Strong Solutions for the Fluid-Particle Interaction Model with Non-Newtonian Potential

Abstract

This paper deals with a mathematical fluid-particle interaction model used to describing the evolution of particles dispersed in a viscous compressible non-Newtonian fluid. It is proved that the initial boundary value problems with vacuum admits a unique local strong solution in the dimensional case. The strong nonlinearity of the system brings us difficulties due to the fact that the viscosity term and non-Newtonian gravitational potential term are fully nonlinear.

1. Introduction

Fluid-particle interaction model arises in many practical applications in science and engineering [1–4] and is of primarily importance in the sedimentation analysis of disperse suspensions of particles in fluids. We focus on the fluid-particle interaction model that describes the evolution of particles dispersed in a viscous fluid.

Carrillo and Goudon first derived a fluid-particle interaction system by formal asymptotics from a mesoscopic description (see [5]). In the microscopic description, the cloud of the particles is related to its distribution function , which is the solution to a dimensionless Vlasov-Fokker-Planck equation. On the other hand, the fluid is described by its density and its velocity field . We assume that the fluid is compressible and isentropic, then solves the compressible Euler equations for the inviscid case or the Navier-Stokes equations for the viscous case, respectively.

In [5], for the inviscid case, the coupling between the kinetic and the fluid equations is obtained through the friction forces that the fluid and the particles exert mutually. The friction force is assumed to follow the Stokes law and thus is proportional to the relative velocity of the fluid and the particles given by . Furthermore, both phases are affected by external forces, which are supposed to derive from a time independent potential . The system was given as follows:where are some related dimensionless parameters, are the pressure, the mass density of particles and fluid, respectively. Setting , , , with , then (1) becomesFinally, letting , then (3) converges to systemFor the viscous case, with the dynamic viscosity terms taken into consideration, then will be have an additional term of . In [6], the viscous stress tensor is assumed to satisfy Newton’s Law for viscosity which requires that where and are constant viscosity coefficients satisfying Then and thus the system turns into the following equationMoreover, if the influence of gravitational potential was taken into consideration, there will be a Poisson equation coupled to the above system, as for the Navier-Stokes-Poisson equation in [7].

Carrillo et al. obtained the global existence and asymptotic behavior of the weak solutions and stability properties to (8). Subsequently, Fang et al. [8] studied the existence of global classical solutions in dimension one. In [9, 10], Balew and Trivisa obtained the existence of global weak solutions and weakly dissipative solutions by entropy method in dimension three. The two-phase flow hydrodynamic models have been proposed in [3]. For some mathematical results on the Navier-Stokes coupled equations such as the nematic liquid crystal flows models where viscous effects are included, for more details, we refer to [11–13] and references therein.

On the other hand, as we know, the viscous stress tensor is depends on the rate of strain , where If the stress and rate of strain satisfy the following linear relation then the fluid is called Newtonian. The coefficient of proportionality is called the viscosity coefficient, and it is a characteristic material quantity for the fluid concerned, which in general depends on density, temperature, and pressure. The governing equations of motions of them will be the Navier-Stokes equations. If the relation is not linear, the fluid is called non-Newtonian. Examples of non-Newtonian fluids are molten plastics, polymer solutions, dyes, varnishes, suspensions, adhesives, paints, greases, paper pulp, and biological fluids like blood. The simplest model of the stress-strain relation for such fluids given by the power laws, which states that for (see [14]). Ladyzhenskaya (see [15]) proposed a special form for on the incompressible model: These models are called For , if then it is a pseudo-plastic fluid, and if then it is a dilatant fluid (see [14]). In the view of physics, the model captures the shear thinning fluid for the case of , and captures the shear thickening fluid for the case of .

Followed by the Ladyzhenskaya model, in this paper, we investigate the compressible non-Newtonian fluid-particle interaction model in one-dimensional case, then system (8) changes to bewith the initial and boundary conditionsand the no-flux condition for the density of particleswhere is a bounded interval, . denote the fluid density, velocity, and the density of particle in the mixture, respectively. is the pressure where , stands for the non-Newtonian gravitational potential, and the given function denotes the external potential. is the viscosity coefficient and , , , are constants.

As to the non-Newtonian fluids, there has been much research both theoretically and experimentally, see ([15–23]). Indeed, we have investigated the existence results of solutions for and in the absence of term for (14) in [22, 23]. However, the influence of non-Newtonian gravitational potential for a practical model was not taken into consideration there. To our knowledge, there seems very few mathematical results for the case of the fluid-particle interaction model systems with non-Newtonian gravitational potential, even in dimension one. The existence results to problem (14)-(16) when which describes the motion of the compressible viscous isentropic gas flow is driven by a non-Newtonian gravitational force is still open up to now. We are interested in the existence and uniqueness of strong solutions on a one-dimensional bounded domain. In fact, the strong nonlinearity of (14) brings us new difficulties in getting the upper bound of and the method used in [8] does not suitable for us. Motivated by Cho etal’s [24, 25] work on Navier-Stokes equations, we establish local existence and uniqueness of strong solutions by the iteration techniques.

Throughout the paper we assume that . In the following sections, we will use simplified notations for standard Sobolev spaces and Bochner spaces, such as , , .

