Stability results of a fractional model for unsteady-state fluid flow problem
- N Thamareerat^{1, 3},
- A Luadsong^{1, 3}Email author and
- N Aschariyaphotha^{2}
https://doi.org/10.1186/s13662-017-1116-3
© The Author(s) 2017
Received: 26 September 2016
Accepted: 13 February 2017
Published: 9 March 2017
Abstract
This paper mainly focuses on a fractional model for unsteady-state fluid flow problem developed based on the meshless local Petrov-Galerkin (MLPG) method with the moving kriging (MK) technique as a background. The contribution of this work is to investigate the stability of a model with fractional order governed by the full Navier-Stokes equations in Cartesian coordinate system both theoretical and numerical aspects. This is examined and discussed in detail by means of matrix method. We show that the scheme is unconditionally stable under the restriction of eigenvalue. The dependence between several of the important parameters that impact on the solution is also studied thoroughly. In discretizing the time domain, an algorithm based on a fixed point method is employed to overcome the nonlinearity. Two selected benchmark problems are provided to validate the stability of the present method, and a very satisfactory agreement with the obtained results can be found.
Keywords
time-fractional Navier-Stokes equations meshless method fixed point iteration stability analysis matrix method unconditionally stable1 Introduction
Recently, increasing interests and considerable researches have been given to fractional differential equations (FDEs) thanks to its applications in wide areas of applied science and engineering. The formulations based on FDEs are more adequate than the previously used classical integer-order models. It is usually recommended to employ the fractional models for describing diverse physical phenomena such as fluid mechanics, plasma physics, electrochemistry, mathematical biology, probability and statistics, finance, electrical networks, rheology, optics, and signal processing to maintain not only the behavior of the original systems but also all of its historical states. It is a well-known fact that there is no generally applicable method to seek the exact solution of most FDEs. The procedures such as linearization or discretization are inevitable. For this reason, it is particularly important to propose and develop a new computationally efficient method for obtaining the numerical solution of FDEs. In order to accommodate the reader who is not acquainted with the concept of fractional calculus, it is perhaps important to recall that fractional calculus is a discipline concerning the possibility of taking non-integer or fractional powers to the derivatives and integrals. As is well known, there are many different types of fractional derivatives. A most frequently and widely used one was proposed by Caputo [1], which will be explained here briefly. The initial conditions for the differential equations with fractional order by Caputo’s definition take on the same form as for the ones with integer order, whose physically meaningful interpretation is very clear. Moreover, an exceptionally good benefit of the fractional derivative interpreted in Caputo’s viewpoint is that, from the physical perspective, some characteristic properties of classical derivative that the derivative of any constant function is zero should be preserved. Of course not all aspects of current or past interest in fractional calculus can be covered in this section, but the aim is at least to provide up to date information in as much detail as possible as well as to generate curiosity and encourage further investigation of the potential applications of this branch of mathematics. To give the reader insight and understanding as regards fractional calculus in more depth, we refer to the books of Oldham and Spanier [2], Miller and Ross [3], Podlubny [4], Kilbas et al. [5] and the references cited therein.
