## 5 Illustration of the Results and Interpretation
to the conventional flat plate hydromagnetic boundary layer flow in a constant free stream (wherein kinetic energy is dissipated as heat in the boundary layer inducing a thicker thermal boundary layer). Again, this deviation is due to the free stream effect in the present model. The impact of the buoyancy ratio (\(N\)r), mixed convection and pressure force parameters (\(K\)) on the dimensionless nanoparticle concentration profiles is illustrated in Figure 8a,b. Figure 8a shows that the dimensionless concentration reduces with buoyancy ratio both in the presence and absence of pressure force. The mixed convection parameter mobilizes a rise in the concentration both with and without buoyancy forces. The higher buoyancy ratio, therefore, decreases concentration boundary layer thickness, whereas, higher mixed convection parameter enhances it. Figure 8: Effects of buoyancy ratio and mixed convection parameters on dimensionless concentration (**a**) with pressure gradient and (**b**) without pressure gradient. Figure 7: Effects of several parameters on dimensionless temperature with pressure gradient. Figure 9a,b have been drawn to show the impact of velocity slip and thermal slip on the concentration profiles. Concentration increases as thermal slip parameter increase both in the presence and absence of velocity slip. The velocity slip parameter leads to an enhancement of the concentration for either an isothermal plate or a plate with a thermal slip. All profiles in Figures 1-9 converge asymptotically confirming that a sufficiently large far-field boundary condition is used during the computations. We now appraise the influences of various thermophysical parameters on friction, the heat transfer rates, and mass transfer rates. Figure 10a,b show the influences of dimensionless time, frequency, and mixed convection parameters on friction factor with/without a pressure gradient, respectively. It is observed that friction is accentuated as mixed convection and frequency parameters increase both with and without a pressure force term. As expected, the dimensionless time reduces friction factor. In other words, with progression in time, the nanoscale boundary layer flow is decelerated, and the momentum boundary layer thickness is increased. Figure 10: Effects of dimensionless time, frequency, and mixed convection parameters on skin friction (**a**) with pressure gradient and (**b**) without pressure gradient. Figure 9: Effects of slip parameters on dimensionless velocity (**a**) with pressure gradient and (**b**) without pressure gradient. Figure 11a,b are portrayed to show the effect of Darcy number, velocity slip, and magnetic field parameters on skin friction for a _moving_ free stream and a _stationary_ free stream, respectively. As expected, slip velocity reduces the friction factor for both a moving and quiescent free stream. This is due to the slope of the velocity at the wall decreasing, which can be perceived from the boundary condition \(f^{\prime}(0)=\delta_{v}f^{\prime\prime}\left(0\right)\). As \(\delta_{v}\rightarrow\infty\) (full slip), the shear stress at the wall is close to zero. Increasing Darcy number decreases friction factor when free stream velocity is moving (\(K=1\)), and the opposite trend is noticed for a stagnant free stream (\(K=0\)). Elevation in the magnetic field parameter increases the friction when the free stream velocity is moving (\(K=1\)), and the opposite trend is observed for a calm, free stream (\(K=0\)). Figure 12a,b display the influence of time, frequency, and mixed convection parameters on heat transfer rates. The heat transfer rate is elevated as mixed convection, and frequency parameters increase for both the moving free stream and stationary free stream. However, Figure 12b indicates that heat transfer rate is decreased with larger values of the dimensionless time. Figure 11: Effects of Darcy number, velocity slip, and magnetic parameters on skin friction (**a**) with pressure gradient and (**b**) without pressure gradient.