Entropy generation in water based aluminum oxide and copper hybrid nanofluid flow over a radially stretching solid and porous disk using semi-analytical methods

• Studied entropy generation in hybrid nanofluid flow over a porous, stretchable disk and a non-porous stretchable disk. • Applied both modified AGM and the hybrid analytical and numerical method (HAN method) for solving the governing equations. • Explored effects of suction, injection, and Eckert number on fluid temperature. • Introduced clear graphical tools and pie charts to visualize parameter impacts. • Validated results via semi-analytical solutions and solved ODEs using Python & Maple. A semi-analytical investigation of entropy generation in an incompressible water based aluminum oxide and copper hybrid nanofluid flowing over a radially stretching porous disk is presented. The governing nonlinear PDEs were reduced via similarity variables to two coupled third-order ODEs and solved semi-analytically by a modified Akbari-Gangi method in Python and the hybrid analytical and numerical method (HAN method) in Maple, with direct substitution into the ODEs providing a novel in-situ validation. Parametric studies reveal that increasing the disk stretching rate accelerates both radial and axial velocities and suppresses temperature while amplifying entropy generation by nearly 360 %; variations in Eckert number from –9 to +9 lead to over 130 % temperature drop and a 936 % surge in entropy irreversibility; fluid injection boosts velocity by up to 664 % and temperature by 53 % but produces a non-monotonic entropy response (–62 % then +684 %); fluid suction raises velocity by 764 %, reduces temperature by 38 %, and drives a dramatic 1 926 % increase in entropy generation. Statistical sensitivity analysis shows that suction most strongly influences velocity and entropy generation, injection dominates the velocity gradient, and Eckert number governs thermal irreversibility. A new graphical framework streamlines visualization of these effects, advancing semi-analytical solution techniques and offering deeper insight for entropy-minimized porous-media heat-transfer applications.