Convective and absolute instability of viscoelastic liquid jets in the presence of gravity

Abstract

The convective and absolute instability of a viscoelastic liquid jet falling under gravity is examined for axisymmetrical disturbances. We use the upper-convected Maxwell model to provide a mathematical description of the dynamics of a viscoelastic liquid jet. An asymptotic approach, based on the slenderness of the jet, is used to obtain the steady state solutions. By considering traveling wave modes, we derive a dispersion relation relating the frequency to the wavenumber of disturbances which is then solved numerically using the Newton-Raphson method. We show the effect of changing a number of dimensionless parameters, including the Froude number, on convective and absolute instability. In this work, we use a mapping technique developed by Kupfer, Bers, and Ram ["The cusp map in the complex-frequency plane for absolute instabilities," Phys. Fluids 30, 3075-3082 (1987)] to find the cusp point in the complex frequency plane and its corresponding saddle point (the pinch point) in the complex wavenumber plane for absolute instability. The convective/absolute instability boundary is identified for various parameter regimes.

Dripping, jetting and tip streaming

“…Leib and Goldstein [167] studied the absolute instability of a jet in a mechanically inert ambient, while Lin and Lian [173] took into account the effect of the surrounding medium. Subsequently, more complex configurations have been considered by several authors, including the effect of a small inner-to-outer density ratio [149,174], a viscous coflowing current [31,175,176], confinement in various geometries [148,177], jet swirling [178], a twofold interface [179,180], viscoelasticity [179][180][181], and gravity [181] among others. In many cases, the dispersion relationship can be derived analytically, while in others the linearized equations are spatially discretized.…”

Non-Newtonian and viscoplastic models of a vertically aligned thick liquid film draining due to gravity

“…The theoretical framework developed here is versatile and can be readily adapted to accommodate other complex liquids, such as viscoelastic liquid films. Indeed, one can use the form of the evolution equations written in terms of the extensional stresses, τ xx , τ zz and the shear stress τ xz , and use a constitutive model appropriate for a viscoelastic liquid, e.g., Oldroyd-B model 32,33 to relate these stresses to their corresponding shear rates. This will be investigated in future.…”

Absolute instability of free-falling viscoelastic liquid jets with surfactants

“…Graphs showing values of ζ 0 , vz 0 , T 0 rr , and T 0 zz against z for different values of F, where the solid lines are with surfactants (present work with η = 0.1 and γ = 0.2) and the dotted lines are without surfactants (obtained by Alhushaybari and Uddin 33 with η = γ = 0). Here, De = 10, Re = 800, We = 2, and α = 0.03.…”

“…We then substitute these expressions into the last set of equations and equate coefficients of O(1) to zero to obtain a system of nonlinear equations. These can be solved to establish a consistent set of boundary conditions as z → 0 (see Alhushaybari and Uddin 33 for more details). We solve the nonlinear ordinary differential equations, (21)- (23), by using the built in MATLAB package (ode45), which is based on the Runge-Kutta method.…”