Cilia serve as sensory organelles extending from cell surfaces, enabling the monitoring of intricate rheological surroundings. The objective of this study is to incorporate the governing equations of the Oldroyd 4-constant model into flows propelled by ciliary motion. Additionally, Maxwell's equations are employed to introduce a body force term within the classical Navier–Stokes equations. The problem is grounded in the assumptions of creeping flow and long wavelengths. The resulting differential equation is simulated using a robust finite difference method in MATLAB R2023a. The obtained solution exhibits convergence and is presented for fluid velocity, pressure rise, and contour lines. The solution is also validated via the shooting method. These results are beneficial in designing artificial magnetic cilia (with similar beating patterns) used for fluid manipulations in lab-on-chip devices.