Starting from the previous description of a monodisperse porous medium (Mono-DPM), one may consider a bidisperse porous medium (BDPM) as a (Mono-DPM) where a different porous medium replaces the corresponding solid phase. A BDPM is thus a standard porous medium consisting of clusters of large particles that appear as agglomerations of small particles [7], [34], [36]. Having in view such a structure, one may characterize a BDPM by two phases, the $f$-phase due to the existing macro-pores (the voids separating the clusters), and the $p$-phase corresponding to the micro-pores (the voids within the clusters). The models of bidisperse porous media with very complex structures are encountered in many biological, medical and engineering applications. There are many references in this sense, among of them we refer to [31], [32], [33], [34], [36].

The first model of Nield and Kuznetsov [33] is linear and refers to the momentum transfer in a BDPM in the steady-case, and is given by the following couple of Brinkman type equations for the velocity field $v_f^{\ast }$ of the $f$-phase and the velocity field $v_p^{\ast }$ of the $p$-phase

where $G=-\nabla p^{\ast }$ is the pressure force due to the pressure field $p^{\ast }$ (the same in both phases). In addition, $\mu$ is the fluid viscosity coefficient, $K_p$ and $K_f$ and ${\tilde{\mu }}_p$ and ${\tilde{\mu }}_f$ are the permeability coefficients and, respectively, the effective viscosity coefficients for the two phases, $\zeta$ is the coefficient for momentum transfer between the phases [33] (see also [34]). All variables are dimensional, as the superscript $\star$ shows.

The next model of BDPM proposed by Nield and Kuznetsov [36] is semilinear and extends the linear model in [33], [34] by considering the semilinear terms of Forchheimer type $|v_p^{\ast }|v_p^{\ast }$ and $|v_f^{\ast }|v_f^{\ast }$ in the following system of coupled equations for the velocity fields $v_f^{\ast }$ and $v_p^{\ast }$

where $c_f$ and $c_p$ are the corresponding Forchheimer coefficients for the two phases, and $\rho$ is the fluid density. See also [6], [42], [43], [45], and [46] for further extensions and models based on forced, natural and mixed convection phenomena. For instance, Straughan [46] analyzed a model, which describes the thermal convection in an anisotropic BDPM, and with different pressures within the macro and micro phases. A similar consideration has been adopted in [6].

The model of Nield and Kuznetsov [36], where the steady-state momentum transfer is described by the semilinear system (1.2), and also the model of Straughan [46] have suggested us in [22] to consider a more general nonlinear system of two coupled Navier-Stokes type equations, as follows

where $\mathrm{\Omega }\subset {ℝ}^N$ is a bounded domain ($N\le 3$), $q\ge 1$, ${\eta }_i,{\kappa }_i,{\alpha }_i\ge 0$ and ${\mu }_i,{\gamma }_i>0$, $i=1,2$, are given constants whose meaning depends on the physical properties of fluid flow and porous media, while $h_i$, $i=1,2$, are given data in some Sobolev spaces.

Note that all along this paper we assume that the fluid density is normalized at $1$. Also, the case $N=1$ is not considered here.

A tridisperse porous medium (TDPM) can be defined as a monodisperse porous medium (Mono-DPM) in which the solid phase is replaced by a bidisperse porous medium (BDPM). Thus, the solid phase contains three types of pores. One type are the macro-pores, but there are also pores of smaller scale called meso-pores, and cracks or fissures of still smaller scale, which are called micro-pores (see [35]). The basic theory for thermal convection in a triple porosity (tridispersive) medium was developed by Nield Kuznetsov [35] (see also [26], [15]). The authors in [8], [16] have considered different velocities $U^f$, $U^m$ and $U^c$ and different pressures $p^f$, $p^m$ and $p^c$ in the macro, meso and micro-pores.

There are many applications where more than two porosity scales are present in a medium, concerning the modeling of fluid flow in cellular biological media, as well as the flow through geological rock formations (cf. [35] and the references therein).

Nield and Kuznetsov [35] followed their model (1.1) (see [33]) and proposed a Darcy model for the steady-state momentum transfer in a TDPM, characterized by three phases determining three levels. At at first level there is a fluid phase, at the second level there is a Mono-DPM, and at the third level there is a BDPM. For each of the three phases there is a volume-averaged velocity (a Darcy velocity), $v_i$ at the $i$th-level, $i=1,2,3$. The authors assumed that the momentum equations for $v_1$ and $v_2$ are coupled, and those for $v_2$ and $v_3$ are coupled, but those for $v_1$ and $v_3$ are not directly coupled since the TDPM is assumed to be structured so that, due to the presence of the second phase, the first and the third phases are geometrically isolated from each other (cf. [26]). Therefore, adopting the Darcy model, Nield and Kuznetsov [35] (see also [26]) considered the following triplet of equations for steady-state momentum transfer

where, as in (1.1), it is assumed that the pressure forces $-\nabla p_i$, $i=1,2,3$, are the same and denoted $G$ in all three phases. In addition, $\mu$ is the fluid viscosity, $K_1$, $K_2$ and $K_3$ are the permeabilities of the three phases, and ${\zeta }_{12}$ and ${\zeta }_{23}$ are some coupling coefficients for momentum transfer. See also [15] for the free convection in a square cavity filled with a tridisperse porous medium.

