  In this paper, we introduce the g-B\'enard equations with time-fractional
derivative of order $\alpha \in (0, 1)$ in domains of $\mathbb R^2$. This
equations model, the memory-dependent heat conduction of liquids in fractal
media considered in g-framework. We aim to study the existence and uniqueness
of weak solutions by means of standard techniques from Navier-Stokes equations
theory and fractional calculus theory.
