Three regimes of inertial focusing for spherical particles suspended in circular tube flows

<jats:p>An experimental and numerical study on the inertial focusing of neutrally buoyant spherical particles suspended in laminar circular tube flows was performed at Reynolds numbers (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003252_inline1" /><jats:tex-math>$Re$</jats:tex-math></jats:alternatives></jats:inline-formula>) ranging from 100 to 1000 for particle-to-tube diameter ratios of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003252_inline2" /><jats:tex-math>${\sim}0.1$</jats:tex-math></jats:alternatives></jats:inline-formula>. In the experiments, we measured the cross-sectional distribution of particles in dilute suspensions flowing through circular tubes several hundreds of micrometres in diameter. In the cross-section located at 1000 times the tube diameter from the tube inlet, all particles were highly concentrated on one annulus or two annuli, depending on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003252_inline3" /><jats:tex-math>$Re$</jats:tex-math></jats:alternatives></jats:inline-formula>. At low <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003252_inline4" /><jats:tex-math>$Re$</jats:tex-math></jats:alternatives></jats:inline-formula>, the particles were focused on the so-called Segré–Silberberg (SS) annulus, in accordance with previous studies (regime (A)). At higher <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003252_inline5" /><jats:tex-math>$Re$</jats:tex-math></jats:alternatives></jats:inline-formula>, two particle focusing annuli appeared, with the outer annulus corresponding to the SS annulus (regime (B)). We call the annulus closer to the tube centre ‘the inner annulus’, although this term was used by Matas <jats:italic>et al.</jats:italic> (<jats:italic>J. Fluid Mech.</jats:italic>, vol. 515, 2004, pp. 171–195) for a significantly broader annulus which included the transient accumulation of particles observed in regime (A). At even higher <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003252_inline6" /><jats:tex-math>$Re$</jats:tex-math></jats:alternatives></jats:inline-formula>, particles were focused on the inner annulus (regime (C)), indicating that the radial position of the SS annulus is no longer a stable equilibrium position. These experimental results were confirmed by a numerical simulation based on the immersed boundary method. The results of this study also indicate that the critical Reynolds numbers between two neighbouring regimes decrease with the increase of the particle-to-tube diameter ratio.</jats:p>