This study provides the exact solutions of couple stress fluid (CSF) in channel. The Couple stress fluid is bounded by two plates in which the lower plate is moving with constant velocity Uo and the upper plate is fixed. The influence of external pressure gradient is considered on the CSF fluid. To transform classical CSF model we have introduced three approaches of fractional derivatives, (a) Caputo (b) Caputo-Fabrizio (CF) and (c) Atangana–Baleanu (AB) fractional definitions. The exact solutions have been obtained using the Laplace and finite Fourier sine transforms jointly. Furthermore, the effect of different fractional derivatives is compared and the results are shown in graphs. Moreover, parameters which effect the CSF motion are discussed in the graph. The most important out come from the given study is the comparison of Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional models with classical CSF model. From the comparison it can be noticed that Atangana-Baleanu fractional model discuss the dynamics of the couple stress fluid with good memory effect as compared to Caputo and Caputo-Fabrizio fractional operators. The CSF model can be reduced to Newtonian fluid as limiting case and also investigated solutions in the absence of external pressure. Finally, Skin friction are evaluated for lower and upper plates and presented the obtained results in tabular form.