## 4 Results and Discussion  
This portion describes the effects of significant parameters, namely Hartmann number \(M\), Helmholtz-Smoluchowski parameter \(U_{HS}\) and electro-osmotic parameter \(m\), in the presence of hafnium and water suspension in three diverse geometries under consideration. In Figures 4-9, the velocities of fluid and hafnium particles are sketched for Hartmann number, Helmholtz-Smoluchowski parameter and electro-osmotic parameter for all three geometries. Due to diverse shapes and configurations, velocity behaves altogether differently. In the first two geometries, velocity of fluid particle suspension exhibits a rapid increase and it is because of Bernoulli's principle of fluid dynamics whereas, the bulging out corrugation at the center of third geometry causes an opposite behavior in velocity as compared to the preceding geometries. Figure 4a-c, indicates the effects of Hartmann number on the velocity profile. A gradual reduction in fluid velocity that emerges corresponding to the Hartmann number is perceived. It is also observed that magnetic fields influence the motion of fluid transversely. It is not unexpected since the induced magnetic field is regarded as the force of hindrance; consequently, it does not permit the fluid to move with its full strength and speed. Figure 5a-c shows the influence of Hartmann number for paramagnetic order of hafnium particles. This figure illustrates that due to the paramagnetic order of hafnium particles along with magnetic susceptibility being greater than zero, the velocity of the particles resists and repels backward. It is mainly due to randomly oriented magnetic movements of the atoms. Figures 6a-c and 7a-c are displayed to see the variation of the velocity for various values of electro-osmotic parameter. Unlike the Hartmann number, \(M\), the flow of fluid and particles is supported by electro-osmotic parameter, \(m\). It is understandable for first two geometries that the velocity of fluid particle suspension would make a rapid upturn in its momentum due to the shapes and configurations of said geometries. However, the velocity of both fluid and particles intensify in the third geometry, despite the bulging out corrugation at the center. This unique pattern of increase in velocity is induced by the electro-kinetic term present in Equation (27). Effects of velocity for fluid and particles are shown in Figures 8a-c and 9a-c, respectively. It is noted that an increase in the velocity for fluid and particles subject to gear up the Helmholtz-Smoluchowski velocity \(U_{HS}\). The variation in Helmholtz-Smoluchowski velocity results alike in all considered geometries regardless of their forms and configurations. One of the most significant features which describes the pattern of the fluid flow can be observed through stream lines. To serve this purpose, stream lines are displayed in Figures 10-12 for different values of Hartmann number, \(M\), electro-osmotic parameter, \(m\), and Helmholtz-Smoluchowski velocity, \(U_{HS}\). A virtually decline in velocities of both phases upon the rise in Hartmann number is a vivid indication of hindrance in the course of the flow. As the literature suggested that magnetic field lines resist any sort of fluid flow, therefore, increase in the Hartmann number causes the emergence of some additional stream lines in the geometry (1) and geometry (3) as shown in Figure 10a,b,e,f. On the other hand, the velocity of the fluid rapidly attains the utmost momentum for Geometry 2 as shown in Figure 10c,d. Thus the lines of magnetic fields only manage to compress the stream lines inwards. These graphs reveal the indicatives of reduction in the velocity of fluid-particle suspension. Furthermore, that the magnetic field lines act similar to resistance force is also verified. In Figure 11, stream lines are drawn against the electro-osmotic parameter. It is seen that the impact of the electro-osmotic parameter on stream lines varies in each geometry and features a different behavior altogether.