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

TL;DR

This paper develops approximate local conical Stokes flow solutions for liquid-liquid tip streaming at vanishing flow rates, providing mathematical insights into the universal scaling laws and supporting microscopic control in flow focusing technologies.

Contribution

It introduces the first approximate local conical solutions in the vanishing flow rate limit, establishing a universal power-law relationship between viscosity ratio and cone angle.

Findings

Universal power-law relationship: α=k λ^{1/2}

Supports microscopic control in flow focusing

Provides mathematical foundation for tip streaming at small scales

Abstract

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 $\alpha$. For a sufficiently small inner-to-outer liquid viscosity ratio $\lambda$, these solutions exhibit a universal power-law relationship between this ratio and the cone angle as $\alpha=k \lambda^{1/2}$, 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.