  Phoretic mechanisms, whereby gradients of chemical solutes induce
surface-driven flows, have recently been used to generate directed propulsion
of patterned colloidal particles. When the chemical solutes diffuse slowly, an
instability further provides active but isotropic particles with a route to
self-propulsion by spontaneously breaking the symmetry of the solute
distribution. Here we show theoretically that, in a mechanism analogous to
B\'enard-Marangoni convection, phoretic phenomena can create spontaneous and
self-sustained wall-driven mixing flows within a straight, chemically-uniform
active channel. Such spontaneous flows do not result in any net pumping for a
uniform channel but greatly modify the distribution of transport of the
chemical solute. The instability is predicted to occur for a solute P\'eclet
number above a critical value and for a band of finite perturbation
wavenumbers. We solve the perturbation problem analytically to characterize the
instability, and use both steady and unsteady numerical computations of the
full nonlinear transport problem to capture the long-time coupled dynamics of
the solute and flow within the channel.
