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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.