The Navier-Stokes equations (NSE) are a set of coupled nonlinear second-order partial differential equations that are conventionally regarded as the appropriate mathematical formulation to describe the numerical simulation being relevant to the unsteady, compressible, and viscous fluid flows. Generally, incompressible flows can be modeled using the NSE in two different formulations which may be based on either primitive (velocity and pressure) or derived (such as velocity and vorticity) variables. The velocity-vorticity approach, generally known as the stream function-vorticity approach, requires the transformation of NSE into equations of velocity and vorticity components and does not include the pressure term. Notwithstanding the advantage that the number of equations to be solved in velocity-vorticity form is two and three for two- and three-dimensional problems, respectively, which is fewer than those in the primitive variables approach, they have lost some of their attractiveness and have received very little attention. Imposition of boundary conditions of the transformed equations is a major drawback that has to be tackled, especially for three-dimensional problems. It may require substantial effort related to the boundary treatment to circumvent this problem, whereas the use of method based on primitive variables is quite common and definitely more straightforward. As a consequence, the incompressible NSE are most frequently solved by the formulation of primitive variables which will be considered here. In 2004 El-Shahed and Salem [6] have proposed the generalized NSE by simply replacing the first-order time derivative term by a derivative of fractional order but still retaining the first- and second-order space derivatives. Afterward, the published research articles with regard to solving a viscous fluid problem in a tube in cylindrical coordinates have emerged continuously. The approximated solutions for the problem as above have been given among others by Momani and Odibat [7] using the Adomian decomposition method (ADM), Ragab et al. [8] using the homotopy analysis method (HAM), Kumar et al. [9] by the ADM and Laplace transform method (LTM), Kumar et al. [10] with the new homotopy perturbation transform method (HPTM), and Wang and Liu [11] using the modified reduced differential transform method (DTM) and new iterative Elzaki transform method. Besides those mentioned above, the new development in solving nonlinear FDEs has been proposed by Kumar and his colleagues. The HAM together with the Laplace transform was applied to solve the nonlinear shock wave equation of fractional order arising in the flow of gases [12]. The implementation of the homotopy analysis Sumudu transform method (HASTM), an inventive coupling of Sumudu transform and well-known homotopy analysis technique, to derive the analytical and numerical solutions of a nonlinear fractional differential-difference problem was shown in [13]. They also pointed out that the most important advantage of the HASTM over the ADM and HPTM is without making use of Adomian’s and He’s polynomials. Other pioneering work can be found in [14, 15].
Meanwhile, numerical methods turn out to be an alternative way of attaining an approximate solution of FDEs. Solving FDEs by the classical mesh-dependent numerical methods such as the finite difference methods (FDM), the finite volume methods (FVM), and the finite element methods (FEM) requires the mesh generation, which appears to be computationally costly. The use of meshes restricted by their applications leads to many difficulties in some specific problems. Attempts to get rid of the aforementioned problem have been devoted to developing the so-called meshless (or meshfree) methods. They have some advantages when compared to the grid based methods by virtue of the flexibility and simplicity of placing nodes at arbitrary locations. The academic work regarding the remarkable progress on meshless methods has thus far received considerable interest and has been published continually both theoretically and numerically (see [16–25]). One of the extensively popularized meshless methods in solving initial and boundary value problems is the meshless local Petrov-Galerkin (MLPG) method originating with Atluri and Zhu [26]. This is one of the truly meshless methods just because the requirement of background cells for integration is not needed.
In meshless procedure, construction of shape function is one of the main challenges that must be taken into account. Originally, the MLPG approach is numerically implemented using the moving least squares (MLS) approximation to the spatially discretized domain. Even though the MLS approximation seems to be one of the most commonly used methods, it is not always advantageous. The only notable imperfection for MLS shape functions is that they do not have the Kronecker delta property, so the techniques like the Lagrange multiplier or penalty method are required to enforce essential (Dirichlet) boundary conditions. On the other hand, the so-called method of kriging is one of the most immensely used techniques in geostatistics for spatial interpolation. This subsequently became another way to enhance the accuracy and efficiency for the meshfree methods. The moving kriging (MK) interpolation was first proposed by Gu [27] and successfully demonstrated the effectiveness in solving steady-state heat conduction problems. In addition to the consistency property, the MK shape functions have the delta property which allows essential boundary conditions to easily be imposed in a similar way to the FEM. The kriging interpolation is shown to be essentially the same as the radial point interpolation method (RPIM) on condition that the same basis functions are used [28].
As mentioned in the second paragraph, virtually all the accomplishments to acquire the solution of fractional NSE in the literature can only be limited to the one dimension. Only the simplest cases are solved analytically so that an exact solution can be obtained. The theoretical study for solving such a problem in higher space dimensions is connected with great difficulties. Evidently, seeking the analytical solution of the time-fractional NSE in multiple dimensions has not been easy by reason of the nonlinearity which makes them very complicated or almost not possible to achieve, not to mention containing unknown function (velocity components and pressure) of several independent variables (space and time) more than one and its partial derivatives with respect to those variables. Up to now, only little attention has been paid to the development of the MLPG method so as to solve the fractional model of fluid flow in Cartesian coordinates. Additionally, as far as the authors are concerned, stability of meshfree methods for a fractional model is rather hard to investigate and needs many efforts, especially for a governing equation considered with nonlinear term. To fill this gap, the main contribution of this paper is to investigate and analyze the stability of a fractional model for unsteady-state flow problem developed based on the meshless methods.
