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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