Motion of a spherical capsule in branched tube flow with finite inertia. (English) Zbl 1383.76569
Summary: We computationally study the transient motion of an initially spherical capsule flowing through a right-angled tube bifurcation, composed of tubes having the same diameter. The capsule motion and deformation is simulated using a three-dimensional immersed-boundary lattice Boltzmann method. The capsule is modelled as a liquid droplet enclosed by a hyperelastic membrane following the Skalak’s law [R. Skalak et al., “Strain energy function of red blood cell membranes”, Biophys. J. 13, No. 3, 245–264 (1973)]. The fluids inside and outside the capsule are assumed to have identical viscosity and density. We mainly focus on path selection of the capsule at the bifurcation as a function of the parameters of the problem: the flow split ratio, the background flow Reynolds number \(Re\), the capsule-to-tube size ratio \(a/R\) and the capillary number \(Ca\), which compares the viscous fluid force acting on the capsule to the membrane elastic force. For fixed physical properties of the capsule and of the tube flow, the ratio \(Ca/Re\) is constant. Two size ratios are considered: \(a/R=0.2\) and 0.4. At low \(Re\), the capsule favours the branch which receives most flow. Inertia significantly affects the background flow in the branched tube. As a consequence, at equal flow split, a capsule tends to flow straight into the main branch as \(Re\) is increased. Under significant inertial effects, the capsule can flow into the downstream main tube even when it receives much less flow than the side branch. Increasing \(Ca\) promotes cross-stream migration of the capsule towards the side branch. The results are summarized in a phase diagram, showing the critical flow split ratio for which the capsule flows into the side branch as a function of size ratio, \(Re\) and \(Ca/Re\). We also provide a simplified model of the path selection of a slightly deformed capsule and explore its limits of validity. We finally discuss the experimental feasibility of the flow system and its applicability to capsule sorting.
References:
| [1] | Barber, J. O.; Alberding, J. P.; Restrepo, J. M.; Secomb, T. W., Simulated two-dimensional red blood cell motion, deformation, and partitioning in microvessel bifurcations, Ann. Biomed. Engng, 36, 10, 1690-1698, (2008) · doi:10.1007/s10439-008-9546-4 |
| [2] | Barber, J. O.; Restrepo, J. M.; Secomb, T. W., Simulated red blood cell motion in microvessel bifurcations: effects of cell – cell interactions on cell partitioning, Cardiovasc. Engng Technol., 2, 4, 349-360, (2011) · doi:10.1007/s13239-011-0064-4 |
| [3] | Barthès-Biesel, D., Motion and deformation of elastic capsules and vesicles in flow, Annu. Rev. Fluid Mech., 48, 25-52, (2016) · Zbl 1356.76459 · doi:10.1146/annurev-fluid-122414-034345 |
| [4] | Barthès-Biesel, D.; Diaz, A.; Dhenin, E., Effect of constitutive laws for two-dimensional membranes on flow-induced capsule deformation, J. Fluid Mech., 460, 211-222, (2002) · Zbl 1066.74023 |
| [5] | Bhujbal, S. V.; De Vos, P.; Niclou, S. P., Drug and cell encapsulation: alternative delivery options for the treatment of malignant brain tumors, Adv. Drug Deliv. Rev., 67-68, 142-153, (2014) · doi:10.1016/j.addr.2014.01.010 |
| [6] | Bouzidi, M.; Firdaouss, M.; Lallemand, P., Momentum transfer of a Boltzmann-lattice fluid with boundaries, Phys. Fluids, 13, 11, 3452-3459, (2001) · Zbl 1184.76068 · doi:10.1063/1.1399290 |
