  In this paper, we show how the two-fluid equations describing the evolution
of a dust and gas mixture can be reformulated to describe a single fluid moving
with the barycentric velocity of the mixture. This leads to evolution equations
for the total density, momentum, the differential velocity between the dust and
the gas phases and either the dust-to-gas ratio or the dust fraction. The
equations are similar to the usual equations of gas dynamics, providing a
convenient way to extend existing codes to simulate two-fluid mixtures without
modifying the code architecture. Our approach avoids the inherent difficulties
related to the standard approach where the two phases are separate and coupled
via a drag term. In particular, the requirements of infinite spatial and
temporal resolution as the stopping time tends to zero are no longer necessary.
This means that both small and large grains can be straightforwardly treated
with the same method, with no need for complicated implicit schemes. Since
there is only one resolution scale the method also avoids the problem of
unphysical trapping of one fluid (e.g. dust) below the resolution of the other.
We also derive a simplified set of equations applicable to the case of strong
drag/small grains, consisting of the standard fluid equations with a modified
sound speed, plus an advection-diffusion equation for the dust-to-gas ratio.
This provides a simple and fast way to evolve the mixture when the stopping
time is smaller than the Courant timestep. We present a Smoothed Particle
Hydrodynamics implementation in a companion paper.
