Extensions to the macroscopic Navier-Stokes equation

S. Sorek, D. Levi-Hevroni, A. Levy, G. Ben-Dor

    Research output: Contribution to journalArticlepeer-review

    11 Scopus citations

    Abstract

    A development is provided showing that for any phase, by not neglecting the macroscopic terms of the deviation from the intensive momentum and of the dispersive momentum, we obtain a macroscopic secondary momentum balance equation coupled with a macroscopic dominant momentum balance equation that is valid at a larger spatial scale. The macroscopic secondary momentum balance equation is in the form of a wave equation that propagates the deviation from the intensive momentum while concurrently, in the case of a Newtonian fluid and under certain assumptions, the macroscopic dominant momentum balance equation may be approximated by Darcy's equation to address drag dominant flow. We then develop extensions to the dominant macroscopic Navier-Stokes (NS) equation for saturated porous matrices, to account for the pressure gradient at the microscopic solid-fluid interfaces. At the microscopic interfaces we introduce the exchange of inertia between the phases, accounting for the relative fluid square velocities and the rate of these velocities, interpreted as Forchheimer terms. Conditions are provided to approximate the extended dominant NS equation by Forchheimer quadratic momentum law or by Darcy's linear momentum law. We also show that the dominant NS equation can conform into a nonlinear wave equation. The one-dimensional numerical solution of this nonlinear wave equation demonstrates good qualitative agreement with experiments for the case of a highly deformable elasto-plastic matrix.

    Original languageEnglish
    Pages (from-to)215-233
    Number of pages19
    JournalTransport in Porous Media
    Volume61
    Issue number2
    DOIs
    StatePublished - 1 Nov 2005

    Keywords

    • Macroscopic momentum balance equation
    • Nonlinear wave propagation
    • Representative elementary volume
    • Saturated porous media
    • Solid-fluid microscopic interface
    • Spatial scaling

    ASJC Scopus subject areas

    • Catalysis
    • Chemical Engineering (all)

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