Supplemental material
“Electro-Capillary Instabilities of Thin Leaky Elastic-Viscous Bilayers”
Kartick Mondal1 and Dipankar Bandyopadhyay1,2
1
Department of Chemical Engineering, Indian Institute of Technology Guwahati, India
2
Centre for Nanotechnology, Indian Institute of Technology Guwahati, India
I. GENERAL LINEAR STABILITY ANALYSIS
A. Base-state analysis
The general linear stability analysis (GLSA) is performed based on the assumptions that the kinematics
of deformation is small. A quiescent base-state is considered with stationary viscous, v0( x )  v0( z )  0 , and
elastic, u0( x )  u0( z )  0 , layers of constant thicknesses, (d – h0 ) and h0 , respectively. The variables with
subscript zero denote the base-state variables. The expression for the base-state potential is,
 i 0  C1i z  C2i , is obtained from the governing equation,  i 0 zz  0 in which the subscript z denotes
differentiation. The following boundary conditions are employed to evaluate the constants, C ji s (i = 1
and 2; j =1 and 2): (i) at the cathode (z = 0),  10 = 0 and at anode (z = d),  20 = ; at the elastic-viscous
interface ( z  h0 ), the balances of the normal (  01 10 z   0 2 20 z  q0 ) and the tangential ( 10   20 )
components of the electric field. The base-state z-momentum balance for the electrohydrodynamic
(EHD) field leads to the expression, pi 0 z  0 , for the for each layer i = 1 and 2, which has a solution,
pi 0  Ci ,
where
Ci s
(i
=
3 and
4)
are
the
constants.
The
normal
stress
balance,
p10  p20  0.5  0 2 20 z   01 10 z  , at the elastic-viscous (z = h0 ) interface is enforced to evaluate basestate pressure for the ith layer. The kinematic equation for the free charge density, s2 20 z
h0
 s1 10 z
h0
 0,
yields the base-state free charge density at the interface, q0   0 2 /  d  h0  for the E L bilayer,
q0   01 / h0 for the V L bilayer, and q0   0  1s2   2 s1  /  ds1  h0  s2  s1   for the L bilayer.
Clearly, the expressions suggest that the applied potential, thickness, and dielectric permittivity of the
leaky layer in the bilayers can influence the base-state charge density at the interface.
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B. Perturbed-state analysis
The governing equations and the boundary conditions shown in the previous section are perturbed
through the normal linear modes, u  ue t  i kx , v  ve t  i kx , i   i 0  i e t  i kx , pi  pi 0  pi e t  i kx ,
h  h0  h e t i kx , and q  q0  q e t  i kx , where the symbols  and k represent the linear growth
coefficient and the wave number of the disturbance, respectively. The tilde symbols denote the perturbed
variables. The linearized Laplace equation for the ith layer is,
 izz  k 2 i  0 .
(1)
The general solution of Eq. (1) is,
 i  D1i e kz  D2i e  kz .
(2)
Here the coefficients D ji (i = 1, 2 and a; j = 1 and 2) are constants. The constants D ji s can be evaluated
by employing the linearized perturbed boundary conditions: (i) at the cathode (z = 0),  1  0 and anode
(z = d),  2  0 ; (ii) at the elastic-viscous interface (z = h0 ), the normal [  01 1z   0 2 2 z  q ] and the
tangential [  1  2   h  10 z  20 z   0 ] component balances of the electric field. The subscript z in
these expressions denotes differentiation
The linearized forms of the governing Eqs. (2.1) and (2.2) in the main manuscript for the elastic film are,
ikp1  G  u zz( x )  k 2u ( x )   0 ,
(3)
 p1z  G  u zz( z )  k 2u ( z )   0 ,
(4)
iku ( x )  u z( z )  0 .
(5)
The linearized forms of the governing Eqs. (2.3) and (2.4) in the main manuscript for the viscous film
are,
ikp2    vzz( x )  k 2 v ( x )   0 ,
(6)
 p2 z    vzz( z )  k 2 v ( z )   0 ,
(7)
ikv ( x )  vz( z )  0 .
(8)
Eliminating pi from the linearized governing Eqs. (3) – (5) and (6 – (8) results in the following
biharmonic equations for the elastic and the viscous layers,
(z)
u zzzz
 2k 2u zz( z )  k 4u ( z )  0 ,
(9)
2
(z)
vzzzz
 2k 2 vzz( z )  k 4 v ( z )  0 .
(10)
The general solutions for the Eqs. (9) and (10) are,
u ( z )  ( A1i  B2i z ) e kz  ( A3i  A4i z ) e  kz ,
(11)
v ( z )  ( B1i  B2i z ) e kz  ( B3i  B4i z ) e  kz .
(12)
Here the coefficients A ji and B ji (i = 1 and 2; j =1 to 4) are constants. The following are the linearized
no slip and impermeability conditions at z = 0,
u( x)  u( z)  0 .
(13)
The linearized continuity of velocities, the tangential and normal stress balances, and the kinematic
conditions at z = h reduce to the forms,
 u ( x )  v ( x ) and  u ( z )  v ( z ) ,
(14)


G  uz( x )  iku ( z )     vz( x )  ikv ( z )   ikq0  2  h 20 z  0 ,
(15)
p1  p2  2Guz ( z )  2vz ( z )   0  2 20 z 2 z  1 10 z 1z   k 2 h  0 ,
(16)
h  u( z) ,
(17)
q   1  s2 2 z  s1 1z   ikq0u ( x) .
(18)
The linearized tangential and normal stress balances, and the kinematic conditions at z = d,
v ( x)  v ( z)  0 .
(19)
The solutions for u ( z ) and v ( x ) , Eqs. (11) and (12), are used to obtain expressions for the linearized
variables, u ( x ) , v ( x ) , and pi , form the linear continuity and x-momentum balance equations. The
expressions for u ( x ) , u ( z ) , v ( x ) , v ( z ) and pi are then replaced in the linear boundary conditions, Eqs. (13
– 19) to obtain a set of homogeneous linear algebraic equations with eight unknown constants, A ji and
B ji (i = 1 and 2; j = 1 to 4). Equating the determinant of the coefficient matrix to zero leads to the
dispersion relation,   f  k  , which is solved analytically with the help of the commercial package
Mathematica™. The critical parameters are obtained by enforcing the neutral stability condition,   0 ,
in the dispersion relation and then solving the algebraic equation for kc and c . The dominant growth
coefficient (  m ) and the corresponding wavelength ( m ) are evaluated by finding out the global maxima
of  .
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