Modified inelastic bouncing ball model of the Brazil nut effect and its reverse

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We developed the modified inelastic bouncing ball model (mIBBM) to describe the emergence of the Brazil nut effect (BNE) and its reverse (RBNE) in a vertically-vibrated binary granular mixture. The mIBBM incorporates the container-to-grains force-transmission efficiency (transmissibility) Tr to quantify the dimensionless mean void lifetime $$ \omega \langle \delta t\rangle $$ that acts as the segregation phase indicator where ω is the vibration angular frequency. The mixture is represented as two non-interacting inelastic balls, Ball A and Ball B. Each ball bounces with a time-of-flight τA (or τB) that depends on transmissibility TrA (or TrB) and the dimensionless container acceleration Γ, i.e., τA = τA(Γ, TrA) and τB = τB(Γ, TrB). The ball dynamics are described by the bifurcation diagrams of the dimensionless times-of-flight, ωτA(Γ, TrA) and ωτB(Γ, TrB). The probability-weighted difference $$ \omega \langle \delta t\rangle $$ between branches of the two diagrams is computed and interpreted as follows: $$ \omega \langle \delta t\rangle > 0 $$ (occurrence of BNE), $$ \omega \langle \delta t\rangle < 0 $$ (RBNE) and $$ \omega \langle \delta t\rangle $$ = 0 (no segregation). Segregation phases are revealed as varying shifts and widths of $$ \omega \langle \delta t\rangle $$ across the Γ axis. The phase boundaries in the $$ \omega \langle \delta t\rangle $$ -versus-Γ diagram are sensitive to changes in TrA, TrB and ΔTr = (TrA − TrB). The mIBBM explains why the BNE is a more likely than the RBNE and predicts a segregation phase sequence that is generally consistent with related experimental results taken over a limited ω-range. Additional experiments are needed to enable a more comprehensive and precise evaluation of the mIBBM.