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Radial Viscous Fingering in Case of Poorly Miscible Fluids

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Abstract

The displacement of a viscous fluid from an annular Hele-Shaw cell with a source of finite radius by a less viscous one is investigated. A special case of poorly miscible fluids is considered when corresponding dimensionless criteria—capillary and Peclet numbers—both tend to infinity. Brinkman model which additionally takes into account small viscous forces in a plane of the cell is used to describe the displacement process. Linear analysis shows a stabilizing effect of viscous forces and reveals a geometrical similarity criterion, namely the ratio of the interface’s radius to the gap between the cell’s plates. The displacement patterns, obtained numerically under Brinkman model, are very sensitive to the discovered criterion. The comparison with available experimental data is acceptable.

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Acknowledgements

The present investigation has been supported by Russian Foundation for Basic Research—Grant 17-08-01032.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow MV Lomonosov State University, Moscow, 119992, Russia

    Oleg A. Logvinov

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  1. Oleg A. Logvinov
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Correspondence to Oleg A. Logvinov.

Appendix

Appendix

The coefficients \(\varPi _j, j=0\ldots 5\) and \(\varPhi \) in dispersion relation (16) have the following form:

$$\begin{aligned} \varPhi= & {} 8 (M-1) (\beta _1 + 1) (\beta _2 + 1) n^2 \\&+\, 2 (\beta _1 + 1) (\beta _2 + 1) k^2\left[ M \left( \langle R_2 \rangle ^{2n} + 1 \right) -\left( \langle R_1 \rangle ^{2n} + 1 \right) \right] n \\&-\, 4 (M-1) k \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) (\beta _1 + 1) n \\&-\, 4 (M-1) k \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \left( \langle R_1 \rangle ^{2n} + 1 \right) (\beta _2 + 1) n \\&-\, M k^3 (\beta _2 + 1) \left( \langle R_1 \rangle ^{2n} + 1 \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \\&+\, k^3 (\beta _1 + 1) \left( \langle R_1 \rangle ^{2n} + 1 \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \\&+\, 2 (M-1) \left( \langle R_1 \rangle ^{2n} + 1 \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) k^2 \\&\times \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) , \\ \varPi _5= & {} 16 (M-1)^2 (\beta _1 + 1) (\beta _2 + 1), \\ \varPi _4= & {} - 8 (M-1)^2 (\beta _1 + 1) k\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) \\&-\, 8 (M-1)^2 (\beta _2 + 1) k\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \left( \langle R_1 \rangle ^{2n} + 1 \right) , \\ \varPi _3= & {} 4 (M-1)^2 k^2\left( \langle R_1 \rangle ^{2n} + 1 \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) \\&\times \,\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \\&+\, 8 (M-1) k^2 (\beta _1 + 1) (\beta _2 + 1)\left[ \left( \langle R_1 \rangle ^{2n} - 1 \right) -M \left( \langle R_2 \rangle ^{2n} - 1 \right) \right] \\&-\, 16 (M-1)^2 (\beta _1 + 1) (\beta _2 + 1), \\ \varPi _2= & {} 4 (M-1) k^3 M (\beta _2 + 1)\left( \langle R_1 \rangle ^{2n} + 1 \right) \left( \langle R_2 \rangle ^{2n} - 1 \right) \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \\&-\, 4 (M-1) k^3 (\beta _1 + 1)\left( \langle R_1 \rangle ^{2n} - 1 \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \\&-\, 8 (M-1) (\beta _1 + 1) (\beta _2 + 1) k^2\left( M \langle R_2 \rangle ^{2n} - \langle R_1 \rangle ^{2n} \right) \\&+\, 8 (M-1) k (\beta _1 + 1)\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \left[ \left( M \langle R_2 \rangle ^{2n} + 1 \right) -\left( \langle R_2 \rangle ^{2n} + 1 \right) \right] \\&+\, 8 (M-1) k (\beta _2 + 1)\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \left[ \left( M \langle R_1 \rangle ^{2n} + 1 \right) -\left( \langle R_1 \rangle ^{2n} + 1 \right) \right] , \\ \varPi _1= & {} - 2 (M-1) (\beta _1 + 1) (\beta _2 + 1) k^4\left[ M \left( \langle R_2 \rangle ^{2n} - 1 \right) -\left( \langle R_1 \rangle ^{2n} - 1 \right) \right] \\&-\, 4 (M-1) k^2\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \\&\times \,\left[ M \left( \langle R_1 \rangle ^{2n} + 1 \right) \left( \langle R_2 \rangle ^{2n} - 1 \right) -\left( \langle R_1 \rangle ^{2n} - 1 \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) \right] \\&-\, 4 (M-1) k^2\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \\&\times \,\left[ 2 \left( M \left( \langle R_1 \rangle ^{2n} + 1 \right) - \left( \langle R_2 \rangle ^{2n} + 1 \right) \right) \right] \\&+\, 4 (\beta _1 + 1) k^3\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \\&\times \left[ M \left( \langle R_1 \rangle ^{2n} + 1 \right) \left( \langle R_2 \rangle ^{2n} - 1 \right) +\left( \langle R_1 \rangle ^{2n} - 1 \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) \right] \\&+\, 4 (\beta _1 + 1) k^3\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \\&\times \left[ M \left( M \left( \langle R_2 \rangle ^{2n} - 1 \right) - \left( \langle R_1 \rangle ^{2n} + 1 \right) \right) -\left( M \left( \langle R_1 \rangle ^{2n} - 1 \right) - \left( \langle R_2 \rangle ^{2n} + 1 \right) \right) \right] \\&+\, 4 (\beta _2 + 1) k^3\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \\&\times \left[ M \left( M \left( \langle R_1 \rangle ^{2n} + 1 \right) \left( \langle R_2 \rangle ^{2n} - 1 \right) -\left( \langle R_1 \rangle ^{2n} - 1 \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) \right) \right] \\&+\, 4 (\beta _2 + 1) k^3\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \\&\times \left[ M \left( M \left( \langle R_1 \rangle ^{2n} + 1 \right) - \left( \langle R_2 \rangle ^{2n} - 1 \right) \right) -\left( M \left( \langle R_2 \rangle ^{2n} + 1 \right) - \left( \langle R_1 \rangle ^{2n} - 1 \right) \right) \right] , \\ \varPi _0= & {} k^5\left[ M \left( \langle R_1 \rangle ^{2n} + 1 \right) \left( \langle R_2 \rangle ^{2n} - 1 \right) -\left( \langle R_1 \rangle ^{2n} - 1 \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) \right] \\&\times \,\left[ M (\beta _2 + 1) \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) -(\beta _1 + 1) \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \right] \\&+\, 2 (M-1) k^4\left[ - M \left( \langle R_1 \rangle ^{2n} + 1 \right) \left( \langle R_2 \rangle ^{2n} - 1 \right) +\left( \langle R_1 \rangle ^{2n} - 1 \right) \left( \langle R_2 \rangle ^{2n} + 1 \right) \right] \\&\times \,\left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _1 \frac{K_{n-1}(k)}{K_{n}(k)} \right) \left( \frac{I_{n-1}(k)}{I_{n}(k)} - \beta _2 \frac{K_{n-1}(k)}{K_{n}(k)} \right) . \\ \end{aligned}$$

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Logvinov, O.A. Radial Viscous Fingering in Case of Poorly Miscible Fluids. Transp Porous Med 124, 495–508 (2018). https://doi.org/10.1007/s11242-018-1081-7

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