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Flow of a semidilute neutrally buoyant and non-colloidal suspension is numerically studied in the Taylor–Couette geometry where the inner cylinder is rotating and the outer one is stationary. We consider a suspension with bulk particle volume fraction $${\phi _b} = 0.1$$ , the radius ratio $$(\eta = {r_i}/{r_o} = 0.877)$$ and two particle size ratios $$\mathrm{\epsilon }\,( = \; d\textrm{/}a) = 60,\;200$$ , where d is the gap width ( $$= {r_o} - {r_i}$$ ) between cylinders, a is the suspended particles’ radius and $$r_i$$ and $$r_o$$ are the inner and outer radii of the cylinder, respectively. Numerical simulations are conducted using the suspension balance model (SBM) and rheological constitutive laws. We predict the critical Reynolds number in which counter-rotating vortices arise in the annulus. It turns out that the primary instability appears through a supercritical bifurcation. For the suspension of $$\mathrm{\epsilon } = 200$$ , the circular Couette flow (CCF) transitions via Taylor vortex flow (TVF) to wavy vortex flow (WVF). Additional flow states of non-axisymmetric vortices, namely spiral vortex flow (SVF) and wavy spiral vortex flow (WSVF) are observed between CCF and WVF for the suspension of $$\mathrm{\epsilon } = 60$$ ; thus, the transitions occur following the sequence of CCF → SVF → WSVF → WVF. Furthermore, we estimate the friction and torque coefficients of the suspension. Suspended particles substantially enhance the torque on the inner cylinder, and the axial travelling wave of spiral vortices reduces the friction and torque coefficients. However, the coefficients are practically the same in the WVF regime where particles are almost uniformly distributed in the annulus by the axial oscillating flow. 

This study explores thermal convection in suspensions of neutrally buoyant, non-colloidal suspensions confined between horizontal plates. A constitutive diffusion equation is used to model the dynamics of the particles suspended in a viscous fluid and it is coupled with the flow equations. We employ a simple model that was proposed by Metzger, Rahli & Yin ( J. Fluid Mech. , vol. 724, 2013, pp. 527–552) for the effective thermal diffusivity of suspensions. This model considers the effect of shear-induced diffusion and gives the thermal diffusivity increasing linearly with the thermal Péclet number ( Pe ) and the particle volume fraction ( ϕ ). Both linear stability analysis and numerical simulation based on the mathematical models are performed for various bulk particle volume fractions $$({\phi _b})$$ ranging from 0 to 0.3. The critical Rayleigh number $$(R{a_c})$$ grows gradually by increasing $${\phi _b}$$ from the critical value $$(R{a_c} = 1708)$$ for a pure Newtonian fluid, while the critical wavenumber $$({k_c})$$ remains constant at 3.12. The transition from the conduction state of suspensions is subcritical, whereas it is supercritical for the convection in a pure Newtonian fluid $$({\phi _b} = 0)$$ . The heat transfer in moderately dense suspensions $$({\phi _b} = 0.2\text{--}0.3)$$ is significantly enhanced by convection rolls for small Rayleigh number ( Ra ) close to $$R{a_c}$$ . We also found a power-law increase of the Nusselt number ( Nu ) with Ra , namely, $$Nu\sim R{a^b}$$ for relatively large values of Ra where the scaling exponent b decreases with $${\phi _b}$$ . Finally, it turns out that the shear-induced migration of particles can modify the heat transfer.