Linear oscillations of axisymmetric viscous liquid bridges

Abstract

Abstract. Small amplitude free oscillations of axisymmetric capillary bridges are considered for varying valúes of the capillary Reynolds number C~1 and the slenderness of the bridge A. A semi-analytical method is presented that provides cheap and accurate results for arbitrary valúes of C~1 and A; several asymptotic limits (namely, C< 1,C> l,ACl and \TT -A| <C 1) are considered in some detail, and the associated approximate results are checked. A fairly complete picture of the (fairly complex) spectrum of the linear problem is obtained for varying valúes of C and A. Two kinds of normal modes, called capillary and hydrodynamic respectively, are almost always clearly identified, the former being associated with free surface deformation and the latter, only with the infernal flow field; when C is small the damping rate associated with both kind of modes is comparable, and the hydrodynamic ones explain the appearance of secondary (steady or slowly-varying) streaming flows. Mathematics Subject Classification (2000). 76E99, 76D05, 76D45, 76M25.

Three-dimensional streaming flows driven by oscillatory boundary layers

“…This expression coincides (up to notation differences) with that obtained for unperturbed plañe walls by Lighthill (1992), who extended a former expression obtained by Hunt and Johns (1963) for the particular case F = 0, and also with those obtained for circular cylinders, in the axisymmetric (Nicolás and Vega, 1996;Lyubimov et al, 1997) and non-axisymmetric (Higuera, 1998;Higuera et al, 2002a,b) cases. If instead the outer flow is at rest and the solid boundary oscillates only tangentially to itself, then U = 0, F = 0, and…”

“…Eq. (5.38) coincides (up to notation differences) with those derived by Nicolás and Vega (1996) for this particular geometry, but not with its counterpart used by Lyubimov et al (1997), where Longuet-Higgins's 2D formula was (somewhat loosely) employed for the cylindrical geometry. For large frequeney the oscillatory flow exhibits a short axial wavelength, much smaller than the transversal curvature radius of the cylinder.…”

“…The parameters s, G, T, and R are defined as in Section 2 and again required to satisfy (2.7), which is the basic assumption and is rewritten here for convenience where A is the intrinsic Laplacian operator along the unperturbed boundary (u.b.). As in Section 2 we assume that the homogeneous versión of (3.6)-(3.8) has only the trivial solution; otherwise, the solution depends on arbitrary complex constants (or amplitudes) whose calculation requires to also derive amplitude equations, involving higher order (viscous and/or non-linear) terms (Nicolás and Vega, 1996;Vega et al, 2001;Martin et al, 2002). The unique solution to this inviscid, linear problem can be obtained upon separation of variables for appropriate geometries (or numerically otherwise) and will be considered below as known.…”

“…And the streaming flow produced in the boundary layer attached to a vibrating free boundary is of interest in water wave theory (Phillips, 1977;Liu and Davis, 1977;Craik, 1982Craik, , 1985Iskandarani and Liu, 1991 and references therein) and has been shown to play a role in the instability of the ocean to Langmuir circulations (Leibovich, 1983). These flows have also been studied in connection with capillary waves (Mollot et al, 1993) and in conjunction with thermal eflects (Nicolás and Vega, 1996;Nicolás et al, 1997Nicolás et al, , 1998Lyubimov et al, 1997), intending to control thermocapillary convection (Anilkumar et al, 1993), which is undesirable in materials processing in microgravity (Kuhlmann, 1999). Most of these works dealt with the two-dimensional (2D) case and used the 2D formulae derived by Schlichting (1968) and Longuet-Higgins (1953) for the boundary conditions at the edge of the boundary layers when solving the mean flow equations in the bulk.…”

Oscillations of a liquid bridge resulting from the coalescence of two droplets

“…Experiments that match theoretical predictions are difficult to realize because of practical difficulties to levitate 15 or attach a droplet 16 and of possible contamination by surfactant 17 . Concerning liquid columns, the eigenmodes of a infinite liquid filament have been known for a long time 18 while those of a bridge of finite length have been determined theoretically 19 and investigated experimentally 20 only recently for oscillations around a cylindrical mean shape.…”