Breaking of non-Newtonian character in flows through a porous medium

T. Chevalier, S. Rodts, X. Chateau, C. Chevalier, and P. Coussot

Phys. Rev. E 89, 023002 – Published 6 February 2014

Abstract

From NMR measurements we show that the velocity field of a yield stress fluid flowing through a disordered well-connected porous medium is very close to that for a Newtonian fluid. In particular, it is shown that no arrested regions exist even at very low velocities, for which the solid regime is expected to be dominant. This suggests that these results obtained for strongly nonlinear fluid can be extrapolated to any nonlinear fluid. We deduce a generalized form of Darcy's law for such materials and provide insight into the physical origin of the coefficients involved in this expression, which are shown to be moments of the second invariant of the strain rate tensor.

Received 7 September 2013

DOI:https://doi.org/10.1103/PhysRevE.89.023002

Authors & Affiliations

T. Chevalier¹, S. Rodts¹, X. Chateau¹, C. Chevalier², and P. Coussot¹

¹Laboratoire Navier (ENPC-IFSTTAR-CNRS), Université Paris-Est, Champs sur Marne
²IFSTTAR, Université Paris-Est, Champs sur Marne

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Vol. 89, Iss. 2 — February 2014

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Images

Figure 1
Distribution function vs velocity scaled by the maximum velocity for flows through a capillary: Newtonian fluid (dashed line) and yield stress fluid (continuous line) following a HB model with an unsheared region (plug) extending over 2/3 of the capillary radius (i.e., $Bi = 5.4$ ). Inset: experimental flow curve of our material (squares) fitted by a HB model [7] (continuous line), and Newtonian flow curve of arbitrary viscosity (dashed line).
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Figure 2
Distribution function vs velocity scaled by the mean velocity (in the small duct) measured for the flow of a yield stress fluid through a 7-cm-diameter and 7-cm-long duct between 3.5-cm-diameter ducts for $Bi = 14$ . The colored regions approximately correspond to those which are visible on the velocity field (see inset) obtained from independent measurements (see [13]) and represented by color scale (in $mm / s$ ): outer arrested region (red and gray dotted lines), transitional region (light green, continuous line), moving plug region (dark blue, dashed line).
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Figure 3
Distribution functions as a function of the velocity scaled by the average velocity through the voids. (i) Computed from literature NMR data for $f$ (at sufficiently small $Δ$ ) with Newtonian fluids (water or oil) through bead packings: (squares) Fig. 1 of [15] $V$ = 3.55 mm $s^{- 1}$ ; (circles) data from Fig. 4 of [16] $V = 1.1 {mm s}^{- 1}$ , and (stars) for a power-law fluid (Xanthan), Fig. 9 of [17], $V = 30 {mm s}^{- 1}$ . (ii) From our tests with a yield stress fluid: ( $D = 2 mm$ ) short dash light blue ( $V = 0.12 {mm s}^{- 1}$ , $Bi = 7.3$ ) and dotted red line ( $V = 4 {mm s}^{- 1}$ , $Bi = 1.8$ ); ( $D = 0.5 mm$ ) black line ( $V = 0.12 {mm s}^{- 1}$ , $Bi = 4.2$ ) and dashed dark blue ( $V = 4 {mm s}^{- 1}$ , $Bi = 1.04$ ). Our data for intermediate velocities fall exactly along the other data. The data for $Bi = 11.4$ are similar although slightly more affected by diffusion. The inset shows the distribution function for the transversal velocity for $D = 0.5 mm,$ and $V = 0.4 {mm s}^{- 1}$ , $Bi = 3$ (black), $V = 1.2 {mm s}^{- 1}$ , $Bi = 1.7$ (dotted red), and $V = 4 {mm s}^{- 1}$ , $Bi = 1.04$ (dashed dark blue).
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Figure 4
Yield stress fluid flow (arriving from a 8-cm-diameter duct) through a die of 1 cm diameter. Distribution function in cross sections at 1 mm (circles) and 16 mm (squares) from the die entrance, calculated from the velocity profiles (see inset) (data from [14]). The velocity field in color scale [highest velocities in red (central region), smallest ones in blue (outside region)] is also shown in the inset.
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