Oscillatory onset and scaling laws of electroconvection in Oldroyd-B fluid at high Weissenberg numbers: A reduced analytical theory

We extend the linear stability analysis of electroconvective instability in a quiescent Oldroyd-B viscoelastic fluid subjected to an electric field into the strongly elastic regime, up to Weissenberg numbers (

) of order

10 cubed 10 3 $10^{3}$

. The oscillatory onset is governed by the interplay among destabilising Coulomb forcing, viscous dissipation and elastic stress relaxation. For the quiescent base state, the linearised constitutive relation allows the polymer stress to be represented through a complex effective viscosity. We then apply a single-mode Galerkin projection based on the Newtonian critical eigenfunction, obtaining a reduced dispersion relation and a closed-form prediction for the neutral oscillation frequency. In the limit of large

italic Wi Wi $\textit{Wi}$

, this analytical solution yields two asymptotic scaling laws, namely a frequency scaling law

omega Subscript c Baseline proportional to StartRoot left parenthesis 1 minus beta right parenthesis slash italic Wi EndRoot ω c ∝ ( 1 − β ) / Wi $\omega _{c} \propto \sqrt {(1-\beta )/\textit{Wi}}$

(

omega Subscript c ω c $\omega _{c}$

is the critical frequency and

beta β $\beta$

is the solvent to total viscosity ratio) and a threshold scaling law

upper T Subscript c Baseline equals beta upper T Subscript c Superscript upper N Baseline plus a left parenthesis beta right parenthesis slash italic Wi T c = β T c N + a ( β ) / Wi $T_{c} = \beta T_{c}^{N} + a(\beta )/\textit{Wi}$

(

upper T Subscript c T c $T_{c}$

is the critical electric Rayleigh number,

upper T Subscript c Superscript upper N T c N $T_{c}^{N}$

is the Newtonian critical electric Rayleigh number and

a left parenthesis beta right parenthesis a ( β ) $a(\beta )$

is the coefficient related to

beta β $\beta$

), which satisfy a self-consistency relation. Here,

upper T Subscript c T c $T_{c}$

exhibits a three-stage evolution with

italic Wi Wi $\textit{Wi}$

, and a frozen-elasticity limit is identified at large

italic Wi Wi $\textit{Wi}$

. With a single geometric constant as input, the values of

omega Subscript c ω c $\omega _{c}$

and

upper T Subscript c T c $T_{c}$

obtained from the analytical solution and from high-resolution spectral computations agree to within 0.3 %, providing mutual corroboration.