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. 2021 Aug 26;118(35):e2108647118. doi: 10.1073/pnas.2108647118

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Fig. 6. — Continuum modeling of structured bubbling: (A–C) Snapshots of local solids volume fraction predicted by continuum simulations using different constitutive models: (A) Schaeffer model (42), (B) Srivastava and Sundaresan model (43), and (C) the proposed model. (D) Solids pressure produced in discrete particle simulations of structured bubbling. (E and F) Solids pressure produced in continuum simulations of structured bubbling using (E) Srivastava and Sundaresan model (43) and (F) the proposed model. (G) Plot of frictional solids pressure versus solids volume fraction for different constitutive models in continuum simulations for flow conditions observed in structured bubbling. The simulated solids phase has properties ρ_p = 2,500 kg/m³ and d_p = 238 μm. The vibration frequency and amplitude are 5 Hz and 4.5 mm, respectively. U/U_mf = 1.37 for the Schaeffer model (42) and the proposed model, and U/U_mf = 1.25 for the Srivastava and Sundaresan model (43).