This study presents a comparative sensitivity analysis of the Hartmann number ( Ha) and Brinkman number ( Br) on magnetohydrodynamic (MHD) flow in rectangular channels and circular pipes. Normalized sensitivity coefficients quantify the response of key metrics, including velocity, wall shear stress, temperature, and convective heat transfer, with validation against recent experimental and numerical studies. The system equations were solved through a coupled analytical–numerical method coded in Python 3.14; velocity field was solved analytically whereas temperature field was discretized using a finite differences scheme and solved numerically using the Thomas algorithm. The entire code was written by the authors. The results show that Ha predominantly governs hydrodynamics, inducing velocity suppression, flow flattening, and enhanced wall shear stress. Rectangular channels experience stronger Hartmann layer effects, while circular pipes exhibit smoother velocity profiles. Conversely, Br primarily controls thermal behavior, with higher values intensifying internal heat generation and elevating centerline temperature, potentially attenuating the average Nusselt number at high Br levels. Nonlinear Ha–Br interactions define distinct operational regimes, from heat transfer enhancement to thermal degradation. Optimal performance windows are identified: Ha ≈ 8–12 and Br ≈ 0.05–0.3 for channels, and Ha ≈ 10–15 and Br ≈ 0.1–0.4 for pipes, balancing thermal and hydraulic efficiency. Deviations from benchmark studies remain within ±5%, confirming predictive reliability. This work provides practical design guidance for advanced MHD thermal systems and establishes a foundation for future studies on temperature-dependent properties, three-dimensional effects, and complex flow regimes. 6. Comparative Sensitivity Analysis and Validation This section presents a rigorous comparative sensitivity analysis of the Hartmann ( Ha) and Brinkman ( Br) number effects on the hydrodynamic and thermal behavior in both rectangular channel and circular pipe configurations. The analysis is contextualized with recent experimental and numerical studies to establish validity ranges and identify critical parameter interactions. 6.1. Methodology for Sensitivity Quantification The sensitivity of key performance metrics to governing parameters is quantified using normalized sensitivity coefficients defined as: S P Q = ∂ Q ∂ P ୍ଠ P r e f Q r e f . (29) This equation represents the dimensionless entropy generation (or energy dissipation) in a magnetohydrodynamic (MHD) flow system. It shows that the total irreversibility ( S ) is composed of two main contributions: viscous dissipation associated with velocity gradients ( ∇ U 2 ) , and magnetic (Joule) dissipation proportional to the square of the velocity field ( U 2 ) and the Hartmann number ( H a ) . Brinkman number ( B r ) scales the relative importance of viscous heating compared to thermal conduction. The equation highlights how both fluid friction and electromagnetic effects contribute to entropy generation in conducting fluid flows under magnetic fields, where ( Q) represents the output metric (e.g., Nusselt number, maximum velocity, centerline temperature), ( P) is the input parameter ( Ha or Br), and the subscript (ref) denotes reference values at Ha = 5, Br = 0.05 for a consistent comparison. Reference Computational Conditions: Reynolds number: Re = 100 (laminar regime); Prandtl number: Pr = 0.71 (air) for channel, Pr = 7.0 (water) for pipe; Wall boundary condition: Constant wall temperature (CWT). Magnetic field orientation: Perfectly transverse and uniform. Effects of Reynolds and Prandtl numbers were evaluated by varying both in the ranges Re = 100–600 and Pr = 0.7–7.0. The velocity field is unaffected by either Re or Pr when assuming fully developed flow, since only Ha appears in the governing momentum equation. Nu demonstrates weak Pr dependence from bulk temperature weighting: fluids with a larger Pr (e.g., water with Pr = 7) have an ~8–12% larger Nu than air ( Pr = 0.7) at the same HaBr, qualitatively aligning with enhanced convection. Re only enters through definition of the Brinkman number with reference to flow velocity scale. At fully developed laminar conditions, the average Nusselt number asymptotes and theoretically becomes independent of Re for a given geometry and boundary condition. In contrast, the effective Br couples to Re through the normalization with velocity. As Re increases, the dimensionless heat generation increases, which leads to increases in the bulk temperature and corresponding decreases in Nu. Thus, this coupling of Re–Br means that simply increasing flow velocities without considering the applied magnetic field strength can unintentionally operate the system from an enhancement regime to one of thermal degradation. 6.2. Parameter Sensitivity Matrix Normalized sensitivity coefficients for key performance metrics are shown in Table 8. 