Distribution of the Dispersed Phase in a Plane Horizontal Channel in Laminar Motion of a Low-Concentration Suspension
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 6, November, 2020
DISTRIBUTION OF THE DISPERSED PHASE IN A PLANE HORIZONTAL CHANNEL IN LAMINAR MOTION OF A LOW-CONCENTRATION SUSPENSION A. V. Ryazhskikh
UDC 621.1.013
The author has formulated the problem of gravity stratification and formation of a sediment of a hydrodynamically low-concentration monodisperse suspension of solid particles not involved in Brownian diffusion and moving in a horizontal plane channel during the laminar flow of a Newtonian dispersed phase. A solution to the initial boundary-value problem for first-order partial differential equations has been obtained in analytical form by applying the one-sided integral Laplace transformation with respect to the axial coordinate. Using the principle of superposition of concentration fields of the fractions, the solution has been generalized to the case of a polydisperse suspension with an arbitrary particle-size-distribution density function. An estimate for the accuracy of physical linearization on replacement of the laminar flow of the carrier medium by an ideal-displacement regime has been given. A comparative analysis of computational experiments with classical experimental data for a wide range of the sedimentation Reynolds number has shown the correctness of the proposed approach. Keywords: concentration field, suspension, laminar flow, plane channel. Introduction. Classical suspensions [1] in which the influence of Brownian diffusion of solid particles is negligible in contrast to nano- and microfluids [2, 3] are rather widespread [4, 6], with suspensions having a low concentration of the dispersed phase being of no practical interest, particularly when they are transported by flow elements of various technical and technological systems [7]. Concrete values of the concentration boundary of the dispersed phase in a suspension, beyond which it can be no longer be considered low-concentrated, are specific and primarily depend on the nature of phases, the form of solid inclusions, temperature, and mechanical actions [8]. If we disregard the action of heat and mass transfer, chemical reactions, and electromagnetic phenomena, this concentration boundary will be determined from the mutual influence of an ensemble of particles through hydrodynamic fields. This estimation can be made, e.g., using a linearized form of Navier– Stokes equations with neglect of inertial terms [9] in the Stimson and Jeffery formulation for the slow motion of two spheres. It follows that if particles of the same diameter are assumed to move in a Newtonian fluid under gravity parallel to each other, the force acting on a particle is ⎛ 3 l⎞ F (1) = − 3πμ f lvs ⎜1 + ⎟, 8 l1 ⎠ ⎝
(1)
and in the case of motion along one generatrix at the distance l2,
⎛ 3 l ⎞ F (2) = − 3πμ f lvs ⎜1 + ⎟. 4 l2 ⎠ ⎝
(2)
Note that for a single particle, expressions (1) and (2) at l1 and l2 → ∞ yield the Stokes formula F = −3πμ f lvs .
(3)
A difference of 1% in the forces (1) and (2) gives respectively l1/l = 37.12 and l2/l = 74.25 (
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