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The EOF of a viscoelastic Maxwell fluid driven by an alternating pressure gradient and electric field in a parallel plate microchannel with sinusoidal roughness has been investigated within the Debye–Hückel approximation based on boundary perturbation expansion and separation of variables. Perturbation solutions were obtained for the potential distribution, the velocity and the mean velocity, and the relation between the mean velocity and the roughness. There are significant differences in the velocity amplitudes of the Newtonian and Maxwell fluids. It is shown here that the velocity distribution of the viscoelastic fluid is significantly affected by the roughness of the walls, which leads to the appearance of fluctuations in the fluid. Also, the velocity is strongly dependent on the phase difference <i>θ</i> of the roughness of the upper and lower plates. As the oscillation Reynolds number Re_Ω increases, the velocity profile and the average velocity <i>u_m</i>(<i>t</i>) of AC EOF oscillate rapidly but the velocity amplitude decreases. The Deborah number <i>De</i> plays a similar role to Re_Ω, which makes the AC EOF velocity profile more likely to oscillate. Meanwhile, phase lag <i>χ</i> (representing the phase difference between the electric field and the mean velocity) decreases when <i>G</i> and <i>θ</i> are increased. However, for larger <i>λ</i> (e.g., <i>λ</i> > 3), it almost has no phase lag <i>χ</i>.