  Electroconvection and its coupling with a morphological instability are
important in many applications, including electrodialysis, batteries and fuel
cells. In this work, we study the effects of a two-dimensional channel flow on
the electroconvective and morphological instabilities using two approaches. In
the bulk analysis, we consider the instability of the electroneutral bulk
region driven by a second kind electroosmosis slip velocity boundary condition
and derive the asymptotic solutions for small and large wavenumbers. In the
full analysis, we consider the entire region of the liquid electrolyte and use
the ultraspherical spectral method to numerically solve the eigenvalue
problems. Both studies show that the imposed flow significantly affects the
electroconvective instability. The imposed flow generates a shielding effect by
deforming the perturbed ion concentration field and hinders the ion transfer
from low- to high- concentration regions which causes the instability. It fully
suppresses the electroconvective instability at small wavenumbers and reduces
the growth rate of the perturbations at large wavenumbers. The direct effect of
the flow on the morphological instability is minor, while the suppression of
the electroconvective instability may change the wavenumber of the most
unstable mode of the coupled instabilities. For the electroconvective
instability, the bulk analysis is qualitatively different from the full
analysis at high wavenumbers. For the morphological instability, good agreement
is found between the two studies at both small and large wavenumbers.