The organization of the remainder of this article is as follows. In Section 2, mathematical preliminaries may be helpful for whoever is about to begin with FDE, so we give a short description and basic concepts of fractional calculus. Also we provide a discrete approximation for the fractional derivative based on a quadrature rule. In Section 3, we first introduce the governing nonlinear time-fractional NSE in two dimensions and then explain how to solve the nonlinear system. We show the stability of the meshfree method, and the sensitivity of several important parameters is also studied and discussed. In Section 4, two numerical tests are examined and discussed to show the validation of stability for the present method. Finally, we end the current work with concluding remarks in Section 5.
2 Mathematical preliminaries
This section can be useful for the reader who is unfamiliar with the fractional derivative. We briefly give some important concepts and definitions of fractional calculus.
Definition 1
A real function \(f ( x )\) with \(x>0\) is said to be in the space \(C_{\mu},\mu\in \mathbf{R}\) if there exists a real number \(p>\mu\) such that \(f ( x ) = x^{p} f_{1} ( x )\), where \(f_{1} ( x ) \in C(0,\infty)\) and for \(m\in \mathbf{N}\) it is said to be in \(C_{\mu}^{m}\) if \(f^{(m)} \in C_{\mu}\).
Definition 2
Definition 3
3 Numerical procedure
We omit the derivation details for the standard local weak formulation, spatial discretization, and temporal discretization here to save space and refer to Thamareerat et al. [25].
3.1 Solving the nonlinear system
3.2 Stability analysis
4 Illustrative examples
Example 1
The maximum and RMSEs of the velocity and pressure for Example 1 with different choices of Δ t on regular points
Δ t | \(\boldsymbol {E_{\infty}}\) | RMSE | ||||
---|---|---|---|---|---|---|
u | v | p | u | v | p | |
1/10 | 2.4645 × 10^{−4} | 2.4645 × 10^{−4} | 6.3263 × 10^{−4} | 9.9623 × 10^{−5} | 9.9623 × 10^{−5} | 4.2763 × 10^{−4} |
1/12 | 2.0197 × 10^{−4} | 2.0197 × 10^{−4} | 6.0606 × 10^{−4} | 8.1507 × 10^{−5} | 8.1507 × 10^{−5} | 4.0973 × 10^{−4} |
1/15 | 1.5990 × 10^{−4} | 1.5990 × 10^{−4} | 5.7928 × 10^{−4} | 6.3486 × 10^{−5} | 6.3486 × 10^{−5} | 3.9171 × 10^{−4} |
1/17 | 1.4065 × 10^{−4} | 1.4065 × 10^{−4} | 5.6661 × 10^{−4} | 5.5126 × 10^{−5} | 5.5126 × 10^{−5} | 3.8318 × 10^{−4} |
1/20 | 1.1888 × 10^{−4} | 1.1888 × 10^{−4} | 5.5232 × 10^{−4} | 4.5932 × 10^{−5} | 4.5932 × 10^{−5} | 3.7355 × 10^{−4} |
The maximum and RMSEs of the velocity and pressure for Example 1 with different choices of Δ t on uneven points
Δ t | \(\boldsymbol {E_{\infty}}\) | RMSE | ||||
---|---|---|---|---|---|---|
u | v | p | u | v | p | |
1/10 | 3.0612 × 10^{−4} | 2.8645 × 10^{−4} | 1.8238 × 10^{−5} | 1.1477 × 10^{−4} | 1.0439 × 10^{−4} | 5.0684 × 10^{−6} |
1/12 | 2.6126 × 10^{−4} | 2.4279 × 10^{−4} | 1.6974 × 10^{−5} | 9.6862 × 10^{−5} | 8.6452 × 10^{−5} | 4.4722 × 10^{−6} |
1/15 | 2.1589 × 10^{−4} | 1.9864 × 10^{−4} | 1.5693 × 10^{−5} | 7.9024 × 10^{−5} | 6.8521 × 10^{−5} | 4.0400 × 10^{−6} |
1/17 | 1.9435 × 10^{−4} | 1.7768 × 10^{−4} | 1.5083 × 10^{−5} | 7.0714 × 10^{−5} | 6.0142 × 10^{−5} | 3.9109 × 10^{−6} |
1/20 | 1.6998 × 10^{−4} | 1.5397 × 10^{−4} | 1.4392 × 10^{−5} | 6.1502 × 10^{−5} | 5.0831 × 10^{−5} | 3.8321 × 10^{−6} |
Example 2
