<p>We discuss a mathematical model of laminar and turbulent shear flows of liquids and gases in rectangular channels based on a system of differential equations describing the longitudinal motion and rotation of vortex tubes. We show that in the case of a steady planar flow, this system of equations has two-parameter analytical solutions for velocity distributions in the cross-section of the channel, which are in good agreement with known experimental data and the results of numerical simulations. Model approximations of the velocity profiles of laminar flows of non-Newtonian liquids and developed turbulent flows of liquids and gases in rectangular channels are discussed as examples.</p>