Viscoelastic potential flow instability theory of Rivlin-Ericksen electrified fluids of cylindrical interface

doi:10.1016/j.joes.2022.06.024

Abstract

Abstract:

A linear electrohydrodynamic Kelvin-Helmholtz instability of the interface between two viscoelastic Rivlin-Ericksen fluids enclosed by two concentric horizontal cylinders has been studied via the viscoelastic potential flow theory. The dispersion equation of complex coefficients for asymmetric disturbance has been obtained by using normal mode technique. the stability criteria are analyzed theoretically and illustrated graphically. The imaginary part of growth rate is plotted versus the wave number. The influences of dynamic viscoelastic, uniform velocities, Reynolds number, electric field, dynamic viscosity, density fluids ratio, dielectric constant ratio and inner fluid fraction on the stability of the system are discussed. The study finds its significance in Ocean pipelines to transfer oil or gas such as Eastern Siberia-Pacific Ocean oil pipeline.

Highlights

● The linear electrohydrodynamic Kelvin-Helmholtz instability analysis of two viscoelastic Rivlin-Ericksen fluids has been studied via the viscoelastic potential flow theory.

● The dispersion equation of complex coefficients has been obtaind by using normal mode technique.

● The effect of various parameters on the stability of the system are discussed.

● some limiting cases are considered and recovered previous works.

Key words: Rivlin-Ericksen fluid, Cylindrical flow, Viscoelastic potential flow, Electrohydrodynamics stability

D.M. Mostafa. Viscoelastic potential flow instability theory of Rivlin-Ericksen electrified fluids of cylindrical interface[J]. Journal of Ocean Engineering and Science, 2024, 9(4): 311-316.

D.M. Mostafa. [J]. Journal of Ocean Engineering and Science, 2024, 9(4): 311-316.

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URL: https://www.qk.sjtu.edu.cn/joes/EN/10.1016/j.joes.2022.06.024

https://www.qk.sjtu.edu.cn/joes/EN/Y2024/V9/I4/311

Figures/Tables 11

Fig.1 Schematic of stability analysis.

Fig.2 Comparison of growth rates between inviscid potential flow and viscoelastic potential flow at

β = 0.5, \tilde{ρ} = 0.012, L = 0.01,

{\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,

Re = 100,

\tilde{σ} = 72.3,

\tilde{ε} = 0.03

and

{\tilde{E}}_{0} = 10

Fig.2 Comparison of growth rates between inviscid potential flow and viscoelastic potential flow at $β = 0.5, \tilde{ρ} = 0.012, L = 0.01,$ ${\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,$ $Re = 100,$ $\tilde{σ} = 72.3,$ $\tilde{ε} = 0.03$ and ${\tilde{E}}_{0} = 10$ .

Fig.3 Effect of inner fluid fraction

β

on the growth rate of instability at

\tilde{ρ} = 0.012, \tilde{μ} = 0.02,

{\tilde{μ}}^{'} = 0.05,

L = 0.01,

{\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,

Re = 100,

\tilde{σ} = 72.3,

\tilde{ε} = 0.03

and

{\tilde{E}}_{0} = 10

Fig.3 Effect of inner fluid fraction $β$ on the growth rate of instability at $\tilde{ρ} = 0.012, \tilde{μ} = 0.02,$ ${\tilde{μ}}^{'} = 0.05,$ $L = 0.01,$ ${\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,$ $Re = 100,$ $\tilde{σ} = 72.3,$ $\tilde{ε} = 0.03$ and ${\tilde{E}}_{0} = 10$ .

Fig.4 Effect of density ratio

\tilde{ρ}

on the growth rate of instability at

β = 0.5, \tilde{μ} = 0.02,

{\tilde{μ}}^{'} = 0.05,

L = 0.01,

{\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,

Re = 100,

\tilde{σ} = 72.3,

\tilde{ε} = 0.03

and

{\tilde{E}}_{0} = 10

Fig.4 Effect of density ratio $\tilde{ρ}$ on the growth rate of instability at $β = 0.5, \tilde{μ} = 0.02,$ ${\tilde{μ}}^{'} = 0.05,$ $L = 0.01,$ ${\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,$ $Re = 100,$ $\tilde{σ} = 72.3,$ $\tilde{ε} = 0.03$ and ${\tilde{E}}_{0} = 10$ .

Fig.5 Effect of electric field

{\tilde{E}}_{0}

on the growth rate of instability at

β = 0.5, \tilde{ρ} = 0.012,

\tilde{μ} = 0.02,

{\tilde{μ}}^{'} = 0.05,

L = 0.01,

{\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,

Re = 100,

\tilde{σ} = 72.3

and

\tilde{ε} = 0.03

Fig.5 Effect of electric field ${\tilde{E}}_{0}$ on the growth rate of instability at $β = 0.5, \tilde{ρ} = 0.012,$ $\tilde{μ} = 0.02,$ ${\tilde{μ}}^{'} = 0.05,$ $L = 0.01,$ ${\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,$ $Re = 100,$ $\tilde{σ} = 72.3$ and $\tilde{ε} = 0.03$ .

