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Stokesian Dynamics Simulation of Suspension Flow in Porous Media

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Abstract

Suspension flow through porous medium was studied using the Stokesian dynamics simulation method. Stokesian dynamics is an efficient tool to carry out numerical simulations for suspension of rigid particles interacting through hydrodynamic and non-hydrodynamic forces. After validating the simulation method for a single particle flowing through an array of fixed grain particles, we have analysed the suspension transport through porous medium. It was observed that the hydrodynamic interactions and the inter-particle non-hydrodynamic forces between the moving and fixed grain particles have a strong influence on the particle trajectories. This was apparent from the change in particle flux with the fractional channel width in the presence of non-hydrodynamic forces. Hydrodynamic interaction between the suspension and grain particles was also studied for large-scale porous system that was generated by a random arrangement of the particles in a periodic cell. It was found that the change in porosity leads to change in the average fluctuation velocity of the suspension. The fluctuation velocity was observed to vary linearly with the particle concentration and average suspension velocity. Finally, a comparative study was performed with suspension flow in a straight channel and it was observed that the shear-induced particle migration in porous medium is altered by the presence of grain particles.

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Abbreviations

\( a \) :

Radius of particle

\( D \) :

Brownian diffusivity of an isolated particle

\( E \) :

Strain tensors of the imposed flow

\( \hat{e} \) :

Unit vector

e〉:

Rate of strain tensor of disturbance flow

\( F \) :

The matrix collectively contains hydrodynamic force and torque on the particles

\( F_{0} \) :

Constant related to the magnitude of the force

\( I \) :

Force density distributed over the particle surface

\( l \) :

Lattice constant

\( M \) :

Mobility tensor

p :

Pressure

\( {\text{Pe}} \) :

Peclet number based on particle size

\( R \) :

Grand resistance tensor

\( T \) :

Torque of the particles

\( u \) :

Translational velocity of particles

u〉:

The average velocity of the suspension

\( U \) :

The matrix collectively contains the translational and rotational velocities of all particles

\( \text{Re}_{\text{p}} \) :

Reynolds number based on particle size

\( R_{\text{FU}}^{ij} \) :

The resistance matrix represents the interactions between \( i \) and \( j \) particles

\( S \) :

The stresslet on the particles

\( V^{'} \) :

Average root-mean-square (RMS) velocity

V〉:

Average velocity

\( \overline{V} \) :

Time-averaged RMS velocity

\( W_{i} \) :

Normalized width of y interval of the channel \( i \)

e:

External force

FE:

Matrix correlates force and strain

FU:

Matrix correlates force and velocity

US:

Correlates velocity and stresslet

ES:

Correlates strain and stresslet

r:

Repulsive force

\( R^{\infty } \) :

Far field

s:

Suspended particle

\( F_{\alpha \beta }^{\text{r}} \) :

Between sphere \( \alpha \) and sphere \( \beta \)

w:

Grain particle

\( \tau \) :

The parameter related to the range of force

\( \phi \) :

Porosity the considered system

\( \varepsilon \) :

The spacing between sphere \( \alpha \) and sphere \( \beta \)

\( \varepsilon_{ijk} \) :

Levi–Civita symbol

\( \varOmega \) :

Rotational velocity

\( \dot{\gamma } \) :

Nominal shear rate

\( \eta \) :

Viscosity of fluid

\( \delta_{ij} \) :

Dirac delta function

\( \rho \) :

Density of fluid

DLVO:

Derjaguin–Landau–Verwey–Overbeek

FCW:

Fractional channel width

FPF:

Fractional particle flux

SD:

Stokesian dynamics

References

  1. Ahfir, N.-D., Benamar, A., Alem, A., Wang, H.: Influence of internal structure and medium length on transport and deposition of suspended particles: a laboratory study. Transp. Porous Med. 76, 289–307 (2009)

  2. Arshadi, M., Zolfaghari, A., Piri, M., Al-Muntasheri, G.A., Sayed, M.: The effect of deformation on two-phase flow through proppant-packed fractured shale samples: a micro-scale experimental investigation. Adv. Water Resour. 105, 108–131 (2017)

  3. Batchelor, G.K., Green, J.T.: The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J. Fluid Mech. 56(2), 375–400 (1972). https://doi.org/10.1017/s0022112072002927

  4. Bedrikovetsky, P.: Upscaling of stochastic micro model for suspension transport in porous media. Transp. Porous Med. 75, 35 (2008)

