## 5 Illustration of the Results and Interpretation  
The transformed Equations (7)-(9) with Equation (10) have been solved using a finite difference method with Bellman-Kalaba quasilinearization, for various values of the involved parameters. The simulated results are plotted in Figures 2-13. The influence of \(Nr\), and \(K\) on velocity profiles is depicted in Figure 2a,b. It is clear from Figure 2a that velocity decreases as the buoyancy ratio rises both with and without pressure force term. Increasing mixed convection parameter leads to acceleration in the flow (increase in velocity) both with and without buoyancy forces. Figure 3a,b show the influences of velocity (momentum) slip, as well as thermal slip on the dimensionless velocity profiles. The dimensionless velocity decreases (the momentum boundary layer is thickened) as the thermal slip parameter rises. The velocity slip parameter, however, leads to an augmentation of the velocity in the presence of a pressure force. The influence of the Darcy number and dimensionless time on the velocity is demonstrated in Figure 4a. Velocity falls as the Darcy number rises for both steady-state and unsteady-state flows. The modification in response is due to the pressure gradient parameter, \(K\), which counteracts the usual accelerating flow accompanying a boost in Darcy number (rise in permeability, \(k_{p}\)) via the corresponding decrease in Darcian body force. The effects of the magnetic field and the Rosseland conduction-radiation parameter are displayed in Figure 4b. Increasing thermal radiation (lower \(R\) values) lessens the velocity for both hydrodynamic (\(M=0\)  
Figure 2: Effects of buoyancy ratio and mixed convection parameters on dimensionless velocity (**a**) with pressure gradient and (**b**) without pressure gradient. and magnetohydrodynamic (\(M\
eq 0\)) flow. It is further noticed that greater magnetic field provides an enhancement of velocity, and this reverses the conventional Lorentz hydromagnetic drag force effect, due to the existence of the free stream which effectively forces the magnetic field to accelerate the flow [50]. Figure 4: Variation of dimensionless velocity with (**a**) Darcy number and dimensionless time and (**b**) with magnetic and radiation parameters. Figure 3: Effects of slip parameters on dimensionless velocity (**a**) with pressure gradient and (**b**) without pressure gradient. this is because as thermal slip parameter \(\delta_{T}\) rises, the fluid at the vicinity of the plate will not be able to sense the influence of heating of the plate, and reduced heat will be released from the plate to the surrounding fluid. Elevation in velocity slip parameter induces an enhancement in the velocity for both the isothermal plate and plate with thermal slip. Figure 7a,b portray the influences of the permeability parameter, dimensionless time, magnetic field and Roseland conduction-radiation parameters, respectively, on temperature distributions. Temperature is enhanced as the dimensionless time increases (Figure 7a). The impact of Darcy number on temperature is insignificant. Figure 7b indicates that _decreasing_ thermal radiation (higher \(R\) values) reduces temperature for both hydrodynamic (\(M=0\)) and magnetohydrodynamic flow (\(M\
eq 0\)). A comparable trend of temperature response was also reported by [PERSON] et al. [51], which provides further corroboration of the physical accuracy of the present simulated results. Increasing the magnetic field yields a reduction in the temperature, i.e., thinner thermal boundary layer, which again is contrary  
Figure 5: Effects of buoyancy ratio and mixed convection parameters on dimensionless temperature (**a**) with pressure gradient and (**b**) without pressure gradient. Figure 6: Effects of slip parameters on dimensionless temperature (**a**) with pressure gradient and (**b**) without pressure gradient.