The temporal capillary instability of a viscoelastic liquid jet is studied by using weakly nonlinear stability analysis.

The jet is supposed axisymmetric, of infinite length and evolving in an isotherm dynamically inert ambient medium. The liquid is considered incompressible and is represented by the Oldroyd-B rheological model. The free surface shape is assumed to be initially characterized by a single-mode perturbation with a small amplitude compared to the radius of the undeformed jet.

The 2D axisymmetric problem is formulated in dimensionless form by using the radius of the undeformed jet as the length scale and the capillary time as the time scale. The set of equations obtained depend on five dimensionless numbers: the dimensionless wavenumber, the initial dimensionless deformation amplitude, an Ohnesorge number and two Deborah numbers. The flow quantities (pressure, velocity components and surface shape) are expanded in series of power of the small parameter of the problem which is the initial dimensionless deformation amplitude.

The ideal case of an inviscid fluid has been treated by Yuen in 1968 (J. Fluid Mech. 33:151-163) up to third order and the particular case of a Newtonian fluid has been recently solved up to second order (Renoult et al. 2018 J. Fluid Mech. 856:169-201). In the continuity of the latter work, the weakly nonlinear stability for the viscoelastic case is performed up to second order.

The first-order equations differ from those of the Newtonian case only by multiplying the Ohnesorge number by a coefficient independent of the spatial and temporal variables. At second order, the viscoelastic case requires the solution of a Poisson equation for the calculation of the pressure as in the Newtonian case, but with an additional right-hand side member.

The first-order contribution to the solution, first derived by Goldin and al. in 1969 (J. Fluid Mech. 38:689-711), is retrieved. The second-order contribution to the solution is determined by following the same methodology as the one employed for the Newtonian case, especially involving the use of a polynomial approximation of modified Bessel functions of the first kind. The additional RHS term in the viscoelastic case leads to an additional approximate contribution to the solution.

The solution will be determined for a large range of the dimensionless parameters and compared to the results of previous studies on the effect of viscoelasticity on liquid jet breakup.