  5. Benamar, A., Ahfir, N.-D., Wang, H., Alem, A.: Particle transport in a saturated porous medium: pore structure effects. ScienceDirect 339(10), 674–681 (2007)

  6. Boccardo, G., Marchisio, D.L., Sethi, R.: Microscale simulation of particle deposition in porous media. J. Colloid Interface Sci. 417, 227–237 (2014)

  7. Bossis, G., Brady, J.F.: Dynamic simulation of sheared suspensions. I. General method. J. Chem. Phys. 80(10), 5141–5154 (1984)

  8. Bradford, S.A., Torkzaban, S., Leij, F., Šimůnek, J., van Genuchten, M.T.: Modeling the coupled effects of pore space geometry and velocity on colloid transport and retention. Water Resour. Res. (2009). https://doi.org/10.1029/2008wr007096

  9. Brady, J.F., Bossis, G.: Stokesian dynamics. J. Fluid Mech. 20, 111–157 (1988)

  10. Bujurke, N.M., Madalli, V.S., Mulimani, B.G.: Laminar flow in a uniformly porous pipe. Indian J. Pure Appl. Math. 31(3), 341–352 (2000)

  11. Chetti, A., Benamar, A., Hazzab, A.: Modeling of particle migration in porous media: application to soil suffusion. Transp. Porous Med. 113(3), 591–606 (2016)

  12. Davis, R.H., Leighton, D.T.: Shear-induced transport of a particle layer along a porous wall. Chem. Eng. Sci. 42(2), 275–281 (1987). https://doi.org/10.1016/0009-2509(87)85057-1

  13. De, N., Singh, A.: Numerical simulation of particle migration in suspension flow through heterogeneous porous media. Part. Sci. Technol. (2019). https://doi.org/10.1080/02726351.2019.1651806

  14. Durlofsky, L., Brady, J.F., Bossis, G.: Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 21–49 (1987)

  15. Elimelech, M.: Predicting collision efficiencies of colloidal particles in porous media. Water Res. 26(1), 1–8 (1992). https://doi.org/10.1016/0043-1354(92)90104-C

  16. Elimelech, M., O’Melia, C.R.: Kinetics of deposition of colloidal particles in porous media. Environ. Sci. Technol. 24(10), 1528–1536 (1990). https://doi.org/10.1021/es00080a012

  17. Fallah, H., Fathi, H.B., Mohammadi, H.: The mathematical model for particle suspension flow through porous medium. Sci. Res. 2, 57–62 (2012)

  18. Galal, S.K., Elgibaly, A.A., Elsayed, S.K.: Formation damage due to fines migration and its remedial methods. Egypt. J. Pet. 25, 515–524 (2016)

  19. Ghidaglia, C., Arcangelis, L.D., Hinch, J., Guazzelli, E.: Transition in particle capture in deep bed filtration. Phys. Rev. E 53, R3028–R3031 (1996)

  20. Granger, J., Dodds, J., Leclerc, D., Midoux, N.: Flow and diffusion of particles in a channel with one porous wall: polarization chromatography. Chem. Eng. Sci. 41(12), 3119–3128 (1986). https://doi.org/10.1016/0009-2509(86)85049-7

  21. Guazzelli, E., Morris, J.: A Physical Introduction to Suspension Dynamics. Cambridge University Press, Cambridge (2012)

  22. Hassonjee, Q., Ganatoss, P., Pfeffers, R.: A strong-interaction theory for the motion of arbitrary three-dimensional clusters of spherical particles at low Reynolds number. J. Fluid Mech. 197, 1–37 (1988)

  23. Hou, B., Chen, M., Cheng, W., Diao, C.: Investigation of hydraulic fracture networks in shale gas reservoirs with random fractures. Arab. J. Sci. Eng. 41, 2681–2691 (2016)

  24. Ikni, T., Benamar, A., Kadri, M., Ahfir, N.-D., Wang, H.-Q.: Particle transport within water-saturated porous media: effect of pore size on retention kinetics and size selection. C. R. Geosci. 345(9–10), 392–400 (2013)

  25. Juanes, R., Sharma, M.M.: Modeling frac-packs and fracture propagation in poorly consolidated sands. Paper presented at the SPE annual technical conference, Texas (2005)

