A unified Stefan–Scott theory for squeeze-film lubrication between non-uniformly deforming boundaries

We present a generalized lubrication theory for the squeeze flow of a fluid film between non-linearly/exponentially stretching or shrinking boundaries, bridging the gap between classical Stefan–Reynolds theory and modern soft-matter kinematics. While existing similarity solutions by Wang [1] and others are restricted to linear boundary velocities ( n = 1 ), they fail to capture the scale-dependent pressure gradients inherent in non-linear stretching ( n ≠ 1 ). By deriving a closed-form analytical solution for the load-carrying force, we demonstrate that the pressure field is governed by a kinematic competition between squeeze-induced compression and stretching-induced suction. We introduce a dimensionless Squeeze–Stretch Number and a modification factor Ω ( n ) that quantify the transition from a positive lubrication load to a suction-induced collapse. Our results reveal a critical radius beyond which stretching-induced suction offsets the squeezing pressure, a phenomenon exacerbated for n > 1 or exponential stretching. Numerical validation using silicone-oil parameters shows that moderate stretching ( a = 5 s − 1 ) can reduce the classic Stefan load capacity by 50% and accelerate film drainage by over 60%. Conversely, we show that centripetal shrinking ( a < 0 ) acts as an inward pump, significantly enhancing film stability. These findings provide a unified theoretical framework for the design of bio-inspired soft robotics, hydrogel implants, and stretchable electronics where boundary deformation and fluid transport are intrinsically coupled.