<p>In this study, we propose an engineering-scale similarity model of magnetohydrodynamics (MHD)-induced unsteady shrinking rotating disk flow subject to Cattaneo–Christov heat flux, Arrhenius chemical reaction, viscous dissipation, joule heating, and temperature-dependent viscosity. The model system constitutes an idealization of thermal machinery operated by mechanical rotation, where flow dynamics, heat transfer on the wall, and entropy generation all govern permissible performance limits and transport efficiency. In the formulation, the shrinking parameter <span class=”math inline”><em>λ</em></span> plays the role of the continuation parameter, whereas the radial wall shear stress <span class=”math inline”><em>F</em><sup>′</sup>(0)</span> is selected as the indicator for the onset of bifurcation. The fold point is analyzed using singularity theory through the identification of singularities, one-dimensional kernels, and transversality conditions; the reduction of the problem through the Lyapunov–Schmidt procedure reveals the canonical square root dependence of the critical parameter around the saddle-node bifurcation point. Using adaptive collocation with continuation, numerical computations are performed to obtain steady-state solutions, wall shears and transfers, and the spectral value associated with the largest time-decay rate of the linearized problem. From the obtained results, a conservative continuation threshold <span class=”math inline”><em>λ</em><sub><em>c</em></sub><sup>scan</sup></span> in agreement with the fold-point behavior is determined, demonstrating the strong influence of the Cattaneo–Christov heat flux model on the temperature and entropy generation distributions but a minor effect on the flow dynamics at the wall. Decomposition of the entropy generation into the thermal, viscous, and magnetic parts and determination of the Bejan number reveal how finite heat transport contributes to entropy generation in the boundary layer.</p>