In this paper we present a topology optimization technique applicable to a broad range of flow design problems. We propose also a discrete adjoint formulation effective for a wide class of Lattice Boltzmann Methods (LBM). This adjoint formulation is used to calculate sensitivity of the LBM solution to several type of parameters, both global and local. The numerical scheme for solving the adjoint problem has many properties of the original system, including locality and explicit time-stepping. Thus it is possible to integrate it with the standard LBM solver, allowing for straightforward and efficient parallelization (overcoming limitations typical for discrete adjoint solvers). This approach is successfully used for the channel flow to design a free-topology mixer and a heat exchanger. Both resulting geometries being very complex maximize their objective functions, while keeping viscous losses at acceptable level.

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The shear-induced migration of dense suspensions with continuously distributed (polydisperse) particle sizes is investigated in planar channel flows for the first time. A coupled lattice Boltzmann–discrete element method numerical framework is employed and validated against benchmark experimental results of bulk shear-induced migration and segregation by particle size. Distinct dependence on the particle size distribution is shown in the flowing (non-plugged) regime (where the bulk solid volume fraction, $\bar{\phi}$ , $\leq 0.3$ ) resulting from a dual dependence on the particle self-diffusivity and local rheology imposed by the particle pressure gradient. Close agreement between statistically equivalent bidisperse and polydisperse suspensions suggests that the bulk migration, and by extension the shear-induced diffusivity, is completely characterised by the first three statistical moments of the particle size distribution. For both bidisperse and polydisperse suspensions in the plugging regime, $\bar {\phi }\geq 0.4$ , the smallest particles preferentially form the plugs, causing the largest particles to segregate to the channel walls. This effect is accentuated as $\bar {\phi }$ increases and has not been reported in the literature hitherto. It is proposed that smaller particles preferentially form the plugs due to their higher shear-rate fluctuations, which completely dominate particle motion near the plug where the mean shear rate vanishes. Finally, increasing inertia causes a greater bulk migration towards the channel walls, but increased mid-plane migration for the largest particles due to the dependence of the particle self-diffusivity on the particle Reynolds number. As $\bar {\phi }$ increases shear-induced migration dominates and these inertial effects disappear, as does dependence on the particle size distribution.