2025/05/30 by Alfonso M. Gañán‐Calvo, Ganan-Calvo, Alfonso M., Miguel A. Herrada +3
Engineering · Mathematics · #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #Particle Dynamics in Fluid Flows
paper · doi:10.48550/arxiv.2505.24741
openalex publication_date 2025/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Steady tip streaming in the vanishing flow rate limit has been evidenced both experimentally and numerically in the literature. However, local conical Stokes flow solutions supporting these results at vanishing small scales around the emitting tip have remained elusive. This work presents approximate local conical solutions in liquid-liquid flow focusing and tip streaming, in general, as the limit of a macroscopic vanishing issued flow rate. This provides mathematical foundations for the existence of an asymptotically vanishing scale at the tip of an intermediate conical flow geometry with angle α. For a sufficiently small inner-to-outer liquid viscosity ratio λ, these solutions exhibit a universal power-law relationship between this ratio and the cone angle as α=k λ1/2, where the prefactor k, of the order of unity, depends on the geometric details of the macroscopic flow. This confirms the existing proposals that anticipate the use of flow focusing and tip streaming technologies for tight control of microscopic scales, down to those where diffuse liquid-liquid interfaces become manifested.