Inertia and Roughness-Induced Effects on the Porous Medium Flow Through a Corrugated Channel
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Inertia and Roughness‑Induced Effects on the Porous Medium Flow Through a Corrugated Channel Eduard Marušić‑Paloka1 · Igor Pažanin1 Received: 26 February 2020 / Accepted: 27 July 2020 © Springer Nature B.V. 2020
Abstract In this paper, we study the flow through a corrugated channel filled with a fluid-saturated sparsely packed porous medium. The porous medium flow is described by the Darcy–Lapwood–Brinkman system taking into account the Brinkman extension of the Darcy law and the flow inertia. We assume the periodicity of the roughness in the longitudinal direction and that the flow is governed by the prescribed pressure drop between the edges of the channel. Introducing average boundary roughness as the small parameter, we propose a higher-order correction of the Darcy–Brinkman filtration velocity via asymptotic analysis. Keywords Darcy–Lapwood–Brinkman model · Periodic rugosities · Inertia effects · Boundary-layer analysis · Higher-order asymptotic approximation
1 Introduction Numerous models have been proposed throughout the literature to describe the flow through a porous media. While the conservation of mass is expressed by the continuity equation (1)
div 𝐔 = 0
for the filtration velocity 𝐔 , different laws have been introduced over the past 16 decades to describe the conservation of the linear momentum. The simplest law and thus the most popular one is the Darcy law (Darcy 1856). The Darcy law states that the driving force necessary to move a specific volume of fluid at certain speed through a porous medium is in equilibrium with the resistance generated by the internal friction between the fluid and the pore structure. It reads:
* Igor Pažanin [email protected] Eduard Marušić‑Paloka [email protected] 1
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
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E. Marušić‑Paloka, I. Pažanin
𝐔=−
K ∇p , 𝜇
(2)
where p denotes the pressure, whereas K and 𝜇 represent the permeability of the porous medium and the (dynamic) viscosity coefficient, respectively. Obviously, the Darcy law (2) does not take into account the usual viscous shear whose existence has been experimentally demonstrated near the boundaries (see, e.g., Beavers et al. 1970). Consequently, (2) is not applicable in many physically relevant settings, for instance, when the no-slip boundary condition is imposed on an impermeable boundary. The Brinkman extension (Brinkman 1947) of the Darcy law includes the macroscopic shear term in the equation leading to
−
𝜇 𝐔 + 𝜇e Δ𝐔 = ∇p , K
(3)
where 𝜇e is the effective viscosity of the fluid in the porous medium. The Darcy–Brinkman equation (3), justified in Lundgren (1972), Saffman (1971) (see also Marušić-Paloka et al. 2012), turns out to be the most suitable governing equation for a creeping flow within a fluid-saturated sparsely packed porous medium wherein there is more window for a fluid to flow so that the distortion of velocity gives rise to usual viscous shear force. For that reason, (3) has been extensively studied
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