| [7] | Carin, M. L.; Barthès-Biesel, D.; Edwards-Lévy, F.; Postel, C.; Andrei, D. C., Compression of biocompatible liquid-filled hsa-alginate capsules: determination of the membrane mechanical properties, Biotechnol. Bioengng, 82, 2, 207-212, (2003) · doi:10.1002/bit.10559 |
| [8] | Carr, R. T.; Kotha, S. L., Separation surfaces for laminar flow in branching tubes—effect of Reynolds number and geometry, Trans. ASME J. Biomech. Engng, 117, 442-447, (1995) · doi:10.1115/1.2794205 |
| [9] | Charrier, J. M.; Shrivastava, S.; Wu, R., Free and constrained inflation of elastic membranes in relation to thermoforming—non-axisymmetric problems, J. Strain Anal., 24, 55-74, (1989) · doi:10.1243/03093247V242055 |
| [10] | Chien, S.; Tvetenstrand, C. D.; Epstein, M. A.; Schmid-Schonbein, G. W., Model studies on distributions of blood cells at microvascular bifurcations, Am. J. Physiol. Heart Circ. Physiol., 248, 4, H568-H576, (1985) |
| [11] | Chu, T. X.; Salsac, A.-V.; Leclerc, E.; Barthès-Biesel, D.; Wurtz, H.; Edwards-Lévy, F., Comparison between measurements of elasticity and free amino group content of ovalbumin microcapsule membranes: discrimination of the cross – linking degree, J. Colloid Interface Sci., 355, 81-88, (2011) · doi:10.1016/j.jcis.2010.11.038 |
| [12] | Cordasco, D.; Bagchi, P., Orbital drift of capsules and red blood cells in shear flow, Phys. Fluids, 25, 9, (2013) · doi:10.1063/1.4820472 |
| [13] | Dabade, V.; Marath, N. K.; Subramanian, G., The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow, J. Fluid Mech., 791, 631-703, (2016) · Zbl 1382.76267 · doi:10.1017/jfm.2016.14 |
| [14] | Di Carlo, D.; Irimia, D.; Tompkins, R. G.; Toner, M., Continuous inertial focusing, ordering, and separation of particles in microchannels, Proc. Natl Acad. Sci. USA, 104, 48, 18892-18897, (2007) · doi:10.1073/pnas.0704958104 |
| [15] | Doddi, S. K.; Bagchi, P., Lateral migration of a capsule in a plane poiseuille flow in a channel, Intl J. Multiphase Flow, 34, 10, 966-986, (2008) · doi:10.1016/j.ijmultiphaseflow.2008.03.002 |
| [16] | Doyeux, V.; Podgorski, T.; Peponas, S.; Ismail, M.; Coupier, G., Spheres in the vicinity of a bifurcation: elucidating the Zweifach-Fung effect, J. Fluid Mech., 674, 359-388, (2011) · Zbl 1241.76467 · doi:10.1017/S0022112010006567 |
| [17] | Dupont, C.; Salsac, A.-V.; Barthès-Biesel, D.; Vidrascu, M.; Le Tallec, P., Influence of bending resistance on the dynamics of a spherical capsule in shear flow, Phys. Fluids, 27, (2015) · doi:10.1063/1.4921247 |
| [18] | Enden, G.; Popel, A. S., A numerical study of the shape of the surface separating flow into branches in microvascular bifurcations, Trans. ASME J. Biomech. Engng, 114, 398-405, (1992) · doi:10.1115/1.2891401 |
| [19] | Enden, G.; Popel, A. S., A numerical study of plasma skimming in small vascular bifurcations, Trans. ASME J. Biomech. Engng, 116, 79-88, (1994) · doi:10.1115/1.2895708 |
| [20] | Fåhraeus, R., The suspension stability of the blood, Phys. Rev., 9, 241-274, (1929) |
| [21] | Fenton, B. M.; Carr, R. T.; Cokelet, G. R., Nonuniform red cell distribution in 20-100 μm bifurcations, Microvasc. Res., 29, 103-126, (1985) · doi:10.1016/0026-2862(85)90010-X |
| [22] | Fung, Y. C., Stochastic flow in capillary blood vessels, Microvasc. Res., 5, 34-48, (1973) · doi:10.1016/S0026-2862(73)80005-6 |
