Boundary layer flow in the industrial applications such as extrusion processes is attributable to impulsive movement of an extensible moving surface. However, the velocity of the extruded surface may not necessarily be linear. Hence, this study proposes a unique model to investigate the unsteady stagnation point flow in hybrid nanofluid (Al₂O₃-Cu/H₂O) induced by exponentially permeable stretching/shrinking sheet with the magnetic field. The system of ordinary differential equations is attained through a simplification of governing partial differential equations by employing appropriate similarity transformation. The numerical computation is determined by utilising the bvp4c procedure in the MATLAB program, which is proficient in generating non-uniqueness solutions. The noteworthy findings revealed that the skin friction coefficient intensifies in conjunction with the local Nusselt number by enhancing the suction/injection parameter and the nanoparticles volume fraction past an exponentially stretching/shrinking sheet. The results also signified that the increment in the unsteady parameter conclusively increases the skin friction coefficient, and the heat transfer rate shows an upsurge attribution resulting from the inclusion of the magnetic parameter towards an exponentially shrinking sheet. Also, the results are evidenced by having dual solutions, which ultimately lead to an investigation of the analysis of the solution stability, thereby validating the feasibility of the first solution. This crucial contribution will help to enhance industrial growth, especially in the manufacturing and processing sectors.

工业应用（例如挤出工艺）中的边界层流动可归因于可扩展移动表面的脉冲运动。然而，挤压表面的速度不一定是线性的。因此，本研究提出了一种独特的模型来研究混合纳米流体 (Al ₂ O ₃ -Cu/H ₂O) 由具有磁场的指数渗透性拉伸/收缩片引起。常微分方程组是通过采用适当的相似变换对偏微分方程的控制进行简化而获得的。数值计算是利用MATLAB程序中的bvp4c程序确定的，该程序擅长生成非唯一性解。值得注意的发现表明，通过增强吸入/注入参数和纳米粒子体积分数超过指数拉伸/收缩片，皮肤摩擦系数与局部努塞尔数一起增强。结果还表明，非定常参数的增加最终会增加皮肤摩擦系数，并且传热率显示出由于包含磁参数而导致呈指数收缩的片材的激增属性。此外，结果由双重解证明，最终导致对解稳定性分析的调查，从而验证第一个解的可行性。这一重要贡献将有助于促进工业增长，特别是在制造和加工部门。