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Linear instability of the electrified free interface between two cylindrical shells of viscoelastic fluids through porous media

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Abstract

In this paper, we have discussed the linear stability analysis of the electrified surface separating two coaxial Oldroyd-B fluid layers confined between two impermeable rigid cylinders in the presence of both interfacial insoluble surfactant and surface charge through porous media. The case of long waves interfacial stability has been studied. The dispersion relation is solved numerically and hence the effects of various parameters are illustrated graphically. Our results reveal that the influence of the physicochemical parameter β is to shrink the instability region of the surface and reduce the growth rate of the unstable normal modes. Such important effects of the surfactant on the shape of interfacial structures are more sensitive to the variation of the β corresponding to non-Newtonian fluids-model compared with the Newtonian fluids model. In the case of long wave limit, it is demonstrated that increasing β, has a dual role in-fluence (de-stabilizing effects) depending on the viscosity of the core fluid. It has a destabilizing effect at the large values of the core fluid viscosity coefficient, while this role is exchanged to a regularly stabilizing influence at small values of such coefficient.

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Abbreviations

(·) j or (·)(j) j :

1 represents a quantity referred to Fluid 1 (inner layer), j = 2 denotes a quantity of Fluid 2 (outer layer)

r :

coordinate transverse to the cylinder surface

z :

coordinate along the cylinder surface

t :

time

p :

fluid pressure

ρ j :

density of the fluid

ρ :

ratio of ρ 2 to ρ 1

µ j :

dynamic viscosity of the inner fluid

µ:

ratio of µ2 to µ1

λ (j)1 :

relaxation time of the fluid

λ (j)2 :

retardation time of the fluid

R (j) :

flow resistance offered by the solid matrix in porous media

Re :

Reynolds number

φ j :

porosity of the medium occupied by the fluid

κ j :

permeability of the porous medium occupied by the fluid

χ j :

permeability parameter of the medium occupied by the fluid

u (j), ν (j) :

velocity components along r and z directions, respectively

τ 0 :

unperturbed-state surface tension

τ :

interfacial surface tension coefficient

τ c :

surface tension of a clean surface interface

Γ :

local surfactant concentration

Γ0 :

unperturbed surfactant concentration

D s :

surfactant diffusivity

R :

ideal gas constant

\(\tilde T\) :

absolute temperature

β :

physicochemical parameter

S mn :

viscoelastic stress tensor of the fluid

k :

wave number of the disturbance

c :

complex wave celerity

η :

interfacial displacement

\(\hat \eta\) :

complex amplitude of the interfacial perturbation

Ψ (j) :

stream function of the fluid

ɛ :

small dimensionless parameter

E (j) :

electric field strength vector

φ (j) :

electric potential function

ε j :

electric permittivity of the fluid

ε :

ratio of ε 2 to ε1

q :

area free charge density on the interface

σ j :

electric conductivity of the fluid

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

    Magdy A. Sirwah

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Correspondence to Magdy A. Sirwah.

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Sirwah, M.A. Linear instability of the electrified free interface between two cylindrical shells of viscoelastic fluids through porous media. Acta Mech Sin 28, 1572–1589 (2012). https://doi.org/10.1007/s10409-012-0208-2

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  • DOI: https://doi.org/10.1007/s10409-012-0208-2

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