  26. Kampel, G., Goldsztein, G.H., Santamarina, J.C.: Particle transport in porous media: the role of inertial effects and path tortuosity in the velocity of the particles. Appl. Phys. Lett. 95(19), 194103 (2009)

  27. Kim, S., Karrila, S.J.: Microhydrodynamics: Principles and selected Application. Butterworth - Heinemann Series in Chemical Engineering. Butterworth-Heinmann, Stoneham (1991)

  28. Lee, J., Koplik, J.: Microscopic motion of particles flowing through a porous medium. Phys. Fluid. 11, 76–87 (1999)

  29. Leighton, D., Acrivos, A.: The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439 (1987)

  30. Liu, Q., Prosperetti, A.: Pressure-driven flow in a channel with porous walls. J. Fluid Mech. 679, 77–100 (2011). https://doi.org/10.1017/jfm.2011.124

  31. Nabovati, A., Sousa, A.C.M.: Fluid flow simulation I in random porous media at pore level using the Lattice Boltzmann method. J. Eng. Sci. Technol. 2(3), 226–237 (2007)

  32. Nott, P.R., Brady, J.F.: Pressure-driven flow of suspensions: simulation and theory. J. Fluid Mech. 275, 157–199 (1994)

  33. Otomo, R., Harada, S.: End effect on permeability of loose particulate bed having different internal structures. Part. Sci. Technol. 29(1), 2–13 (2011). https://doi.org/10.1080/02726351.2010.488715

  34. Otomo, R., Harada, S.: Fluid permeability in stratified unconsolidated particulate bed. Transp. Porous Media 96(3), 439–456 (2013). https://doi.org/10.1007/s11242-012-0098-6

  35. Ranjith, P.G., Perera, M.S.A., Perera, W.K.G., Wub, B., Choi, S.K.: Effective parameters for sand production in unconsolidated formations: an experimental study. J. Pet. Sci. Eng. 105, 34–42 (2013)

  36. Sahimi, M., Imdakm, A.O.: Hydrodynamics of particle motion in porous media. Phys. Rev. Lett. 66(9), 1169–1172 (1991)

  37. Santos, A., Barros, P.H.L.: Multiple particle retention mechanisms during filtration in porous media. Environ. Sci. Technol. 44(7), 2515–2521 (2010). https://doi.org/10.1021/es9034792

  38. Santos, A., Bedrikovetsky, P.: A stochastic model for particulate suspension flow in porous media. Transp. Porous Media 62(1), 23–53 (2006). https://doi.org/10.1007/s11242-005-5175-7

  39. Santos, A., Bedrikovetsky, P., Fontoura, S.: Analytical micro model for size exclusion: pore blocking and permeability reduction. J. Membr. Sci. 308(1), 115–127 (2008)

  40. Schwarzer, S.: Sedimentation and flow through porous media: simulating dynamically coupled discrete and continuum phases. Phys. Rev. E 52(6), 6461–6475 (1995)

  41. Sen, T.K., Mahajan, S.P., Khilar, K.C.: Colloid-associated contaminant transport in porous media: 1. Experimental studies. AIChE J. 48(10), 2366–2374 (2002)

  42. Shapiro, A.A., Bedrikovetsky, P.G., Santos, A., Medvedev, O.O.: A stochastic model for filtration of particulate suspensions with incomplete pore plugging. Transp. Porous Media 67(1), 135–164 (2007). https://doi.org/10.1007/s11242-006-0029-5

  43. Sharma, M.M., Yortos, Y.C.: Fine migration through porous media. AIChE J. 33(10), 1654–1662 (1987)

  44. Sharma, M.M., Yortsos, Y.C.: Transport of particulate suspensions in porous media. AIChE J. 33(10), 1636–1643 (1987)

  45. Shen, W., Xu, Y., Li, X., Huang, W., Gu, J.: Numerical simulation of gas and water flow mechanism in hydraulically fractured shale gas reservoirs. J. Nat. Gas Sci. Eng. 35, 726–735 (2016)

  46. Singh, A.: Rheology of Non-colloidal Suspension. Phd. Thesis, Indian Institute of Science (2000)

  47. Singh, A., Nott, P.R.: Normal stresses and microstructure in bounded sheared suspensions via Stokesian dynamics simulations. J. Fluid Mech. 412, 279–301 (2000)

  48. Siyyam, H., Hamdan, M.H.: Analysis of particulate behaviour in porous media. Paper presented at the international conference on FLUID MECHANICS, Mexico, January 25–27 (2008)

