  We write down the most general membrane equations dual to black holes for a
general class of gravity theories, up to sub-leading order in $1/D$ in large
$D$ limit. We derive a "minimal" entropy current which satisfies a local form
of second law from these membrane equations. We find that consistency with
second law requires the membrane equations to satisfy certain constraints. We
find additional constraints on the membrane equations from the existence of
membrane solutions dual to stationary black holes. Finally we observe a tension
between second law and matching with Wald entropy for dual stationary black
hole configurations, for the minimal entropy current. We propose a simple
modification of the membrane entropy current so that it satisfies second law
and also the stationary membrane entropy matches the Wald entropy.
