Brinkman's term for the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in perforated pipes

Technical Summary: Homogenization of Steady-State Navier-Stokes Equations in Perforated Pipes with Mixed Boundary Conditions

Problem Statement
This work investigates the asymptotic behavior of the steady motion of a viscous, incompressible fluid within a distorted pipe containing a large number of small solid particles. The domain $\Omega_\varepsilon$ is a perforated subset of a bounded, admissible domain $\Omega \subset \mathbb{R}^3$, obtained by removing $N(\varepsilon)$ particles of diameter $\varepsilon^3$ separated by a mutual distance $\varepsilon$. This configuration corresponds to the critical regime ($\alpha = 3$) in homogenization theory, where the particle size scales with the cube of the inter-particle distance.

The fluid flow is governed by the steady-state Navier-Stokes equations with mixed boundary conditions. Specifically:

1. No-slip conditions are imposed on the physical pipe walls ($\Gamma_W$) and on the surfaces of the particles ($\partial K_n^\varepsilon$), where particles may move with prescribed constant velocities $\mu_n^\varepsilon$.
2. Bernoulli pressure conditions are applied at the inlet ($\Gamma_I$) and outlet ($\Gamma_O$). These involve the normal derivative of the velocity and the Bernoulli pressure ($\Phi = p + \frac{1}{2}|u|^2$).
3. Two distinct scenarios are analyzed:
   • Prescribed Net Flux: The total flux $F$ through the outlet is fixed, while the pressure drop is an unknown constant determined by the solution.
   • Prescribed Pressure Drop: The pressure drop between inlet and outlet is fixed, while the resulting flux is an unknown quantity.

The primary mathematical challenge lies in deriving effective (homogenized) equations for the limit as $\varepsilon \to 0$ without imposing smallness assumptions on the external force, flux rate, or pressure drop. This is complicated by the fact that standard energy estimates for the Navier-Stokes equations often fail to provide uniform bounds in such mixed boundary value problems without additional constraints.

Methodology
The authors employ the energy method within the framework of homogenization theory, following the general strategy of Tartar. The proof proceeds in three main stages:

1. Uniform Bounds ($\varepsilon$-independent estimates):
   Establishing uniform bounds for the solutions $(u_\varepsilon, \Phi_\varepsilon)$ is the most critical and novel step. Unlike standard cases where simple energy estimates suffice, the authors cannot rely solely on testing the momentum equation with the solution itself due to the mixed boundary conditions and the unknown pressure drop.

   • Instead, they utilize a contradiction argument based on Bernoulli's law for solutions of the stationary Euler equations.
   • They assume the norms of the solutions blow up as $\varepsilon \to 0$, normalize the sequence, and show that the limit satisfies the Euler equations.
   • By analyzing the boundary behavior of the limit Bernoulli pressure on the pipe walls (using local regularity results and the specific geometry of the inlet/outlet), they demonstrate that the limit pressure must be constant on the walls. This leads to a contradiction with the flux or pressure drop conditions, thereby proving the uniform boundedness of the sequence.
   • For the prescribed flux problem, an additional geometric assumption is made that particles do not accumulate near the inlet and outlet cylinders to facilitate the boundary analysis of the limit pressure.

2. Construction of Correctors and Restriction Operators:
   To handle the perforated domain, the authors construct Stokes correctors $(\omega_{n,1}^\varepsilon, \omega_{n,2}^\varepsilon)$ in the exterior of the particles and in the annular regions surrounding them. These correctors are used to define a restriction operator $R_\varepsilon$.

   • Unlike previous works in the subcritical regime where scalar capacitary correctors sufficed, the critical regime requires these vector-valued Stokes correctors to account for the non-vanishing relative capacity of the particle swarm.
   • The restriction operator $R_\varepsilon(\phi)$ maps test functions from the homogenized domain $\Omega$ to the perforated domain $\Omega_\varepsilon$ by modifying them near the particles to satisfy the zero boundary conditions, while preserving the divergence-free property.

3. Passage to the Limit:
   Using the uniform bounds and the restriction operator, the authors pass to the limit in the weak formulation of the Navier-Stokes equations.

   • The nonlinear convective term converges via strong convergence in $L^4$ (guaranteed by the uniform bounds and compact embeddings).
   • The interaction between the fluid and the particles generates an additional term in the limit equation. This term arises from the limiting behavior of the sum of the drag forces exerted by the particles, characterized by the Stokes resistance matrices $G_n$.

Key Contributions and Results
The paper establishes two main theorems (Theorems 2.4 and 2.5) describing the homogenized limit of the system:

• Emergence of the Brinkman Term: The homogenized equations in the limit domain $\Omega$ are the Navier-Stokes equations augmented by a Brinkman term. The effective equation takes the form:
  $$-\Delta u + (\nabla \times u) \times u + \nabla \Phi + Gu - J = f$$
  where:

  • $Gu$ represents a friction term proportional to the fluid velocity $u$, arising from the collective drag of the particles (with $G$ being the limit of the scaled sum of Stokes resistance matrices).
  • $J$ is a source term arising from the limiting flux of the moving particles themselves.
  • The boundary conditions on the inlet and outlet remain of the mixed type involving the Bernoulli pressure.

• Extension to Critical Regime with Non-Zero Data: The results extend previous homogenization findings (specifically those in [59] for the subcritical regime) to the critical regime ($\alpha=3$). Crucially, these results hold without any smallness assumption on the external force, flux rate, or pressure drop. This is a significant advancement, as many homogenization results for Navier-Stokes rely on small data assumptions to ensure uniqueness or existence of bounds.

• Handling of Moving Particles: The framework accommodates particles with non-zero constant velocities $\mu_n^\varepsilon$, provided their kinetic energy is uniformly bounded (Assumption 2.11). This allows for a more realistic modeling of particle-fluid interactions compared to fixed particle models.

Significance
The paper claims that its most significant contribution is the rigorous derivation of Brinkman's law for the steady Navier-Stokes equations in the critical regime under unrestricted data. The authors emphasize that the derivation of uniform bounds via the contradiction argument involving Bernoulli's law is a delicate matter, particularly because the complex structure of the perforations must be accounted for to ensure the argument holds at the $\varepsilon$-level.

By successfully constructing a genuine restriction operator that does not modify test functions vanishing on the particles and by handling the convergence of the pressure term in a polydisperse setting with non-zero particle velocities, the work provides a robust mathematical foundation for modeling fluid flow in perforated pipes where the particle density is high enough to induce a Brinkman-type drag but not so high as to result in Darcy's law. The results are directly applicable to fields such as hemodynamics, petroleum engineering, and aerodynamics where fluid flow through arrays of solid obstacles is a fundamental phenomenon.