## 4 Results and Discussion  
In the present work, the flow of the Oldroyd-B fluid was investigated, and the exact expressions of the analytical solutions to non-dimensional equations of temperature and velocity were explored by the application of the Laplace integral transformation with the slipping boundary conditions under Newtonian heating. For many reasons, these exact analytical solutions for dimensionless velocity and temperature are very important. For instance, in order to explore the accuracies of many computed approximate solutions by using numerical techniques, for complex flow phenomena and these solutions in various fields of applied sciences and engineering which have great importance, exact analytical results are mandatory. Thus, closed-form solutions are essential to describe the non-Newtonian fluids' behavior. Many graphs have been portrayed to examine the effects of various physical parameters \(\lambda_{1}\), \(M\), \(Pr\), \(Gr\) and \(\lambda_{2}\). The graphical demonstration for the temperature profile and velocity field were generated to correspond to several connected parameters by using the Mathcad software. For the velocity field solution, all diagrams are plotted corresponding to the slip and no-slip boundary conditions for the Oldroyd-B fluid model. In Figure 2, the effects of the Prandtl number for the temperature profile are displayed. It can be easily deduced from these graphs that the temperature profile decreases in function of the increasing \(Pr\). Generally, the thickness of the thermal outline layer rapidly decreases as the values of \(Pr\) linearly increase due to the deceleration of this temperature curve. In Figures 3-7, the graphs for the velocity field for the slip and no-slip boundary conditions are portrayed for the function \(f(t)=e^{at}\) by taking the values \(a=0.25\) and \(t=1.5\) for all diagrams. Moreover, Figures 8-12 illustrate the graphical behavior of the velocity field for the slip and no-slip boundary conditions for the function \(f(t)=sint\). In Figures 3 and 8, the effects of the Prandtl number \(Pr\) on the velocity field under slip conditions and the corresponding velocity profile under no-slip conditions are shown. It can be noted in these graphs that both the velocity under slip conditions as well as the velocity under no-slip conditions decline with the advancement of the Prandtl number. In Figures 4 and 9, the graphs for the velocity field under slip conditions and the corresponding velocity profile under no-slip conditions are illustrated for the related role of viscosity and the buoyancy forces in the movement of the fluid are depicted. It can be observed that the velocity profile escalated in function of the enhancement of the valuesof \(Gr\) for both cases. When the positive values of \(Gr\) are considered, this causes the fluid temperature to increase, i.e., by turning into free convection currents in the movement of the fluid region. A strong buoyancy force is produced in the flowing region relative to the increasing values of \(Gr\). Due to this strong buoyancy force, all the viscous forces become powerless, leaving one to appreciate the fluid velocity. In Figures 5 and 10, the impacts of the magnetic field \(M\) on the velocity field under slip conditions and the corresponding velocity profile under no-slip conditions are illustrated. It can be easily perceived from these graphs that both the velocity under slip conditions as well as the velocity under no-slip conditions decelerate in function of the increase in the strength of the magnetic field. Resistive type forces are termed Lorentz forces and these are generated due to the imposition of a magnetic field. These forces behave similarly to dragging forces which suppress the forces that help the fluid flow. Consequently, retardation in the fluid flow ultimately causes the deceleration in the motion of the fluid and in due course the fluid comes to a halt.