One-dimensional reduction of viscous jets. I. Theory

Abstract

We build a general formalism to describe thin viscous jets as one-dimensional objects with an internal structure. We present in full generality the steps needed to describe the viscous jets around their central line, and we argue that the Taylor expansion of all fields around that line is conveniently expressed in terms of symmetric trace-free tensors living in the two dimensions of the fiber sections. We recover the standard results of axisymmetric jets and we report the first and second corrections to the lowest order description, also allowing for a rotational component around the axis of symmetry. When applied to generally curved fibers, the lowest order description corresponds to a viscous string model whose sections are circular. However, when including the first corrections, we find that curved jets generically develop elliptic sections. Several subtle effects imply that the first corrections cannot be described by a rod model since it amounts to selectively discard some corrections. However, in a fast rotating frame, we find that the dominant effects induced by inertial and Coriolis forces should be correctly described by rod models. For completeness, we also recover the constitutive relations for forces and torques in rod models and exhibit a missing term in the lowest order expression of viscous torque. Given that our method is based on tensors, the complexity of all computations has been beaten down by using an appropriate tensor algebra package such as xAct, allowing us to obtain a one-dimensional description of curved viscous jets with all the first order corrections consistently included. Finally, we find a description for straight fibers with elliptic sections as a special case of these results, and recover that ellipticity is dynamically damped by surface tension. An application to toroidal viscous fibers is presented in the companion paper [Pitrou, Phys. Rev. E 97, 043116 (2018)10.1103/PhysRevE.97.043116].

One-dimensional reduction of viscous jets. II. Applications

“…In the next section, we briefly review the main features of our one-dimensional model for viscous fibers which is built in Ref. [1]. We then emphasize the differences with the rod model in § II B.…”

“…However, our model for curved fibers in Ref. [1] consists in including the first corrections that are O( 2 R ). In particular, when including these higher order effects, the shape of the sections cannot simply be described by the radius R as it deforms.…”

“…We gathered the rod model equations and a discussion on their shortcomings in § VII.G.4 of Ref. [1]. To summarize, for the rod model one assumes that fiber sections remain i) circular, ii) orthogonal to the fiber central line, and iii) that the fluid velocity on the central line is such that the fluid particle on the central line stays on the central line [P ⊥ (V ) = 0].…”

“…In our companion article [1], we developed a formalism to describe a viscous fiber as a one-dimensional object with an internal structure. Given the numerical complexity for solving the full set of fluid dynamics equations, taking into account junction conditions at the fiber side for the stress tensor, this description allows for a simpler route for the numerical resolution of viscous fiber dynamics.…”

Formation of Twisted Liquid Jets

“…The major and minor axes of the jet's cross section interchange alternately [1][2][3]. The axis switching of elliptical jets have been studied extensively using theory [4][5][6][7], experiments [8][9][10], and a combination of both [10,11]. For a co-moving observer with the fluid, the axis switching appears as a standing wave of the boundary curve of the cross section [12].…”