Gentile and Straughan [16] considered different pressures in their analysis of the thermal convection in a tridisperse porous medium when only one temperature is employed and the horizontal layer, which contains the saturated porous medium, is heated from below.

A multidisperse porous medium (MDPM) can be viewed as a standard porous medium consisting of a fluid phase ($f$-phase) and a solid phase containing several types of pores (provided by different porous media) located within each other.

Taking into account the (TDPM) model of Nield and Kuznetsov [35], where the steady-state momentum transfer is described by the linear system (1.4), the (BDPM) model of the same authors [36] described by the semilinear system (1.2), and the model of Gentile and Straughan [16], we consider a more general nonlinear coupled type Navier-Stokes system with reaction terms arising in the analysis of fluid flows in anisotropic multidisperse porous media (MDPM). This system has the form

where $m\in \mathbb{N}$, $\mathrm{\Omega }\subset {ℝ}^N,N\le 3,$ is an open bounded set, ${\eta }_{0i},{\kappa }_{0i},{\delta }_i\in L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ are given functions, such that ${\eta }_{0i}\left(x\right)\ge \eta$, for some constant $\eta \ge 0$, and ${\kappa }_{0i}\left(x\right)\ge 0$. In addition, $h_i\in H^{-1}{\left(\mathrm{\Omega }\right)}^N$ and the entries $a_{j\mathrm{\ell }}^{(i)}$ of the matrix $A_i$ are nonnegative functions in $L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ with $a_{j\mathrm{\ell }}^{(i)}=a_{\mathrm{\ell }j}^{(i)}$ and

for some constant $\mu >0$ related to the viscosity tensors $A_i$, $i=1,2,\mathrm{\dots },m$. In addition, the coupling terms are

We assume that $1\le q_i,r_{ij}\le 2^{\ast }/2=N/\left(N-2\right)$ if $N=3$, and in case $N=2.$ Also ${\gamma }_{ij}\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ are given non-negative functions, $i,j=1,2,\mathrm{\dots },m$.

The unknowns of system (1.5) are the velocity and pressure fields $u_i$ and $p_i$, $i=1,2,\mathrm{\dots },m$.

In order to analyze this system, we provide a deep localization analysis of two important related energies, the kinetic energy and the enstrophy (see Subsection 1.4).

Let $\mathrm{\Omega }\subset {ℝ}^N$ be an open bounded set, $N\le 3$.

The kinetic energy of a flow velocity field $u$ in $\mathrm{\Omega }$ has the following expression

and the enstrophy is given by [13, p. 28]

The significance of the enstrophy is that it determines the rate of dissipation of the kinetic energy

Indeed, by assuming that the flow velocity and pressure fields $u$ and $p$ satisfy the unsteady Navier-Stokes system in $\mathrm{\Omega }$,

and the boundary condition ${u|}_{\partial \mathrm{\Omega }}=0$, multiplying in $L^2(\mathrm{\Omega })$ the Navier-Stokes equation with $u$, integrating by parts, and using the boundary condition, we obtain that

which is relation (1.8). This relation shows that the kinetic energy decays due to the viscosity with the rate $-\nu E(u)$. In the case $\mathrm{\Omega }={ℝ}^3$, this result has been obtained in [13, (1.9)]. Let us add that the enstrophy has an important role in the study of turbulent flows (see [13, Chapter 2] for further details).

In addition, if $\mathrm{\Omega }={ℝ}^N$, $N\le 3$, and the velocity field $u$ decays rapidly at infinity, then the enstrophy $E(u)$ becomes

where $\omega =\nabla \times u$ is the vorticity field. Note that this formula remains true even if $\mathrm{\Omega }\ne {ℝ}^N$, but the boundary terms that follow from the integration by parts vanish (see [13, Chapter 2, p.28]).

With reference to system (1.5), there are two matters of physical interest in relation to the velocities.

