Abstract
In this paper, we will study the existence of strong solutions for a nonlinear system of partial differential equations arising in convective flow, modeling a phenomenon of mixed convection created by a heated and diving plate in a porous medium saturated with a fluid. The main tools are Schäfer’s fixed-point theorem, the Fredholm alternative, and some theorems on second-order elliptic operators.
1. Introduction
In recent years, many authors have studied the case of the semi-infinite vertical plane plate immersed in a porous medium saturated with a fluid. The following problem is derived from this phenomenon:with mixed boundary conditionswhere is the rectangular Cartesian coordinates system. The constants and depend on the density, the viscosity, and the thermal expansion coefficient of the fluid. They also depend on the permeability and the thermal diffusivity of the saturated porous medium.
In the framework of boundary layer approximations, by introducing similarity variables, we can transform the system of partial differential equations into a system of ordinary differential equations of the third order with appropriate boundary values (see [1], p.3). These two-point boundary value problems can be studied by using a shooting method (see for example [2, 3]). For the auxiliary dynamical system, we refer the reader to [4]. For an integral equation, we refer the reader to [5]. For nonstandard analysis techniques, we refer the reader to [6]. For numerical method, we refer the reader to [7]. The second natural way of dealing with this problem, which is the framework of this paper, is straightly related to the coupled partial differential equations (see [1, 8, 9]).
The aim of this paper is to generalize the existence of strong solutions of the problem introduced in [1, 8, 9]. More precisely, we will give new results about the existence of strong solutions, in particular, without assuming that is small (as in [8], Theorem 4). We choose the data (the function ) in the larger space inspired by other works, and we will give conditions which make each solution in .
Let us introduce now the problem in which we are interested. Let be a bounded domain of whose boundary is sufficiently smooth and divided into two parts such that and
On , we consider the boundary value problem defined bywith boundary conditionswhere is a strictly positive constant, is the unit outward normal vector on , and .
2. Preliminaries
In this section, we will introduce the notations, definitions, and preliminary facts which are used throughout the paper.
2.1. Sobolev Spaces and Notations
For a positive integer and for , the Sobolev space is defined as follows:where is the -derivative of in the sense of distributions. For , we denote by the space . The boundary space is defined as, equipped with the norm
Throughout the paper, we use the following notations:(i), , and is Poincaré’s constant of (ii)(iii), and (iv), is the norm of (or the norm of )(v) is the norm of , i.e., the space of linear operators from into (normed spaces)
2.2. Variational Formulation
Let us assume
Then, from the existence and uniqueness theorem for the Dirichlet problem for strong solutions (see [10], Theorem 5.13), there exists a unique function satisfying
Furthermore, and . For the coefficient , it is supposed thatwhere
Let us notice that is dense in (see [11], Proposition 3.57). Let us set . Then, is the solution of problem (5)–(7) if and only if is the solution of the problemwith boundary conditions
Let us assume that is a solution of problem (15)–(18). Multiplying equation (15) by a function and equation (16) by a function , integrating on , and using Green’s formula, we yieldwhere
3. Main Results
We start with the following lemma which plays a key role in our main results.
Lemma 1. Let . The function is the unique solution of problemin the space .
Proof. It is clear that is the solution of problem (21). Then, if satisfies , multiplying the members of this equation by , we getBy integration on , we getOn the other hand, we havebecause , in the sense of distributions. Since , , and hence, by the density of in (when is at least a Lipschitz open set), we havethat is, . Thus, .
Remark 1. From the proof of the previous lemma, we observe that for all and , we haveThus, by the density of in and by the embedding , we can easily show that for all and , we have
Now, we will prove the following proposition which plays an important role in proving our main results.
Proposition 1. Let us assume that is of class and . Then, for all , the equationhas exactly one solution in the space .
Proof. Let us considerdefined bywhere the operator is defined byThen, it is clear that for all , we haveFrom the existence and uniqueness theorems for the Dirichlet problem for strong solutions (see [12], Theorem 9.15, Lemma 9.17, problem 9.8) combined with the open mapping theorem (see [13], Corollary 2.7), the operator is invertible. On the other hand, we can write wheredefined bywith as the operator from into defined by and , where , anddefined byThen, is compact. Indeed, is continuous because are continuous, and the operator is compact (Rellich–Kondrachov theorem); it is clear that is continuous. Now, it is easy to show that for all , (32) holds, if and only ifholds. By Lemma (21), is the unique solution of equationin the space . Hence, from the Fredholm alternative, equation (28) has exactly one solution . Besidesas needed.
