One-dimensional models for slender axisymmetric viscous liquid jets

Abstract

A one-dimensional analysis of slender axisymmetric viscous liquid jets is considered. A set of one-dimensional models is derived by substituting a truncated Taylor series in the radial coordinate into the Navier–Stokes equations and boundary conditions at the interface. The relative error, defined as the order of magnitude of the neglected terms divided by the order of the retained ones, is small if the dimensionless wave number k is small enough. The Lee slice model is generalized to take into account viscosity, the relative error being k2. A new model having a parabolic radial dependence for the axial velocity is developed, with a relative error k4. The Cosserat model comes from the introduction of the mean axial velocity into the previous one, but an inconsistency arises from neglecting some viscous terms of the same order as those retained. A new model for the mean axial velocity is derived. It conserves the same inertial contribution but avoids the above-mentioned problem by estimating the involved terms instead of neglecting them. Therefore the relative error is k4 for any value of viscosity. Linear stability analysis is performed for the infinite jet. Results are compared with the exact linear solution given by Lord Rayleigh. The main features predicted in the derivation of the one-dimensional models manifest themselves in the linear case.

On the breakup of viscous liquid threads

“…The latter analysis has been given in [15] where unique scaling laws were established by solution of a nonlinear eigenvalue problem (see later). In what follows we show that the similarity solutions of the model equations are identical to those found by a direct analysis of the Stokes system and we test the analytical self-similar structures and in particular the unique scaling exponents, with numerical solutions obtained by solving the initial value problem (19) and (20). This is done in a later Section with excellent agreement.…”

“…The evolution equations (19) and (20) have been proven by Renardy [16] to possess singularities with the jet radius vanishing after a nite time. A Langrangian formulation was used to prove the theorem and in particular it is established that arbitrarily small initial conditions can lead to breakup.…”

“…S 0t + w 0 S 0z + 1 2 w 0z S 0 = 0 ; (19) 3 2 Sw 0 z + 3 S 0 z w 0 z + S 0 z 2 S 0 = 0 : (20) Asymptotic and numerical solutions of (19) and (20) are given in later Sections but some comments on the physical origin of these equations are useful. An integrated form of the evolution equations has been derived previously by M. Renardy [16] by use of physical arguments.…”

“…Our approach is based on working with a simpli ed set of evolution equations which arise from the governing equations after a slender jet ansatz is adopted. One-dimensional slender jet models have recently been used by many authors in the modeling of viscous liquid jets (see for example Renardy [16], Eggers and Dupont [17], Eggers [18], [19], and Garcia and Castellanos [20]). Such approaches are unsteady extensions of the steady ber extrusion problem considered by previous investigators (see for example Schultz and Davis [21] and references therein, as well as the review article by Denn [22]).…”

Capillary breakup of suspensions near pinch-off

“…When the fluid thread is long and thin, the flow is predominantly directed along the axis, so the velocity field is essentially one-dimensional. By expanding the velocity field in the radial direction and taking only the leading-order terms into account, a one-dimensional version of the momentum equations is deduced 11,12 . Since the flow is without a typical length scale near breakup, the concept of selfsimilarity can be successfully applied to this problem to retrieve an analytical solution 13,14 .…”