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

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.


Introduction
Fluid-particle interaction model arises in many practical applications in science and engineering [1][2][3][4] and is of primarily importance in the sedimentation analysis of disperse suspensions of particles in fluids.We focus on the fluidparticle 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 (, ) ≥ 0 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 Φ().The system was given as follows: Finally, letting  → 0, then (3) converges to system + div  () = 0 ()  + div  ( ⊗ ) + ∇  ( +  ()) + (sign ()  + ) ∇  Φ = 0 For the viscous case, with the dynamic viscosity terms taken into consideration, then (4) 3 will be have an additional term of div  S. In [6], the viscous stress tensor S = S(∇  ) is assumed to satisfy Newton's Law for viscosity which requires that S =  (∇ + ∇  ) +  div  I, where  and  are constant viscosity coefficients satisfying Then div  S (∇  ) = Δ   + ∇  div  (7) and thus the system turns into the following equation Moreover, if the influence of gravitational potential Ψ = Ψ(, ) 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][12][13] and references therein.
On the other hand, as we know, the viscous stress tensor S is depends on the rate of strain E  (∇  ), 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 stressstrain relation for such fluids given by the power laws, which states that for 0 <  < 1 (see [14]).Ladyzhenskaya (see [15]) proposed a special form for S on the incompressible model: These models are called ,   0 ,  1 > 0,  > 2; − ,   0 = 0,  1 > 0,  > 1; ℎ,   0 ,  1 > 0,  = 1. ( For  0 = 0, if  < 2 then it is a pseudo-plastic fluid, and if  > 2 then it is a dilatant fluid (see [14]).In the view of physics, the model captures the shear thinning fluid for the case of 1 <  < 2, and captures the shear thickening fluid for the case of  > 2.
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 be with the initial and boundary conditions and the no-flux condition for the density of particles where Ω is a bounded interval, Ω  = Ω×[0, ]., ,  denote the fluid density, velocity, and the density of particle in the mixture, respectively.() =   is the pressure where  > 0,  > 1, Ψ(, ) stands for the non-Newtonian gravitational potential, and the given function Φ() denotes the external potential. > 0 is the viscosity coefficient and  1 > 0,  2 > 0,  > 2,  > 2 are constants.
As to the non-Newtonian fluids, there has been much research both theoretically and experimentally, see ( [15][16][17][18][19][20][21][22][23]).Indeed, we have investigated the existence results of solutions for  > 2 and 1 <  < 2 in the absence of term Ψ() 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 ,  > 2 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.

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  0 ≥ , where 0 <  ≪ 1 is a positive number.We denote by We deal with the term of Multiplying (14) 3 by Ψ and integrating over Ω, we obtain then we have Hence, we deduce that From (14) From ( 30) and (31), by Gronwall's inequality, it follows that sup Besides, we can also get the following estimates.By using From ( 14) 3 , we have Differentiating (14) 3 with respect to time , multiplying it by Ψ  , integrating over Ω to , and using Young's inequality, we have thus, we get where  is a positive constant, depending only on  0 .

Estimate for
Therefore, taking a limit on  in (53), as  → 0, we conclude that where  is a positive constant, depending only on  0 .

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  0 = 0 and assuming that  −1 was defined for  ≥ 1, let   ,   ,   be the unique smooth solution to the following problems: with the initial and boundary conditions where We directly construct approximate solutions of the problem (59)-(63).More precisely, we first find   from ( 59) and ( 63) with smooth function  −1 , i.e., and It follows from the classical linear hyperbolic theory that there is a unique solution   on this above initial problem.Using the method of characteristics, we have By ( 68) and (69), we have Using (67), then which means for all  ∈ (0,  1 ) .
Next, combining the classical stableness results of the elliptic equation, the existence of Ψ  can be obtained by ( 61) and (63), then by ( 62) and (63) we get   .The last, with   , Ψ  ,   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  0 , 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  +1 ,  +1 ,  +1 satisfy the system of equations where  is a positive constant, depending only on  0 .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.