Lemma 2. Under the assumptions of the previous proposition, the mapping from into is defined by where is continuous. Moreover, there exists such that for all , if ,
Proof. Let and let be the unique solution of problem (40). Let be a sequence of elements in converging to in . Then, there exists a sequence of elements in satisfyingThen, for all is the solution of the problemBy the weak maximum principle theorem (see [12], Theorem 8.1), for all , we haveand by variational formulation (system (19)), we haveReplacing by and using Remark 1, we giveOn the other hand, for , we haveSimilarly, . We replace the previous inequality in inequality (46), and we use Remark 1, to getThus,withThen, there exists a subsequence of the sequence converging almost everywhere to . That is,From inequality (49), we get, for all ,where depends only on , and , as needed.
Remarks 2. (1)The main advantage of Proposition 1 (particularly, Fredholm’s alternative) is the generalization of the results ([8], Theorem 1 and Theorem 4, and [1], Theorem 4), and more precisely, it leads us to show that there exist strong solutions to problem (15)–(18) without assuming that is bounded by certain constants and without assuming that is quite small.(2)The previous lemma is similar to Lemma 3.1 in [8], but in this paper, we have seen a new technique for its proof (it is proved by a weak maximum principle).
The following proposition is useful.
Proposition 2. Let us assume, in addition to assumptions (11)–(13), that is of class , and . Then, problem (15)–(18) have at least one solution in the space .
Proof. For all , we have . Then, from the existence and uniqueness theorem for the Dirichlet problem for strong solutions ([12], Theorem 9.15), the problemhas exactly one solution . Besides, there exists such that(see [12], Lemma 9.17). Let us denote by the application from into defined by , and let us consider the application defined bywith given in Lemma 2 and . Then, we havethat is, the set of the solutions of problem (15)–(18) isThen, is continuous and compact. Indeed, are continuous because for , we haveand the continuity of has been proved in Lemma 2. By the Rellich–Kondrachov theorem, we can easily show that is compact. Now, let be in the setThen, there exists such thatBy the variational formulation, the previous problem is equivalent toFrom the first equation in system (61), we havewith . Hence,Now, from the second equation in system (61), since (see Remark 1) and (the same method as the one used to get inequality (47)), we haveFrom inequalities (63) and (64), we deduce that , i.e., is bounded. From Schäfer’s fixed-point theorem (see [14], Theorem 11.1, p.59), we deduce that has at least one fixed point . That is, is the strong solution of problem (15)–(18). Furthermore, . Thus, , i.e., . Hence, , as needed.
Remark 3. From the previous proof, it is observed that under the assumptionsthe previous result (the previous proposition) is still valid, which is the generalization of ([8], Theorem 1 and Theorem 4). In addition, Proposition 1 helps us to generalize the previous result in the case where .
Now, we are going to generalize the previous result in the case where . To this end, we start with the following remark.
Remark 4. Let be a solution of the problemThen, from the first equation with boundary conditions, for all , we haveNow, from the second equation in system (66) with boundary conditions, for all , we haveOn the other hand, we havethe same method as the one used to get inequality (47). Then, we haveThereby, if is a solution of problem (66), then for all , we have
Now, we will prove the following result.
Theorem 1. Let us assume, in addition to assumptions (11)–(13), that is of class and . Then, problem (15)–(18) have at least one solution in the space .
Proof. Let be a sequence converging to in . Then, from Proposition 2, there exists which is bounded in and satisfyingBy the compactness of the embedding , there exists a subsequence of the sequence converging weakly in and strongly in , to an element denoted , and we have . Now, we shall complete the proof in three steps.
Step 1. converges in to . Indeed, from Remark 4, for all , we haveThen, for , we haveBy Lemma 2 and the fact that is convergent in , we get that there are strictly positive constants , and , such thatThat is, converges strongly to . On the other hand, for all , we haveBy the same process as the one used to obtain inequality (52), we havethat is, converges strongly to .