1.1. Main Results

Theorem 1. Let be a positive constant and , and assume that the initial data satisfy the following conditions:andfor some , where is a constant. Then there exist a time and a unique strong solution to (14)-(16) such that

2. A Priori Estimates for Smooth Solutions

In this section, we will prove the local existence of strong solutions. Provided that is a smooth solution of (14)-(16) and , where is a positive number. We denote by and introduce an auxiliary function Then we estimate each term of in terms of some integrals of and apply arguments of Gronwall-type and thus prove that is locally bounded.

2.1. Estimate for

First we need to do the following estimates. Using , we rewritten asBy virtue of Then Taking the above inequality by norm, we getWe deal with the term of .

Multiplying by and integrating over , we obtain then we haveHence, we deduce thatFrom , taking it by norm, we getMultiplying by , integrating over , we have Integrating by parts, using Sobolev inequality, we deduce thatDifferentiating with respect to , and multiplying it by , integrating over , using Sobolev inequality, we haveFrom (30) and (31), by Gronwall’s inequality, it follows thatBesides, we can also get the following estimates. By using ,From , we have thenDifferentiating with respect to time , multiplying it by , integrating over to , and using Young’s inequality, we have thus, we getwhere is a positive constant, depending only on .

2.2. Estimate for

Multiplying (21) by , integrating (by parts) over , we haveWe deal with each term as follows: From we getSubstituting the above into (38), and using Young’s inequality, we obtainUsing , we haveCombining (42)-(43), yieldswhere is a positive constant, depending only on .

2.3. Estimate for and

Multiplying by , integrating the resulting equation over , and using the boundary conditions (16), Young’s inequality, we haveMultiplying by , integrating (by parts) over , and using the boundary conditions (16), Young’s inequality, we haveDifferentiating with respect to , multiplying the resulting equation by , and integrating (by parts) over , we getCombining (45)-(47), we get

2.4. Estimate for

Differentiating (21) with respect to , multiplying the result equation by , and integrating it over with respect to , we haveNote that Combining (40), (49) can be rewritten intoBy using Sobolev inequality, Hölder inequality and Young’s inequality, (35), (37), we estimate each term of as follows: Substituting into (51), and integrating over on the time variable, we haveTo obtain the estimate of , we need to estimate . Multiplying (21) by and integrating over , we get According to the smoothness of , we obtain Therefore, taking a limit on in (53), as , we conclude thatwhere is a positive constant, depending only on .

Combining the estimates of (27), (28), (32), (33), (44), (48), (56) and the definition of , we conclude thatwhere are positive constants, depending only on . This means that there exist a time and a constant , such that

3. Proof of the Main Theorem

In this section, our proof will be based on the usual iteration argument and some ideas developed in [24, 25]. We construct the approximate solutions, by using the iterative scheme, inductively, as follows: first define and assuming that was defined for , let be the unique smooth solution to the following problems:with the initial and boundary conditionswhere We directly construct approximate solutions of the problem (59)–(63). More precisely, we first find from (59) and (63) with smooth function , i.e., andIt follows from the classical linear hyperbolic theory that there is a unique solution on this above initial problem. Using the method of characteristics, we haveBy (68) and (69), we have Using (67), then which meansNext, combining the classical stableness results of the elliptic equation, the existence of can be obtained by (61) and (63), then by (62) and (63) we get . The last, with being given, by virtue of (72), from (60) and (63), according to the classical theorem of quasi-linear parabolic equation (see [17], Chapter VI, Theorem 5.2), there exists a unique smooth solution . With the process, the nonlinear coupled system has been deduced into a sequence of decoupled problems and each problem admits a smooth solution. And the following estimates hold:where is a generic constant depending only on , but independent of .

Next, we have to prove that the approximate solution converges to a solution to the original problem (14) in a strong sense. To this end, let us define then we can verify that the functions satisfy the system of equationsMultiplying (75) by , integrating over , and using Young’s inequality, we obtainwhere is a positive constant, depending on and for all and .

Multiplying (76) by , integrating over , and using Young’s inequality, we obtainLet and then We estimate the second term of (80) as follows:Similarly, we have and then we have Substituting (83) into (80) and using Young’s inequality, we havewhere , for all and . Using (73) we derive Multiplying (78) by , integrating over , and using (73) and Young’s inequality, we haveCollecting (79), (86), and (88), we obtainwhere depends only on and , for all and . Using (73), we have Integrating (89) over with respect to t, using Gronwall’s inequality, we have from the above recursive relation, choose and such that , using Gronwall’s inequality, we deduce thatSince all of the constants do not depend on , as , we conclude that sequence converges to a limit in the following convergence:and there also holdsFor each small , let , is a mollifier on , and is a smooth solution of the boundary value problem:where and satisfies , .

We deduce that is a solution of the following initial boundary value problem: where .

By the proof of Lemma 2.3 in [20], there exists a subsequence of , as , in , in , Hence, satisfies (18) of Theorem 1. By virtue of the lower semi-continuity of various norms, we deduce that satisfies the following uniform estimate:where is a positive constant, depending only on .

The uniqueness of solution can be obtained by the same method as the above proof of convergence; we omit the details here. This completes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11526105; 11572146) and the Funds of Education Department of Liaoning Province (JQL201715411).