Abstract
The continuous dependence of the two fluids which interface with each other in a bounded domain is derived. It is first shown how to establish the a priori results for the solutions for the flow in a bounded domain occupied by a viscous fluid in contact with a porous solid. It is also proved that the solution depends continuously on the Forchheimer coefficient.
1. Introduction
In this paper, we study two fluids in one bounded domain when they interface with each other. We want to know what effect they can give to each other. Let an appropriate part of the plane denotes the boundary between a porous medium which occupies a bounded region and a nonlinear viscous fluid which occupies a bounded region . The interface is denoted by . The remaining parts of the boundaries of and are denoted, respectively, by and . We also denote and . We also note that is above the plane and is below the plane .
Let and denote the velocity, temperature, and pressure in and , respectively. The Forchheimer system consists of the partial differential equations (see [1]) where is the gravity force function. The coefficient is a positive constant which is named as the Forchheimer coefficient. The viscosity variation in (1) is accounted for by the term , i.e., we are considering a viscosity like .
The Darcy equations are (see Nield and Bejan [2]) where and are the bounded, simply connected, and star-shaped domains and is a given number satisfying . We impose the following boundary conditions: for prescribed functions and , and denotes the unit outward normal of . We also let be the unit outward normal of . Obviously, on . The initial conditions are written as for prescribed functions , , and . On the interface , the conditions are
The purpose of this paper is to study the continuous dependence on the coefficients of problems (1)–(5). This type of stability is often called structural stability to distinguish it from continuous dependence on the initial data, on the boundary data, or even on the partial differential equation themselves. In continuum mechanics problems, it is necessary to be able to establish continuous dependence on the model; this is discussed in terms of differential equations by Hirsch and Smale [3]. Such stability estimates are fundamental in that one wishes to know if a small change in a coefficient in an equation or boundary data, or a small change in the equations themselves, will lead to a drastic change in the solution. When we study the continuous dependence or convergence, structural stability expresses the changes in the model itself rather than the original data. Many references to work of this nature are discussed in the monograph of Ames and Straughan [4] and the monograph of Straughan [5].
In the area of porous media, there have been many studies of continuous dependence, or structural stability, including those of Scott and Straughan [6, 7], Franchi and Straughan [8], Hoang and Ibragimov [9], Lin and Payne [10, 11], Liu [12, 13], Liu et al. [14–16], Scott [6], Scott and Straughan [7], Li [17], Cicho et al. [18], Ma and Liu [19], Ciarletta et al. [20], and Payne et al. [18–21]. These results about structural stability mainly focus on one fluid in the domain. But in reality, there are usually two or more fluids in the domain. They can interface with each other. So the study of such type of problems may be interesting and meaningful. It is worth drawing attention to this type of study. Several quantitative results in physical problems were obtained in [22–25].
In this paper, we derive an a priori convergence result which compares the solution to the Forchheimer system of partial differential equations with that of the Darcy equations. The purpose of this paper is to study the continuous dependence of a solution to the Forchheimer system to a solution to Darcy equations on the Forchheimer coefficient and the effective viscosity coefficient . Different from [14, 26], there are two nonlinear terms and and there is no Laplace term in (1). So, some Sobolev inequalities do not hold for our problem. This will bring great difficulty. To get our result, we must seek a new method to overcome the difficulty. In the next section, we derive a number of a priori bounds which will be used in establishing the continuous dependence result in Section 3.
2. A Priori Bounds
In this section, we want to drive bounds for the various norms of and .
2.1. Bounds for and
Multiplying with and integrating over , we obtain
Integrating by parts and using Young’s inequality and the arithmetic-geometric mean inequality now lead to where By using the divergence theorem and Equation (2), we get
So, we have
2.2. Bounds for and
Payne et al. [27] have obtained the following result: with and is the maximum temperature on the interface . Similarly, in the , with . However, in the area , the maximum temperature cannot be reached on the interface . Therefore, we observe that where . To derive the bounds for and , we introduce another two functions and which for each satisfy
Then, from the identities we have
So, (20) leads to that
It follows by Lemma 1 that where
Since we have
In computing the bounds for and , we introduce another two functions and which for each satisfy
Then, from the identities it follows from (26) that
From (28), we have where
Upon integration of (29), we have
Inserting (31) into (28), we get
So
The terms as well as may be bounded by boundary data by using a Rellich identity, cf. [6, 7]. Combining (13), (22), (24), and (33), we may have where
After integrating (34), we have
Inserting (36) back into (34), we have the following lemma.
Lemma 1. Let and be the solutions of (1)-(5) with . Then, where
Combining (13) and (34), we can obtain the following lemma.
Lemma 2. Let and be the solutions of (1)-(5) with . Then,
3. Continuous Dependence on the Forchheimer Coefficient
In this section, we want to derive an a priori estimate showing how and depend continuously on the Forchheimer coefficient . Let and be solutions of (1)-(5) with , and and be solutions of (1)-(5) with , respectively. We define
Then, satisfy the following equations and satisfy equations
The boundary conditions are
The initial conditions can be written as
The interface conditions are
We observe for later convenience that may be rearranged as
Our main result is the following theorem.
Theorem 3. Let and be the the classical solutions to the initial-boundary value problem (1)–(5) corresponding to , and and also be the the classical solutions to the initial-boundary value problem (1)–(5) but corresponding to . Then, for any , we have as . The differences of velocities satisfy Furthermore, there is a positive function , given specifically in (58), such that where have been defined in (40) and (41).
Proof. We begin with the identity From (51), it follows that
Using the divergence theorem, we have
We use the Cauchy-Schwarz inequality to have
Using Hölder’s inequality and the A-G mean inequality, we have
We find that the result given in Appendix B of Liu [13] for or for arbitrary positive constant . We use this inequality for with and (2.30), (2.32) in (55) to get for arbitrary We also have
Combining (53), (54), (55), (58), and (59), we obtain
We follow a similar procedure starting from (47) to obtain
We add (60) and (61) to find where
To bound and , we multiply and by and , respectively, and integrate by parts to find or
Combining (62) and (4.26), we find that or where
Thus, after integration, we may derive from (67) the estimate:
Using (65) and (66) and choosing in (69), we have
Inequalities (70) and (71) are a priori bounds demonstrating continuous dependence of the solution on the Forchheimer coefficient .
Data Availability
All data generated or analyzed during this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This work is supported by the Key Projects of Universities in Guangdong Province (Natural Science) (2019KZDXM042).