Axial flow of incompressible Burgers fluids in an infinite circular cylinder that slides along its symmetry axis with an arbitrary time-dependent velocity is analytically and numerically investigated in the presence of a constant magnetic field. Analytic expressions are established for the dimensionless velocity field and the non-trivial shear stress. For validation, distinct expressions are determined for the fluid velocity and their equivalence in a concrete case is graphically proved. The influence of relaxation time and of Burgers and magnetic parameters on the fluid velocity is graphically highlighted and discussed. It was found that, outside a small vicinity of the symmetry axis of cylinder, the fluid flows slower in the presence of the magnetic field. The oscillatory translational movement of the cylinder induces a motion with oscillating velocity of the fluid. If the cylinder velocity tends to a constant value for large values of the time t, the fluid motion becomes steady in time and the corresponding steady solutions are determined.
