Cone-jet Stokes solutions in strong viscous flows: the vanishing flow rate limit

Steady tip streaming in the limit of vanishing flow rate has been experimentally and numerically documented, yet theoretical solutions describing local conical Stokes flows have remained elusive. Here, we derive approximate analytical solutions for local conical flows in liquid–liquid flow focusing scenarios, addressing the limit of negligible emitted flow rate. Our analysis demonstrates the existence of a universal relationship between the inner-to-outer liquid viscosity ratio and the cone angle, establishing the theoretical underpinning for precise control of microscopic jet formation. A posteriori comparison with previously published experiments reveals that digitised cusp-like meniscus profiles collapse quantitatively onto the predicted slender-body similarity solution. These findings pave the way for technologies that require exact manipulation of fluid flows at nearly molecular dimensions.