  49. Song, W., Jinzhou, Z., Yongming, L.: Hydraulic fracturing simulation of complex fractures growth in naturally fractured shale gas reservoir. Arab. J. Sci. Eng. 39, 7411–7419 (2014)

  50. Taheri, S., Ghomeshi, S., Kantzas, A.: Permeability calculations in unconsolidated homogeneous sands. Powder Technol. 321, 380–389 (2017)

  51. Tufenkji, N., Elimelech, M.: Deviation from the classical colloid filtration theory in the presence of repulsive DLVO interactions. Langmuir 20(25), 10818–10828 (2004). https://doi.org/10.1021/la0486638

  52. Vaz, A., Maffra, D., Carageorgos, T., Bedrikovetsky, P.: Characterisation of formation damage during reactive flows in porous media. J. Nat. Gas Sci. Eng. 34, 1422–1433 (2016)

  53. Vilarrasa, V., Koyama, T., Neretnieks, I., Jing, L.: Shear-Induced flow channels in a single rock fracture and their effect on solute transport. Transp. Porous Med. 87, 503–523 (2010)

  54. Vittal, S., Sharma, M.M.: A Stokesian dynamics model for particle deposition and bridging in granular media. J. Colloid Interface Sci. 153, 313–336 (1992)

  55. Wang, F., Pan, Z., Zhang, Y., Zhang, S.: Simulation of coupled hydro-mechanical-chemical phenomena in hydraulically fractured gas shale during fracturing-fluid flowback. J. Pet. Sci. Eng. 163, 16–26 (2018)

  56. Wang, M., Brady, J.F.: Short-time transport properties of bidisperse suspensions and porous media: a Stokesian dynamics study. J. Chem. Phys. 142(9), 094901 (2015). https://doi.org/10.1063/1.4913518

  57. Wua, K., Chen, Z., Li, X., Xu, J., Li, J., Wang, K., Wang, H., Wang, S., Dong, X.: Flow behavior of gas confined in nanoporous shale at high pressure: real gas effect. Fuel 205, 173–183 (2017)

  58. Yang, X.M., Sharma, M.M.: Formation damage caused by cement filtrate in sandstone cores. SPE Prod. Eng. 23, 399–406 (1991)

  59. Yousif, O.S.Q., Karakouzian, M., Rahim, N.O.A., Rashed, K.A.: Physical clogging of uniformly graded porous media under constant flow rates. Transp. Porous Med. 643(3), 643–659 (2017)

  60. Zhai, Z., Sharma, M.M.: A new approach to modelling hydraulic fractures in unconsolidated sands. Paper presented at the SPE annual technical conference, University of Texas at Austin (2005)

  61. Zhang, Q., Prosperetti, A.: Pressure-driven flow in a two-dimensional channel with porous walls. J. Fluid Mech. 631, 1–21 (2009). https://doi.org/10.1017/s0022112009005837

  62. ZhaoQin, H., Jun, Y., YueYing, W., Ke, T.: Numerical study on two-phase flow through fractured porous media. Sci. China Technol. Sci. 54(9), 2412–2420 (2011)

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Acknowledgements

We would like to acknowledge the anonymous reviewers for providing valuable suggestions. The authors also acknowledge the Indian Institute of Technology Guwahati for providing the high performance computing facilities to carry out the simulations.

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Correspondence to Anugrah Singh.

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Appendix: Formulation of Grand Mobility Matrix

Appendix: Formulation of Grand Mobility Matrix

The velocity field in the fluid around a particle moving at low Reynolds number is given by the Stokes equation:

$$ - \nabla p + \eta \left( {\nabla^{2} u} \right) = 0. $$
(13)
$$ \nabla \cdot u = 0, $$
(14)

where \( p \) is the pressure, \( \eta \) is the viscosity of the fluid, and \( u \) is particle velocity. The momentum equation for a single-point force of strength F located at x is:

$$ - \nabla p + \eta \nabla^{2} u = - F\delta \left( x \right), $$
(15)

where \( \delta \) is Dirac delta function. The disturbance velocity field at any point in the fluid due to the presence of point force is obtained by the solution of Eqs. (14) and (15):

$$ u\left( x \right) = F \cdot \frac{\zeta \left( x \right)}{8\pi \eta }, $$
(16)

where \( \zeta \left( x \right) \) is the free-space Green function or propagator for Stokes flow, known alternatively as the Stokeslet or Oseen tensor, and is given by:

$$ \zeta \left( x \right) = \frac{1}{r}\delta_{ij} + \frac{1}{{r^{3} }}x_{i} x_{j} . $$
(17)