6.3. Comparative Analysis of Ha and Br Effects Plotting various transport metrics of interest against the Hartmann ( Ha) and Brinkman (Br) numbers reveals distinct hydrodynamic and thermal regimes governed by the interplay of magnetic forces and internal heating. Analysis of the Hartmann number indicates three primary flow regimes common to both channel and pipe geometries depending on the applied magnetic field strength. In the low Hartmann regime (0 15) drives both configurations toward a plug-flow state, exceeding 85% velocity flattening. Here, channel flows develop exceedingly thin Hartmann boundary layers where δ/L 20. In parallel, the Brinkman number exhibits a strong, non-linear coupling with the Hartmann number, resulting in three distinct thermal operating regimes based on the intensity of internal heat generation. Under weak internal heating conditions ( Br≤ 0.01), Nu increases monotonically with Ha for both configurations, with pipe flows experiencing a heating rate approximately 15–20% greater than channel flows. Because the overall temperature rise remains suppressed below 5% of the reference temperature T r e f , either geometry proves highly effective for precision cooling applications. Under moderate internal heating (0.01 0.5), the dominant internal heat generation causes Nu to decrease with increasing Br across all Hartmann numbers. In this severe regime, the temperature rise exceeds 50% of T r e f , and the system becomes highly susceptible to catastrophic thermal runaway if Br surpasses 2.0, unless auxiliary cooling or active thermal management strategies are integrated. 6.4. Validation Against Recent Experimental and Numerical Studies The regime map presented in Figure 7 is physically consistent with trends reported in previous experimental and numerical investigations of magneto hydrodynamic (MHD) heat transfer with viscous dissipation and Joule heating effects. In particular, the observed dependence of heat transfer enhancement on the Hartmann number ( Ha) and Brinkman number ( Br) aligns well with the analytical and numerical findings are demonstrated that the interaction between magnetic damping, viscous dissipation, and Joule heating leads to non-monotonic variations in the Nusselt number ( Nu) and the emergence of optimal operating conditions. Furthermore, recent studies on MHD microchannel flows under thermal confirm that increasing viscous dissipation (higher Br) can significantly alter thermal performance and entropy generation, particularly in regimes where Joule heating becomes dominant. These findings support the existence of high- Br regimes (Regions III and IV), where thermal degradation or complex multi-physics interactions are observed. The enhancement of heat transfers at moderate Hartmann numbers, as identified in Region II, is also consistent with the broader literature on MHD convection, where magnetic fields are known to suppress velocity fluctuations while simultaneously enhancing thermal gradients under certain conditions. This balance leads to an optimal range of Ha in which heat transfer is maximized before excessive magnetic damping reduces flow effectiveness. Although the exact regime boundaries presented in Figure 7 are specific to the current formulation and geometry, the overall trends and physical interpretations are strongly supported by existing studies. Therefore, the proposed operational map provides a consistent and physically grounded framework for understanding the coupled effects of magnetic field strength and viscous dissipation in MHD channel and pipe flows. 6.5. Uncertainty Analysis and Error Propagation Uncertainty analysis and error propagation studies showed that the dominant sources of uncertainty for the predicted heat transfer behavior include uncertainties due to temperature-dependent variations of thermophysical properties (~±5%), magnetic field non-uniformity due to practical limitations (~±3%), effects of finite wall conductivity (~±4%) and measurement errors (~±2%) under normal operating conditions. The combined uncertainty associated with the predicted Nusselt number was determined using the root-sum-square (RSS) method as: δ N u t o t a l = ∑ i = 1 n δ N u i 2 (30) This equation defines the total uncertainty (or overall error propagation) in the Nusselt number evaluation. It indicates that the combined uncertainty ( δ N u t o t a l ) is obtained by the root-sum-square of the individual uncertainties ( δ N u i ) , assuming that the errors from different sources are statistically independent. This formulation is commonly used in experimental and numerical heat transfer analysis to quantify the overall reliability of calculated heat transfer coefficients. With this method, the overall predicted uncertainty in Nu was about ±6.2% for channels and ±5.8% for pipes under normal operating conditions. Uncertainties of this size are acceptable for most engineering applications.
