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The analysis of the double-diffusion model and$\mathbf{H}(\mathrm{div})$-conforming method introduced in [B\"urger, M\'endez,Ruiz-Baier, SINUM (2019), 57:1318--1343] is extended to the time-dependentcase. In addition, the efficiency and reliability analysis of residual-based{\it a posteriori} error estimators for the steady, semi-discrete, and fullydiscrete problems is established. The resulting methods are applied to simulatethe sedimentation of small particles in salinity-driven flows. The methodconsists of Brezzi-Douglas-Marini approximations for velocity and compatiblepiecewise discontinuous pressures, whereas Lagrangian elements are used forconcentration and salinity distribution. Numerical tests confirm the propertiesof the proposed family of schemes and of the adaptive strategy guided by the{\it a posteriori} error indicators.

Divergence-conforming methods for transient doubly-diffusive flows: A priori and a posteriori error analysis