Here \( r = \left| {x_{i} - x_{j} } \right| \) is the resultant distance between the source and field point. The velocity field due to a finite-sized particle can be considered as equivalent to the Stokeslets distributed over the surface of that particle. Now, the disturbance velocity field at any point in the fluid due to N such particles is simply:

$$ u_{i} (x) - u_{i}^{\infty } \left( x \right) = - \sum\limits_{\alpha = 1}^{N} {\int\limits_{{s_{\alpha } }} {\frac{{\zeta_{ij} (x - y)}}{8\pi \eta }(\sigma_{jk} n_{k} )(y){\text{d}}S(y),} } $$
(18)

where \( u_{i}^{\infty } (x) \) is the velocity field at a point \( x \) in the absence of particles, \( S_{\alpha } \) denotes the surface of the particle \( \alpha \), and \( y \) is the position of the point force on the particle surface. \( \sigma_{jk} (y) \) is the fluid stress on the surface, and \( n_{k} (y) \) is the surface normal vector pointing into the fluid. When the particles are in close arrangement, there will be large numbers of surfaces and calculation of velocity will be computationally difficult. To avoid this difficulty, Stokesian dynamics does not evaluate the integral in Eq. (18); instead, a multipole expansion of the integral about the centre of each particle is carried out. This technique starts with the Taylor series expansion of Eq. (18) about the centre of each particle:

$$ \begin{aligned} u_{i} (x) - \mathop u\nolimits_{i}^{\infty } \left( x \right) & = - \frac{{\zeta_{ij} (x - x^{\alpha } )}}{8\pi \eta }\sum\limits_{\alpha = 1}^{N} {\int\limits_{{s_{\alpha } }} {(\sigma_{jk} n_{k} )(y){\text{d}}S(y)} } \\ & + \frac{{{\rm I}_{j} }}{8\pi \eta }\frac{{\partial \zeta_{ij} (x - y)}}{{\partial x_{k} }}(x_{k} - x_{k}^{\alpha } ) + \cdots , \\ \end{aligned} $$
(19)

where \( I_{j} \) is the force density, and \( \eta \) is fluid viscosity. The force density at any point y can be expressed in terms of fluid stress tensor:

$$ I_{j}^{\alpha } = (\sigma_{jk} n_{k} )(y). $$
(20)

The zeroth moment of the multipole expansion is the total force (\( F^{\alpha } \)) exerted by a particle \( \alpha \) on the fluid:

$$ F_{j}^{\alpha } = - \int\limits_{{S_{\alpha } }} {(\sigma_{jk} n_{k} )(y)} {\text{d}}S(y). $$
(21)

The first moment has both symmetric and antisymmetric parts. The symmetric part is called as stresslet \( \left( {S_{jk} } \right) \), and the antisymmetric component is the total torque \( (T_{i}^{\alpha } ) \):

$$ S_{ij}^{\alpha } = - \frac{1}{2}\int\limits_{{S_{\alpha } }} {\left\{ {\left( {y_{i} - x_{i}^{\alpha } } \right)I_{j} + (y_{j} - x_{j}^{\alpha } )I_{i} - \frac{2}{3}\delta_{ij} (y_{k} - x_{k}^{\alpha } )I_{k} } \right\}} {\text{d}}S_{y} , $$
(22)
$$ T_{i}^{\alpha } = - \int\limits_{{S_{\alpha } }} {\varepsilon_{ijk} (y_{j} - x_{j}^{\alpha } )(\sigma_{jk} n_{k} )(y){\text{d}}S_{y} } , $$
(23)

where \( \varepsilon_{ijk} \) is Levi–Civita symbol. In the case of dilute suspension, the hydrodynamic interactions are well represented by the force–torque–stresslet approximation. Nonetheless, the series in Eq. (19) can be easily extended to include the higher order multipole moments for the case of a dense suspension. The nth-order multipole moment for particle α is given as:

$$ Q_{i \ldots j}^{\alpha } = \int\limits_{{s_{\text{p}} }} {(y_{i} - x_{i}^{\alpha } )^{n} (\sigma_{jk} n_{k} )(y){\text{d}}S(y)} . $$
(24)