| [23] | Green, A. E.; Adkins, J. E., Large Elastic Deformations, (1960), Oxford University Press · Zbl 0090.17501 |
| [24] | Gubspun, J.; Gires, P.-Y.; De Loubens, C.; Barthès-Biesel, D.; Deschamps, J.; Georgelin, M.; Léonetti, M.; Leclerc, E.; Edwards-Lévy, F.; Salsac, A.-V., Characterization of the mechanical properties of cross-linked serum albumin microcapsules: effect of size and protein concentration, Colloid Polym. Sci., 294, 1381-1389, (2016) · doi:10.1007/s00396-016-3885-8 |
| [25] | Guo, Z.-L.; Zheng, C.-G.; Shi, B.-C., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E, 65, (2002) · Zbl 1244.76102 |
| [26] | Guo, Z.-L.; Zheng, C.-G.; Shi, B.-C., Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method, Chin. Phys., 11, 366-374, (2002) · doi:10.1088/1009-1963/11/4/310 |
| [27] | Helmy, A.; Barthès-Biesel, D., Migration of a spherical capsule freely suspended in an unbounded parabolic flow, J. Méc. Théor Appl., 1, 5, 859-880, (1982) · Zbl 0517.76107 |
| [28] | Hu, X.-Q.; Salsac, A.-V.; Barthès-Biesel, D., Flow of a spherical capsule in a pore with circular or square cross-section, J. Fluid Mech., 705, 176-194, (2012) · Zbl 1250.76194 · doi:10.1017/jfm.2011.462 |
| [29] | Koleva, I.; Rehage, H., Deformation and orientation dynamics of polysiloxane microcapsules in linear shear flow, Soft Matt., 8, 3681-3693, (2012) · doi:10.1039/c2sm07182g |
| [30] | Lefebvre, Y., Leclerc, E., Barthès-Biesel, D., Walter, J. & Edwards-Lévy, F.2008Flow of artificial microcapsules in microfluidic channels: a method for determining the elastic properties of the membrane. Phys. Fluids20, 123102,1-10. doi:10.1063/1.3054128 · Zbl 1182.76444 |
| [31] | De Loubens, C.; Deschamps, J.; Georgelin, M.; Charrier, A.; Edward-Lévy, F.; Leonetti, M., Mechanical characterization of cross – linked serum albumin microcapsules, Soft Matt., 10, 4561-4568, (2014) · doi:10.1039/c4sm00349g |
| [32] | Masaeli, M.; Sollier, E.; Amini, H.; Mao, W.; Camacho, K.; Doshi, N. T.; Mitragotri, S.; Alexeev, A.; Di Carlo, D., Continuous inertial focusing and separation of particles by shape, Phys. Rev. X, 2, 3, (2012) |
| [33] | Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 3, 220-252, (1977) · Zbl 0403.76100 · doi:10.1016/0021-9991(77)90100-0 |
| [34] | Popel, A. S.; Johnson, P. C., Microcirculation and hemorheology, Annu. Rev. Fluid Mech., 37, 43, (2005) · Zbl 1117.76078 · doi:10.1146/annurev.fluid.37.042604.133933 |
| [35] | Pozrikidis, C., Axisymmetric motion of a file of red blood cells through capillaries, Phys. Fluids, 17, 3, (2005) · Zbl 1187.76425 · doi:10.1063/1.1830484 |
| [36] | Pozrikidis, C., Numerical simulation of cell motion in tube flow, Ann. Biomed. Engng, 33, 2, 165-178, (2005) · doi:10.1007/s10439-005-8975-6 |
| [37] | Pries, A. R.; Secomb, T. W.; Gaehtgens, P., Biophysical aspects of blood flow in the microvasculature, Microvasc. Res., 32, 654-667, (1996) |
| [38] | Rachik, M.; Barthès-Biesel, D.; Carin, M. L.; Edwards-Lévy, F., Identification of the elastic properties of an artificial capsule membrane with the compression test: effect of thickness, J. Colloid Interface Sci., 301, 1, 217-226, (2006) · doi:10.1016/j.jcis.2006.04.062 |
| [39] | Ramanujan, S.; Pozrikidis, C., Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities, J. Fluid Mech., 361, 117-143, (1998) · Zbl 0921.76058 · doi:10.1017/S0022112098008714 |
