Back transport of particles by inertial lift effect

When liquid flow moves upward near the membrane surface, velocity gradient forms as shown in Fig. 1 here, where liquid velocity is zero in the interface and maximum in the center of the flow channel. Since the liquid in the right side of the particle moves faster than that in the left side in the figure, particle spins counter clockwise in the figure. Particle spinning is the most vigorous near the membrane surface due to the steepest velocity gradient. On the contrary, in the center of the channel, particles do not spin at all, but it moves fastest along the stream line.

In terms of the energy, the particles in the center have the highest motion energy, but no spin energy. On the contrary, the particles travelling near the membrane surface have the highest spin energy, but little motion energy. Therefore, the particles travelling somewhere in between the membrane surface and the center of the channel will have the lowest total energy. Once particles happens to get into the lowest energy zone by Brownian motion, shear induced migration, etc., they tend to stay in that area. As a result, particles tend to move away from the membrane surface toward the low energy zone.

Inertial lift can be experimentally verified by monitoring particle distribution in a pipe filled with a flowing medium. As shown in Fig. 1, neutrally buoyant particles flowing in a pipe tend to gather somewhere in between wall and the center and form a donut-shaped cloud. This phenomenon is also known as “tubular pinch effect”.

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Fig. 1. Experimental result demonstrating “tubular pinch effect” in the fluid flowing in pipe at Reynolds number 1,650, where r and R are distance from the center and pipe radius, respectively (Matas, 2004).

Mathematically, the particle Reynolds number should not be negligible to have significant inertial lift effect so that the nonlinear inertial terms in the Navier-Stokes equations play a role (Belfort, 1994). The lift velocity of a rigid, neutrally buoyant, freely rotating particle in a plain Poiseuille flow, vL, is given by

1                ————————————- (1)

where
v= lift velocity (m/s)
rp = particle radius (m)
Um= maximum flow velocity in channel (m/s)

 

© Seong Hoon Yoon