First requirement: Find suitable forces $h_i,$ in order to guarantee that the kinetic energies $\frac{1}{2}{\int }_{\mathrm{\Omega }}{\left|u_i\right|}^2𝑑x$ or the enstrophies ${\int }_{\mathrm{\Omega }}{\left|\nabla u_i\right|}^2𝑑x$ stay bounded between a priori given bounds. Such a requirement is important in practice in case that these energies have to be limited.

Second requirement: The forces $h_i$ being given, find the bounds of the corresponding kinetic energies and enstrophies.

Obtaining localization results for solutions to nonlinear problems in general requires the manipulation of various inequalities such as Poincaré’s and Harnack’s inequalities and also abstract localization techniques such as Krasnoselskii-type methods and those that use the properties of the topological degree, in particular of the fixed point index. For such kind of results concerning various classes of nonlinear problems, we refer the reader to the papers [5, 37, 38, 39] and [41].

The paper is structured as follows. First, in Section 2, we mention some well-known but useful results regarding the stationary Navies-Stokes equations in the incompressible case. We introduce the corresponding solution operator of the stationary Navier-Stokes equation and state some of its useful properties. The next Section 3 is devoted to localization results for the anisotropic Navier-Stokes equations with velocity-dependent reaction terms. By using the fixed point index theory we obtain localization results of the enstrophy and of kinetic energy under some suitable conditions. The Section 4 is devoted to the analysis of coupled systems of anisotropic Navier-Stokes equations. We obtain localization results for the corresponding enstrophy and kinetic energies.

We conclude this Introduction by some results about the fixed point index, which are used in the reasonings that follow.

Recall that, if $X$ is a Banach space, $U$ is a bounded open subset of $X$ and $T:\overline{U}\to X$ is a compact operator (continuous with $T\left(\overline{U}\right)$ relatively compact), fixed point free on the boundary $\partial U$ of $U$, then by the fixed point index $i(T,U)$ one means the Leray-Schauder degree $D(I-T,U)$(see [11, 17]). Among the properties of the index, we mention the following ones which will be useful further:

• (I1)

  if $i(T,U)\ne 0$, then $T$ has at least one fixed point in $U.$

• (I2)

  if $V\subset U$ is open and $T$ has no fixed points on the boundary of $V,$ then $i(T,U\setminus \overline{V})$ $=i(T,U)-i(T,V).$

• (I3)

  if $0\in U$ and $x\ne \lambda T\left(x\right)$ for all $(x,\lambda )\in \partial U\times (0,1),$ then $i(T,U)=1$ (see [17, Theorem 12.7.3]).

• (I4)

  if there exists a point $x_0\in X\setminus \left\{0\right\}$ such that $x\ne T\left(x\right)+\lambda x_0$ for all $x\in \partial U$ and $\lambda >0,$ then $i(T,U)=0$ (see [17, Theorem 12.7.11(ii)]).

Assume that

(H1):

$\mathrm{\Phi }:\mathrm{\Omega }\times V\to$ $L^2{\left(\mathrm{\Omega }\right)}^N$ is $L^2$-Carathéodory and

$${\left|\mathrm{\Phi }(x,u)\right|}_{L^2}\le f\left({\left|u\right|}_{H_0^1}\right),\text{ for }u\in V,$$

and some increasing function $f:[0,R_0]\to {ℝ}_{+},$ where $R_0\in (0,\frac{\sqrt{\mu {\lambda }_1+\eta }}{2\kappa M\mu \sqrt{{\lambda }_1}})$ and

(3.3)

$$f\left(R_0\right)\le \mu \sqrt{{\lambda }_1}{R}_0.$$

Then for ${\left|u\right|}_{H_0^1}\le R_0,$ one has

Therefore, the operator $T\left(u\right)=S\left(\mathrm{\Phi }(\cdot ,u)\right)$ is well-defined and continuous on the set

In addition $T$ is compact due to the compact embedding of $L^2{\left(\mathrm{\Omega }\right)}^N$ in $V^{\prime }.$

Assume now that

(H2):

there exists $e\in V$ with ${\left|e\right|}_{L^2}=1$ such that

$${(\mathrm{\Phi }(\cdot ,u),e)}_{L^2}\ge g\left({\left|u\right|}_{H_0^1}\right),\text{ for }u\in V,$$

and some function $g:{ℝ}_{+}\to {ℝ}_{+}.$

Obviously, from Poincaré’s inequality, an upper bound for the enstrophy immediately gives an upper bound for the kinetic energy. We now try to obtain a lower bound of the kinetic energy. To this aim we introduce a new condition.

(H2’):

One has

$${(\mathrm{\Phi }(\cdot ,u),e)}_{L^2}\ge g_0\left({\left|u\right|}_{L^2}\right),\text{ for }u\in V,$$

and some function $g_0:{ℝ}_{+}\to {ℝ}_{+}.$