Step 2. satisfies systems (71) and (72). Indeed, for all , we haveFor all , we haveTaking the limit as in (71) when givesNow, for all , we havebecauseBy the dominated convergence theorem, we haveObviously,Taking the limit in (77) as yields
Step 3. . IndeedSince , we have . Then, from the existence and uniqueness theorems for the Dirichlet problem for strong solutions and since the applicationis continuous and coercive, we deduce that . Thus, in which (see [11], Theorem 2.72), that is, . Then, from Proposition 1, there exists a unique satisfyingOn the other hand, from Remark 1, the bilinear formis continuous and coercive. Thus, from the Lax–Milgram theorem, we deduce that . Hence, , that is, , and thereby, from the existence and uniqueness theorems for the Dirichlet problem for strong solutions ([12], Theorem 9.15, problem 9.8) and from the Lax–Milgram theorem, is the unique function in obeyingBy the same method as the one used in the previous proposition, we getThis completes the proof.
Remark 5. Notice that in Step 3 of the previous proof, Proposition 1 is necessary. Also, we need Lemma 1 and Remark 1, particularly, for and which is a different result from ([8], Lemma 2.1) and it is proved by the generalized Green formula.
In addition to the previous results, we will study the existence of solutions in .
Theorem 2. Let us assume, in addition to the hypotheses of the previous theorem, that is of class , , and with . Then, problem (15)–(18) have at least one solution in the space .
Proof. Let us assume , and let be a solution of problem (15)–(18). Then, we haveOn the other hand, from (12) and by the regularity theorem of elliptic problem ([12], Theorem 8.13), is in . It then follows thatSince , we haveFrom the regularity theorem of elliptic problem ([12], Theorem 8.13), we deduce that is in . Hence,that is, . Continuing the above process, we obtain, after steps, .
Let us denote by the trace operator on . Then, we have the following result.
Corollary 1. Let us assume, in addition to the hypotheses of the previous theorem, that is of class and and there exists such that . Then, problem (15)–(18) have at least one solution in the space .
Remark 6. If are closed and open subsets and is sufficiently smooth (being able to apply Poincaré’s inequality), then by applying Theorem 2.24 given in ([15], p. 132) or Theorem 5.8 given in ([16], p. 146), we can prove that the previous results of the existence of strong solutions are still valid in the general case where .
4. Conclusion
In this paper, we study a problem given by two strongly coupled partial differential equations in a two-dimensional bounded domain, modeling a phenomenon of convective flow created by a heated and diving plate in a porous medium saturated with a fluid. We have established the existence of strong solutions without assuming that is bounded by a certain constant (nor is small enough) and by choosing in . Besides, we have added a new result (which does not exist in the literature) which is the existence of solutions of class (particularly, solutions of class ).
Appendix
Numerical Example
Let us consider a semi-infinite vertical permeable or impermeable flat plate embedded in a fluid-saturated porous medium at the ambient temperature . The rectangular Cartesian co-ordinates system is applied with the origin fixed at the leading edge of the vertical plate. The -axis is directed upward along the plate, and the -axis normal to it. It is assumed that the porous medium is homogeneous and isotropic where all the properties of the fluid and the porous medium are constants except the density. Also, the flow is incompressible and follows the Darcy–Boussinesq law. Based on these assumptions, the following governing equations are introduced:with boundary conditionswith and . The parameter is the mass transfer coefficient where refers to the impermeable wall. Also, the permeable surface is represented by for the fluid suction and for the fluid injection.
Now, a brief discussion for the similarity solutions is given. Assuming that the convection takes place in a thin layer around the plate (we can neglect and ) and hence the boundary layer approximation is obtained:with same boundary conditions (A.3)-(A.4).
For the case of prescribed heat, we introduce the new dimensionless similarity variableswhereis the local Rayleigh number; equations (A.1) and (A.2) with boundary conditions (A.3) and (A.4) lead to the third-order ordinary differential equationon subjected towhere
In the case of prescribed heat flux, we introduce the new dimensionless similarity variables:where the Rayleigh number is
Equations (A.5) and (A.6) with boundary conditions (A.3) and (A.4) lead to the third-order ordinary differential equationon subjected to
In order to show the effectiveness of main results, numerical results are presented in this part and discussed. Here, the transformed equations governing the case of prescribed constant heat and the prescribed heat flux at are solved, numerically, using the fourth-order Runge–Kutta method based on the shooting technique. The obtained data are presented in Figures 1 and 2 for , as well as values of and are included in Tables 1 and 2. It is seen that a strong solution is existed and the behaviors of for all values of are asymptotic. In addition, the growth in the power index has negative impacts on the values of , and due to the decrease in the temperature distributions at the surface and thickness of the thermal boundary layers.
Data Availability
No external data were used to support the study in this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.