The RMSEs of the velocity and pressure with regular nodes and fixed \(\pmb{\alpha=0.99}\) , \(\pmb{N=121}\) , \(\pmb{\Delta t=0.1}\) , \(\pmb{{Re}=100}\) at some time levels t for Example 2
t | u | v | p |
---|---|---|---|
0.1 | 1.7176 × 10^{−5} | 2.5966 × 10^{−5} | 4.9654 × 10^{−3} |
0.3 | 2.2558 × 10^{−5} | 5.2199 × 10^{−5} | 6.1420 × 10^{−3} |
0.5 | 2.7272 × 10^{−5} | 7.8739 × 10^{−4} | 6.4973 × 10^{−3} |
0.7 | 3.0502 × 10^{−5} | 1.0357 × 10^{−4} | 6.0118 × 10^{−3} |
1 | 3.3760 × 10^{−5} | 1.3715 × 10^{−4} | 4.8912 × 10^{−3} |
1.2 | 3.5518 × 10^{−5} | 1.5722 × 10^{−4} | 4.2616 × 10^{−3} |
1.5 | 3.7882 × 10^{−5} | 1.8404 × 10^{−4} | 3.6934 × 10^{−3} |
1.7 | 3.9256 × 10^{−5} | 1.9976 × 10^{−4} | 3.5532 × 10^{−3} |
2 | 4.0961 × 10^{−5} | 2.2016 × 10^{−4} | 3.5617 × 10^{−3} |
The RMSEs of the velocity and pressure with different values of correlation parameter ω for Example 2
ω | u | v | p |
---|---|---|---|
0.2 | 3.3760 × 10^{−5} | 1.3715 × 10^{−4} | 4.8912 × 10^{−3} |
0.5 | 8.2165 × 10^{−5} | 2.9563 × 10^{−4} | 4.2259 × 10^{−3} |
1 | 1.1378 × 10^{−4} | 3.0463 × 10^{−4} | 2.4659 × 10^{−3} |
1.2 | 1.1291 × 10^{−4} | 2.8097 × 10^{−4} | 2.3875 × 10^{−3} |
1.5 | 1.2407 × 10^{−4} | 2.4424 × 10^{−4} | 1.0728 × 10^{−3} |
5 Concluding remarks
In the numerical experiment, an important point that is worth mentioning here is that due to the fact that the exact solution of the system of equations (3.1)-(3.3) is unavailable, the obtained results have to be compared with analytical solution of classical NSE, which are congruous with what we expected when \(\alpha\rightarrow1\). The most important feature of fractional models is the convergence of the approximation to the classical model. The solution for the integer-order system must be recovered. The reliability of the solutions with α not approaching 1 is guaranteed by the theoretical study shown in Section 3.2. We prove the unconditional stability using a technique based on eigenvalue of the matrix, and the effect of many important parameters on the solution is also investigated thoroughly. The present study is motivated by lack of detailed experimental study and stability analysis in the literature related to a fractional model of full NSE in Cartesian system of coordinate. As we can see, the applications of FDE are manifold and important, but for the multidimensional unsteady-state flow problem it is in its beginning stage and needs further work. In the end, the authors definitely believe that the insights and source of information drawn in this paper with emphasis on the theoretical analysis and numerical study will be of use to the readers interested in fluid flow problem and to the science and engineering community.
Declarations
Acknowledgements
The financial support of this research by Science Achievement Scholarship of Thailand (SAST) is sincerely appreciated. The authors gratefully acknowledge the encouragement from the Department of Mathematics, Faculty of science, King Mongkut’s University of Technology Thonburi (KMUTT). We also wish to thank the anonymous reviewers for their many constructive comments and invaluable suggestions, which have led to an improved version of the manuscript.
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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