# Mathematical modeling of particle aggregation and sedimentation in the inclined tubes

### Abstract

Sedimentation of the aggregating particles in the gravity field is widely used as an easy and cheap test of the suspension stability of different technical suspensions, blood and nanofluids. It was established the tube inclination makes the test much faster that is known as the Boycott effect. It is especially important for the very slow aggregating and sedimenting blood samples in medical diagnostics or checking the ageing of the nanofluids. The dependence of the sedimentation rate on the angle of inclination is complex and poorly understood yet. In this paper the two phase model of the aggregating particles is generalized to the inclined tubes. The problem is formulated in the two-dimensional case that corresponds to the narrow rectangle vessels or gaps of the viscosimeters of the cone-cone type. In the suggestion of small angles of inclination the equations are averaged over the transverse coordinate and the obtained hyperbolic system of equations for is solved by the method of characteristics. During the sedimentation the upper region (I) of the fluid free of particles, the bottom region (III) of the compactly located aggregates without fluid, and the intermediate region of the sedimenting aggregates (II) appear. The interface between I and II can be registered by any optic sensor and its trajectory is the sedimentation curve. Numerical computations revealed the increase in the initial concentration of the particles, their aggregation rate, external uniform force and inclination angle accelerate the sedimentation while any increase in the fluid viscosity decelerates it that is physically relevant. Anyway, the behaviors of the acceleration are different. For the elevated force the interfaces I-II and II-III shifts uniformly, while for the elevated concentration or aggregation rate the interface I-II or II-III moves faster. Small increase of the inclination angle accelerates the sedimentation while at some critical angles is starts to decelerate due to higher shear drag in the very viscous mass of the compactly located aggregates. Based on the results, a novel method of estimation of the suspension stability is proposed.

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### References

A. E. Boycott. Sedimentation of blood corpuscles, Nature, - 1920. -– V. 104, 532 p.

S. A. P. Moys. Sedimentation of polydisperse particles at low Reynolds numbers in inclined geometries, PhD Thesis. Santiago de Chile, 2016. - 78 p.

P. Hanson, T. Trigg, G. Rachal, M. Zamora. Investigation of Barite Sag in Weighted Drilling Fluids in Higlu Deviated Wells, SPE paper 20423 presented at the SPE Annual Conference and Exhibition, New Orlean, 1990. - Sept. - P. 23-26.

J. Duran, T. Mazozi. Granular boycott effect: how to mix granulates, Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics, 1999. - Nov. –- 60 (5 Pt B):6199 –- 201 p.

T. Peacock, F. Blanchette, J. W. M. Bush. The stratified Boycott effect, J. Fluid Mech, 2005. - 529. - P. 33-49.

R. S. MacTaggart, D. H-S. Law, J. H. Masliyah, K. Nandakumar. Gravity separation of concentrated bidisperse suspensions in inclined plate settlers, Intern. J. Multiphase Flow, 1988. - 14(4):519. - 532 p.

E. Fukada, T. Azuma. Erythrocyte sedimentation rate II. Effects of tilt angle in saline solution, Biorheology, 1988. - 1-2. V. 25. - P. 157-164.

A. Acrivos, E. Herbolzheimer. Enhanced sedimentation in settling tanks with inclined walls, J. Fluid Mech., 1979. –- 3. V.92. - P. 435–457.

H. Lamb. Hydrodynamics, 1932. Cambridge University Press.

Eric Ponder. On Sedimentation and rouleaux formation, Experimental Physiology, 1926. - 2. V. 16. - P. 173–194.

N. Nakamura, K. Kuroda. La cause de l’acceleration de la vitesse de sedimentation des suspensions dans les recipients inclines, Keijo J. Med., 1937. - V. 8. - P. 256–296.

W. D. Hill, R. R. Rothfus, K. Li. Boundary-enhanced sedimentation due to settling convection, Int. J. Multiphase Flow, 1977. - V. 3. - P. 561–583.

R. F. Probstein, R. E. Hicks. Lamella settlers: a new operating mode for high performance, Ind. Water Eng., 1978. - V. 15. - P. 6–8.

I. Rubinstein. A steady laminar flow of a suspension in a channel, International Journal of Multiphase Flow, 1980. - 5. V. 6. - P. 473–490.

W. F. Leung, R. F. Probstein. Lamella and tube settlers. 1. Model and operation, Industrial & Engineering Chemistry Process Design and Development, 1983. - 1. V. 22. - P. 58–67.

Y. Toyama, T. Dobashi, A. Sakanishi, S. Oka. Enhanced erythrocyte sedimentation rate and upflow layer in inclined rectangular vessel, Jap. J. Appl. Phys., 1990. - 2. V. 29. - P. 453-454.

J. Zhang, Q. Guo, M. Liu, J. Yang. A lab-on-CD prototype for high-speed blood separation, J. Micromech. Microeng., 2008. - 12. V. 18:125025.

S. Chakraborty. Microfluidics and Microscale Transport Processes, 2012. CRC Press., - 366 p.

Z.-J. Xu, E.E. Michaelides. A Numerical Simulation of the Boycott Effect, Chemical Engineering Communic., 2005. - 4. V. 192. - P. 532-549.

N. Kizilova, L. Batyuk, V. Cherevko. Human Red Blood Cell Properties and Sedimentation Rate: a Biomechanical Study, Biomechanics in Medicine and Biology: Proceedings of the International Conference of the Polish Society of Biomechanics, Zielona Gora, 2018. - September 5-7. - Poland. K. Arkusz, R. Bedzinski, T. Klekiel, S. Piszczatowski, eds. Springer Series “Advances in Intelligent Systems and Computing.” - V. 831, - 2019. - P. 3-22.

N. N. Kizilova, V. A. Cherevko. Gravitational sedimentation of erythrocytes: experiments and theoretical model, Visnyk of V.N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, 2009. - V. 875. - P. 80-94.

Y. S. Losev. Some problems of hydromechanics of suspensions with varying densities: application to the blood. PhD Thesis, 1984. - Moscow University. - 135 p.

O. M. Datsok, Ye. N. Zholonsky, N. N. Kizilova. Two-phase model of the erythrocytes sedimentation in a non-uniform force field, Visnyk Kharkov Polytechnic University, 2002. - V. 135. - P. 61-66.

N. N. Kizilova. Effect of radial motion of erythrocytes on their sedimentation in a tube in an external magnetic field, Fluid Dynamics, 1991. - 5. V. 26. - P. 737-744.

V. A. Baranets, N. N. Kizilova. Discrete modeling of aggregation and sedimentation of nanoparticles in suspensions, Visnyk of V.N. Karazin Kharkov National University. Ser. “Mathematical modeling. Information technologies. Automated control systems”, 2018. - V. 38.

*Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics*,

*90*, 42-59. https://doi.org/10.26565/2221-5646-2019-90-03

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