In the critical dilute regime where particle diameter scales as ε^3 and mutual distance as ε, weak limits of steady Navier-Stokes solutions in a perforated pipe satisfy the Brinkman equation with a friction term, for both prescribed flux and prescribed pressure drop, without small-data assumptions.
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Brinkman's term for the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in perforated pipes
In the critical dilute regime where particle diameter scales as ε^3 and mutual distance as ε, weak limits of steady Navier-Stokes solutions in a perforated pipe satisfy the Brinkman equation with a friction term, for both prescribed flux and prescribed pressure drop, without small-data assumptions.