oldroyd-B model as \(\alpha ,\beta\to1\). Moreover, (16) refers to a fractional Maxwell fluid as \(\lambda_{2}\to0\).
3 Numerical approximation scheme
3.1 Functional spaces and norms
3.2 Finite difference approximation
3.3 Galerkin finite element approximation
Consider the following weak formulation of the flow problem (19).
Weak Form
In the sequel, we derive a space and time discrete weak formulation of the problem (19) using (28). Let \(\varphi _{h}\) be an approximate solution to (28) in \(\mathcal {C}^{1} ([0,T]; V^{h}_{0} )\), that is, the solution to following problem.
Semidiscrete weak form
4 Analysis of numerical scheme
This section is dedicated to the stability and error analysis of the numerical scheme established in Section 3. In the sequel, C represents a generic constant independent of τ and h but dependent on φ, α, β, \(\lambda_{1}\), \(\lambda_{2}\), u, and T and may differ from step to step.
4.1 Truncation error
Lemma 4.1
As an immediate consequence of Lemma 4.1, the following result is evident.
Lemma 4.2
(Truncation error)
4.2 Stability of discrete problem
Theorem 4.3
Proof
- 1.Initial step: (\(j=2\)). Taking \(\chi_{h}=\varphi _{h}^{2}\) and \(k=1\) in (32) yieldsBy initial conditions and the definition of operator \(\mathcal {M}_{1}\) we obtain$$ \bigl\Vert \varphi _{h}^{2}\bigr\Vert _{1,*}^{2} \leq \mathcal {M}_{1}\bigl[\varphi _{h}^{1},\varphi _{h}^{2} \bigr]+C\Vert g\Vert _{0}\bigl\Vert \varphi _{h}^{2} \bigr\Vert _{0}. $$(34)Consequently, inequality (34), together with the inequality \(\|\varphi _{h}^{2}\|_{0}\leq\|\varphi _{h}^{2}\|_{1}\), yields$$ \mathcal {M}_{1}\bigl[\varphi _{h}^{1},\varphi _{h}^{2} \bigr] = (1+2 \tau C_{\alpha}) \bigl(\varphi _{h}^{1}, \varphi _{h}^{2}\bigr)+\tau C_{\beta}\bigl\langle \varphi _{h}^{1},\varphi _{h}^{2}\bigr\rangle +\tau C_{\alpha}\bigl(\varphi _{h}^{0},\varphi _{h}^{2} \bigr)= 0. $$$$ \bigl\Vert \varphi _{h}^{2}\bigr\Vert _{1,*}\leq C \Vert g\Vert _{0}. $$
- 2.
Supposition step: (\(j < m\)). Assume that estimate (33) holds for \(2< j< m\), that is, there exists a constant C independent on τ and h such that \(\|\varphi _{h}^{j}\|_{1,*}\leq C \|g\|_{0}\).
- 3.Induction step: (\(j=m\)). It is evident from the definition of \(b_{k}^{\gamma}\) thatTaking \(\chi_{h}=\varphi _{h}^{m}\) and \(k=m-1\) in (32) yields$$ 1=b_{0}^{\gamma}> b_{1}^{\gamma}> \cdots> b_{k}^{\gamma}\to0\quad \text{as } k\to +\infty. $$(35)Again, by the definition of the operator \(\mathcal {M}_{m-1}\) we have$$ \bigl\Vert \varphi _{h}^{m}\bigr\Vert _{1,*}^{2} \leq \mathcal {M}_{m-1}\bigl[\varphi _{h}^{m-1}, \varphi _{h}^{m}\bigr] +C\Vert g\Vert _{0}\bigl\Vert \varphi _{h}^{m}\bigr\Vert _{0}. $$(36)To get the last inequality, we have used the assumption step and the fact that$$\begin{aligned}& \mathcal {M}_{m-1} \bigl[\varphi _{h}^{m-1},\varphi _{h}^{m} \bigr] \\& \quad = (1+2 \tau C_{\alpha}) \bigl(\varphi _{h}^{m-1}, \varphi _{h}^{m}\bigr)+\tau C_{\beta}\bigl\langle \varphi _{h}^{m-1},\varphi _{h}^{m}\bigr\rangle + \tau C_{\alpha}\bigl(\varphi _{h}^{m-2},\varphi _{h}^{m} \bigr) \\& \qquad {}+\tau C_{\alpha}\bigl(\zeta^{\alpha}_{m-1}[ \varphi _{h}]-\zeta_{m-2}^{\alpha}[\varphi _{h}], \varphi _{h}^{m}\bigr)+\tau C_{\beta}\bigl\langle \zeta^{\beta}_{m-1}[\varphi _{h}],\varphi _{h}^{m} \bigr\rangle \\& \quad \leq (1+2\tau C_{\alpha})\bigl\Vert \varphi _{h}^{m-1} \bigr\Vert _{0} \bigl\Vert \varphi _{h}^{m}\bigr\Vert _{0} + \tau C_{\beta}\biggl\Vert \frac{\partial \varphi _{h}^{m-1}}{\partial y} \biggr\Vert _{0} \biggl\Vert \frac{\partial \varphi _{h}^{m}}{\partial y} \biggr\Vert _{0} \\& \qquad {}+ \tau C_{\alpha}\bigl\Vert \varphi _{h}^{m-2} \bigr\Vert _{0} \bigl\Vert \varphi _{h}^{m}\bigr\Vert _{0} + \tau C_{\beta}\biggl\Vert \frac{\partial\zeta_{m-1}^{\beta}[\varphi _{h}]}{\partial y} \biggr\Vert _{0} \biggl\Vert \frac{\partial \varphi _{h}^{m}}{\partial y} \biggr\Vert _{0} \\& \qquad {}+ \tau C_{\alpha}\bigl\Vert \zeta^{\alpha}_{m-1}[ \varphi _{h}]\bigr\Vert _{0} \bigl\Vert \varphi _{h}^{m} \bigr\Vert _{0} + \tau C_{\alpha}\bigl\Vert \zeta^{\alpha}_{m-2}[\varphi _{h}]\bigr\Vert _{0} \bigl\Vert \varphi _{h}^{m}\bigr\Vert _{0} \\& \quad \leq C\max \biggl(1+2\lambda_{1}^{\alpha}\frac{T^{-\alpha}}{\Gamma (2-\alpha)}, \lambda_{2}^{\beta}\frac{T^{1-\beta}}{\Gamma(2-\beta )} \biggr) \Vert g\Vert _{0} \bigl\Vert \varphi _{h}^{m} \bigr\Vert _{1}. \end{aligned}$$for all \(p< m\), which holds again by the assumption step and relation (35). This completes the proof together with (36).$$\begin{aligned} \bigl\Vert \zeta^{\gamma}_{p}[\varphi ]\bigr\Vert _{1,*} &= \Biggl\Vert \sum_{s=1}^{p} b_{s}^{\gamma}\bigl(\varphi _{h}^{p+1-s}-\varphi _{h}^{p-s} \bigr)\Biggr\Vert _{1,*} \\ &\leq \sum_{s=1}^{p} \bigl(\bigl\Vert \varphi _{h}^{p+1-s}\bigr\Vert _{1,*}+\bigl\Vert \varphi _{h}^{p-s}\bigr\Vert _{1,*} \bigr) \leq C\Vert g\Vert _{0} \end{aligned}$$
4.3 Convergence of numerical scheme
Lemma 4.4
Proof
Theorem 4.5
(Convergence)
Proof
The convergence estimate (39) can be proved by arguments analogous to those in the proof of [41], Theorem 3.2, and that of [43], Theorem 2.1. The key ingredients of the proof are further presented for completeness.
5 Numerical results and discussion
In this section, we present some numerical results showing the validity of the approximation scheme and discuss the approximate velocity profile.
5.1 Validation of numerical scheme
Convergence rates in \(\pmb{L^{2}}\) and \(\pmb{H^{1}}\) norms
Number of elements | \(\boldsymbol{\|u - u_{\mathrm{ex}} \|_{L^{2}(\Omega)}}\) | \(\boldsymbol{\| u - u_{\mathrm{ex}} \|_{H^{1}(\Omega)}}\) |
---|---|---|
60 | 5.0077 × 10^{−3} | 9.5930 × 10^{−2} |
70 | 4.7797 × 10^{−3} | 8.2534 × 10^{−2} |
80 | 4.6342 × 10^{−3} | 7.2526 × 10^{−2} |
90 | 4.5357 × 10^{−3} | 6.4777 × 10^{−2} |
100 | 4.4659 × 10^{−3} | 5.8608 × 10^{−2} |
110 | 4.4146 × 10^{−3} | 5.3588 × 10^{−2} |
120 | 4.3758 × 10^{−3} | 4.9430 × 10^{−2} |
130 | 4.3458 × 10^{−3} | 4.5934 × 10^{−2} |
140 | 4.3220 × 10^{−3} | 4.2958 × 10^{−2} |
150 | 4.3029 × 10^{−3} | 4.0398 × 10^{−2} |
5.2 Characteristic behavior of velocity profile
6 Concluding remarks
In this article, we presented a Galerkin finite element method blended with a finite difference scheme for time fractional derivative to approximate flow velocity in an anomalous Oldroyd-B fluid confined between two infinite horizontal plates. The flow is induced by variable acceleration of the lower plate. No slip condition at the boundary is imposed. Convergence analysis of the numerical scheme is performed, and error bounds are presented. Numerical results are discussed, and the influence of pertinent flow parameters on the velocity field is delineated. The results presented in this investigation generalize those for Brownian Oldroyd-B fluids and anomalous Maxwell fluids in analogous flow configurations. In the present study, the pressure gradient is considered to be negligible for simplicity. The results contained in this paper can be extended for Burgers fluids and will be discussed in a forthcoming investigation.
Declarations
Acknowledgements
This research was supported by the Korea Research Fellowship Program funded by the Ministry of Science, ICT and Future Planning through the National Research Foundation of Korea (NRF-2015H1D3A106240).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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