Thus, the above formulations can be used to determine the disturbance velocity in the fluid due to the presence of large number of particles. Now, the interest is to find out the velocity field of the particles in the fluid due to the presence of other particles. The motion of a spherical particle \( \alpha \) immersed in the suspension is given by Faxén’s formulae (Batchelor and Green 1972):

$$ \mathop u\nolimits_{i}^{\alpha } - \mathop u\nolimits_{i}^{\infty } \left( {\mathop x\nolimits^{\alpha } } \right) = \frac{{\mathop F\nolimits_{i}^{\infty } }}{6\pi \eta a} + \left( {1 + \frac{1}{6}a^{2} \nabla^{2} } \right)\mathop u\nolimits_{i}^{'} \left( {\mathop x\nolimits^{\alpha } } \right), $$
(25)
$$ \mathop \varOmega \nolimits_{i}^{\alpha } - \mathop \varOmega \nolimits_{i}^{\infty } \left( {\mathop x\nolimits^{\alpha } } \right) = \frac{{\mathop T\nolimits_{i}^{\alpha } }}{{8\pi \eta a^{3} }} + \frac{1}{2}\mathop \varepsilon \nolimits_{ijk} \mathop \nabla \nolimits_{j} \mathop u\nolimits_{k}^{'} \left( {\mathop x\nolimits^{\alpha } } \right), $$
(26)
$$ \mathop { - E}\nolimits_{ij}^{\infty } = \frac{{\mathop S\nolimits_{ij}^{\alpha } }}{{\frac{20}{3}\pi \eta a^{3} }} + \left( {1 + \frac{{\mathop a\nolimits^{2} }}{10}\mathop \nabla \nolimits^{2} } \right)\mathop e\nolimits_{ij}^{'} \left( {\mathop x\nolimits^{\alpha } } \right), $$
(27)

where \( u_{i}^{'} (x^{\alpha } ) \) is the velocity disturbance at the centre of particle \( \alpha \) caused by the other particles, i.e., other than the particle \( \alpha \) itself, and relative to the imposed flow \( u_{i}^{\infty } \). Also, \( e_{ij}^{'} \) is the rate of strain of the disturbance flow, \( E_{ij}^{\infty } \) is the rate of strain of the imposed flow, and \( \varOmega^{\infty } \) is the angular velocity of the imposed flow. \( u^{\alpha } \) and \( \varOmega^{\alpha } \) are the translational and angular velocities of particle \( \alpha \), respectively.

Thereafter, applying Eqs. (25), (26), and (27) on each particle in the suspension, a grand mobility matrix, \( M \), can be obtained which relate the translational velocity, angular velocity, and rate of strain tensor of each particle to the force, torque, and stresslet due to N number of particles:

$$ \left( {\begin{array}{*{20}c} {U - U^{\infty } } \\ { - E^{\infty } } \\ \end{array} } \right) = M \cdot \left( {\begin{array}{*{20}c} F \\ S \\ \end{array} } \right), $$
(28)

where \( U - U^{\infty } \) is a 6N vector containing the translational and angular velocities of all particles relative to the disturbed flow, and \( - E^{\infty } \) is a 9N vector that represents the rate of strain for the flow field at the centre of the particles. Besides, \( F \) is a 6N vector containing the force and torque exerted by the particles on the fluid and \( S \) is the 9N stresslet vector. The grand mobility matrix is a positive definite and symmetric matrix, and it can be decomposed into the following submatrix:

$$ M = \left( {\begin{array}{*{20}c} {M_{\text{UF}} } & {M_{US} } \\ {M_{\text{EF}} } & {M_{\text{ES}} } \\ \end{array} } \right), $$
(29)

where the correlation between various components is indicated by the subscripts. \( M_{\text{UF}} \) relates particle velocity to the forces, \( M_{\text{US}} \) relates velocity to stresslet, \( M_{\text{EF}} \) relates the rate of strain tensor to forces, and \( M_{\text{ES}} \) relates the rate of strain tensor to stresslet.

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De, N., Singh, A. Stokesian Dynamics Simulation of Suspension Flow in Porous Media. Transp Porous Med 131, 473–502 (2020). https://doi.org/10.1007/s11242-019-01354-3

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Keywords

  • Stokesian dynamics
  • Lubrication forces
  • Porous media
  • Particle trapping
  • Suspension flow