Fig.6 Comparison of growth rates between Asymmetric disturbance and Axisymmetric disturbance at

β = 0.5, \tilde{ρ} = 0.012,

\tilde{μ} = 0.02,

{\tilde{μ}}^{'} = 0.05,

L = 0.01,

{\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,

Re = 100,

\tilde{σ} = 72.3,

\tilde{ε} = 0.03

and

{\tilde{E}}_{0} = 10

Fig.6 Comparison of growth rates between Asymmetric disturbance and Axisymmetric disturbance at $β = 0.5, \tilde{ρ} = 0.012,$ $\tilde{μ} = 0.02,$ ${\tilde{μ}}^{'} = 0.05,$ $L = 0.01,$ ${\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,$ $Re = 100,$ $\tilde{σ} = 72.3,$ $\tilde{ε} = 0.03$ and ${\tilde{E}}_{0} = 10$ .

Fig.7 Effect of dielectric constant ratio

\tilde{ε}

on the growth rate of instability at

β = 0.5, \tilde{ρ} = 0.012,

\tilde{μ} = 0.02,

{\tilde{μ}}^{'} = 0.05,

L = 0.01,

{\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,

Re = 100,

\tilde{σ} = 72.3

and

{\tilde{E}}_{0} = 20

Fig.7 Effect of dielectric constant ratio $\tilde{ε}$ on the growth rate of instability at $β = 0.5, \tilde{ρ} = 0.012,$ $\tilde{μ} = 0.02,$ ${\tilde{μ}}^{'} = 0.05,$ $L = 0.01,$ ${\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,$ $Re = 100,$ $\tilde{σ} = 72.3$ and ${\tilde{E}}_{0} = 20$ .

Fig.8 Effect of Reynolds number Re on the growth rate of instability at

β = 0.5, \tilde{ρ} = 0.012,

\tilde{μ} = 0.5,

{\tilde{μ}}^{'} = 0.05,

L = 0.01,

{\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,

\tilde{σ} = 72.3,

\tilde{ε} = 0.03

and

{\tilde{E}}_{0} = 20

Fig.8 Effect of Reynolds number Re on the growth rate of instability at $β = 0.5, \tilde{ρ} = 0.012,$ $\tilde{μ} = 0.5,$ ${\tilde{μ}}^{'} = 0.05,$ $L = 0.01,$ ${\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,$ $\tilde{σ} = 72.3,$ $\tilde{ε} = 0.03$ and ${\tilde{E}}_{0} = 20$ .

Fig.9 Effect of dynamic viscosity ratio

\tilde{μ}

on the growth rate of instability at

β = 0.5, \tilde{ρ} = 0.012,

{\tilde{μ}}^{'} = 0.05,

L = 0.01,

{\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,

Re = 10,

\tilde{σ} = 72.3,

\tilde{ε} = 0.03

and

{\tilde{E}}_{0} = 10

Fig.9 Effect of dynamic viscosity ratio $\tilde{μ}$ on the growth rate of instability at $β = 0.5, \tilde{ρ} = 0.012,$ ${\tilde{μ}}^{'} = 0.05,$ $L = 0.01,$ ${\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,$ $Re = 10,$ $\tilde{σ} = 72.3,$ $\tilde{ε} = 0.03$ and ${\tilde{E}}_{0} = 10$ .

Fig.10 Effect of dynamic viscoelastic ratio

{\tilde{μ}}^{'}

on the growth rate of instability at

β = 0.5, \tilde{μ} = 0.02,

\tilde{ρ} = 0.012,

\tilde{ρ} = 0.012,

{\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,

Re = 100,

\tilde{σ} = 72.3,

\tilde{ε} = 0.03

and

{\tilde{E}}_{0} = 10

Fig.10 Effect of dynamic viscoelastic ratio ${\tilde{μ}}^{'}$ on the growth rate of instability at $β = 0.5, \tilde{μ} = 0.02,$ $\tilde{ρ} = 0.012,$ $\tilde{ρ} = 0.012,$ ${\tilde{U}}_{1} = 100, {\tilde{U}}_{2} = 900,$ $Re = 100,$ $\tilde{σ} = 72.3,$ $\tilde{ε} = 0.03$ and ${\tilde{E}}_{0} = 10$ .

Fig.11 Effect of uniform velocities

{\tilde{U}}_{1}

and

{\tilde{U}}_{2}

on the growth rate of instability at

β = 0.5, \tilde{ρ} = 0.012, \tilde{μ} = 0.02,

{\tilde{μ}}^{'} = 0.05,

L = 0.01,

Re = 100,

\tilde{σ} = 72.3,

\tilde{ε} = 0.03

and

{\tilde{E}}_{0} = 10

Fig.11 Effect of uniform velocities ${\tilde{U}}_{1}$ and ${\tilde{U}}_{2}$ on the growth rate of instability at $β = 0.5, \tilde{ρ} = 0.012, \tilde{μ} = 0.02,$ ${\tilde{μ}}^{'} = 0.05,$ $L = 0.01,$ $Re = 100,$ $\tilde{σ} = 72.3,$ $\tilde{ε} = 0.03$ and ${\tilde{E}}_{0} = 10$ .

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