| [40] | Risso, F.; Carin, M., Compression of a capsule: mechanical laws of membranes with negligible bending stiffness, Phys. Rev. E, 69, 6, (2004) |
| [41] | Roberts, B. W.; Olbricht, W. L., The distribution of freely suspended particles at microfluidic bifurcations, AIChE J., 52, 199-206, (2006) · doi:10.1002/aic.10613 |
| [42] | Rong, F. W.; Carr, R. T., Dye studies on flow through branching tubes, Microvasc. Res., 39, 186-202, (1990) · doi:10.1016/0026-2862(90)90069-4 |
| [43] | Rosén, T.; Lundell, F.; Aidun, C. K., Effect of fluid inertia on the dynamics and scaling of neutrally buoyant particles in shear flow, J. Fluid Mech., 738, 563-590, (2014) · doi:10.1017/jfm.2013.599 |
| [44] | Secomb, T. W.; Styp-Rekowska, B.; Pries, A. R., Two-dimensional simulation of red blood cell deformation and lateral migration in microvessels, Ann. Biomed. Engng, 35, 5, 755-765, (2007) · doi:10.1007/s10439-007-9275-0 |
| [45] | Shrivastava, S.; Tang, J., Large deformation finite element analysis of non-linear viscoelastic membranes with reference to thermoforming, J. Strain Anal., 28, 31-51, (1993) · doi:10.1243/03093247V281031 |
| [46] | Skalak, R.; Tozeren, A.; Zarda, R. P.; Chien, S., Strain energy function of red blood cell membranes, Biophys. J., 13, 3, 245-264, (1973) · doi:10.1016/S0006-3495(73)85983-1 |
| [47] | Sui, Y.; Chew, Y. T.; Roy, P.; Low, H. T., A hybrid method to study flow-induced deformation of three-dimensional capsules, J. Comput. Phys., 227, 12, 6351-6371, (2008) · Zbl 1160.76028 · doi:10.1016/j.jcp.2008.03.017 |
| [48] | Sui, Y.; Low, H. T.; Chew, Y. T.; Roy, P., Tank-treading, swinging, and tumbling of liquid-filled elastic capsules in shear flow, Phys. Rev. E, 77, (2008) |
| [49] | Sui, Y.; Teo, C. J.; Lee, P. S., Direct numerical simulation of fluid flow and heat transfer in periodic wavy channels with rectangular cross-sections, Intl J. Heat Mass Transfer, 55, 73-88, (2012) · Zbl 1235.80028 · doi:10.1016/j.ijheatmasstransfer.2011.08.041 |
| [50] | Svanes, K.; Zweifach, B. W., Variations in small blood vessel hematocrits produced in hypothermic rats by micro-occlusion, Microvasc. Res., 1, 210-220, (1968) · doi:10.1016/0026-2862(68)90019-8 |
| [51] | Woolfenden, H. C.; Blyth, M. G., Motion of a two-dimensional elastic capsule in a branching channel flow, J. Fluid Mech., 669, 3-31, (2011) · Zbl 1225.76100 · doi:10.1017/S0022112010004829 |
| [52] | Wu, W.-Y.; Weinbaum, S.; Acrivos, A., Shear flow over a wall with suction and its application to particle screening, J. Fluid Mech., 243, 489-518, (1992) · doi:10.1017/S0022112092002799 |
| [53] | Yan, Z.-Y.; Acrivos, A.; Weinbaum, S., A three-dimensional analysis of plasma skimming at microvascular bifurcations, Microvasc. Res., 42, 17-38, (1991) · doi:10.1016/0026-2862(91)90072-J |
| [54] | Ye, H.; Huang, H.; Lu, X.-Y., Numerical study on dynamic sorting of a compliant capsule with a thin shell, Comput. Fluids, 114, 110-120, (2015) · Zbl 1390.76787 · doi:10.1016/j.compfluid.2015.02.021 |
| [55] | Zhang, L.; Salsac, A.-V., Can sonication enhance release from liquid-core capsules with a hydrogel membrane?, J. Colloid Interface Sci., 368, 648-654, (2012) · doi:10.1016/j.jcis.2011.11.038 |
| [56] | Zhong-Can, O.-Y.; Helfrich, W., Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders, Phys. Rev. A, 39, 10, 5280, (1989) · doi:10.1103/PhysRevA.39.5280 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
