## 4 Discussion  
Due to the large uses of shear thinning fluids in medicines, especially for the preparation of drugs using oscillatory phenomena to mix the fluid particles to maintain uniformity and other engineering applications, the Williamson fluid is selected from the class of pseudoplastic fluids. Therefore, the recent work has been carried out considering the time-dependent thin film fluid flow of the Williamson fluid model. The study has been carried out on an inclined oscillating plate which is moving with constant velocity \(U\) and a constant magnetic field is applied to the plate perpendicularly in the presence of heat transmission. The physical configuration of the problem has been shown in Figure 1. The modelled equations of velocity and temperature fields have been solved analytically by using OHAM. The effect of various embedded parameters has been discussed in Figures 2-10. The physical presentation of the oscillation effect for the velocity profile has been shown in Figure 2. The 3D figure demonstratesthat initially the fluid oscillates with the plate jointly and, due to the strong cohesive forces of the non-Newtonian Williamson fluid, its amplitude towards the free surface decreases gradually. Figure 3 describes the 3D presentation of the temperature field. Small amplitude has been observed due to the effect of oscillating shear stresses, which has been combined with the energy equation. Fluid distribution during oscillation \(\omega t\) in 2D has been shown in Figure 4. This shows that the fluid flow oscillates together with the plate and this oscillation decreases towards the free surface. Physically, the complexity of non-Newtonian behavior and the strong internal forces among the fluid molecules produce friction forces and, as a result, the amplitude of the fluid flow falls towards the free surface. The effect of the temperature field during oscillation has been shown in Figure 5. The amplitude of the flow increases gradually when the temperature increases. In fact, the cohesive forces among the fluid molecules, which keep these molecules closely compacted, reduce and, as a result, the amplitude of the fluid flow rises. The magnetic effect during fluid motion has been shown in Figure 6. Increasing the magnetic parameter results in decreasing the fluid velocity. The reason for this is that the Lorentz force acts against the flow and, as a result, resistance force increases which reduces the flow motion. The effect of the oscillating parameter \(\omega\) has been revealed in Figure 7. The rising values of the oscillating parameter increase the fluid flow and the effect is similar as in [26; 27]. The effect of Williamson number has been shown in Figure 8. Fluid motion reduces with the rise in Williamson number. The reason for this is that the rise in relaxation time causes higher resistance in the fluid flow and, as a result, the velocity field reduces. Figure 9 shows the effect of the gravitational parameter \(m\) on the velocity field. Increasing the gravitational parameter \(m\) decreases the velocity field. The plate moves in an upward direction and carries a fluid layer in its own direction while the gravitational force acts in the opposite (downward) direction. Therefore, the gravitational force opposes the motion of the fluid and tends to reduce the motion of the fluid particles. Increasing the Eckert number results in an increase in fluid motion because the thermal boundary layer thickness increases by the said increase in the Eckert number. As a result, the cohesive forces decrease which in turn increases the velocity of fluid shown, as shown in Figure 10. Increasing the Williamson number during temperature distribution results in the motion of the fluid increasing as well, as shown in Figure 11. It is noted that the increase is small near the plate and rises more rapidly towards the free surface, in agreement with [26; 27].