- Email: [email protected]
The maxwell stress at a deformable surface
Journal of Electrostatics, 4 (1978) 303--306
303
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
Short Communication
THE MAXWELL STRESS AT A D E F O R M A B L E S U R F A C E
J.M.H. PETERS*
Department of Applied Mathematics and Theoretical Physics, University of Liverpool, Liverpool (Gt. Britain) (Received October 26, 1977; in revised form December 9, 1977)
A dielectric under electric stress is subject to three electrical bulk forces. Two result from the translatory action of the electric field on free and polarisation charges which are present in the medium, and the third is associated with a very small volume change caused by the reorientation of dipoles in the direction of the external field [ 1 ]. With incompressible dielectrics, however, this latter force assumes the character of a pressure force and it may be included with the dynamic pressure when a total force balance is made [2]. For this reason, it is excluded from the present analysis. The t w o remaining forces associated with free and polarisation charge will be called the Coulomb and polarisation forces, respectively. Taken together, they may be written as the gradient of a dyadic (tensor) quantity when the external electric field is irrotational. Integration across an interface between two different dielectric media with the aid of Gauss's divergence theorem, then yields an expression known as the Maxwell stress. The Maxwell stress is important in problems on interfacial electrohydrodynamics since it gives the electrical contribution to the boundary conditions at an interface between two different fluid media. Previous work in this field has been mainly concerned with the analysis of the effects of electric fields directed either perpendicular to, or parallel to, a fluid--liquid interface and the evaluation of the Maxwell stress in these special cases has been greatly simplified. The present analysis considers, as an example, the more general case in which the electric field is directed obliquely to an initially plane interface in hydrostatic equilibrium. It is clear that the method suggested here may also be applied to other interfaces requiring curvilinear coordinates for their description, so that, for example, the first-order components of the Maxwell stress at a perturbed ellipsoidal interface may be evaluated systematically. If the volumic density of free charge present in the bulk of a dielectric of permittivity e be denoted by qv, then the s u m f of the Coulombic and polarisation volume forces is: 0
f = q v E _ - - ~1 _E2 V e, *Present address: The Polvtechnic. Livernnnl C'.t Rrlt~in
(1)
304
where E is the external electric field vector. With the aid of Gauss's law in differential form, namely [ 1 ] = V" ( e E ) ,
qv
(2)
and the fact that the electric field E is irrotational when magnetic effects are negligible, eqn. (1) may be written as 1 2 f = V-(eEE-~-E /).
(3)
The term in brackets, which is here most conveniently expressed in dyadic notation - - / b e i n g the unit dyad -- is the Maxwell stress dyad. The total-force _F acting on a thin disc which straddles the interface is the integral of eqn. (3) over the volume of the disc, namely:
F = f ;'(¢_EE_-2_E2Ddv V
--
f
d,,
(4)
$
where h is the unit normal to the interface between the dielectric media -labelled (1) and (2), respectively -- which points from region (1) to region (2). When the disc is made infinitely thin, the surface integral in eqn. (4) reduces to:
where the notation [H ~ represents the difference H (2) - H (1) and the quantity is evaluated at the interface. If_Et denotes the tangential component of the electric field -- it is a vector lying in the plane of the interface -- and E n denotes the magnitude of the normal c o m p o n e n t of the field, then_E may be written as: _E = Et + En_ .
(6)
Introducing this into eqn. (5) gives: F
=
e
~
h_ + E n E t
.
(7)
If the interface is deformable and n denotes its normal displacement from equilibrium, then when 77 is small, the unit normal h to the interface may be approximated to the first order in small quantities, by:
£w --
3u
hu
3v
hv
hw '
305
where u, v and w are orthogonal coordinates with u and v in the plane of the initially unperturbed interface and w directed normally to it, hu, hv and hw are the corresponding scale factors and eu, ev and ~w are unit vectors. In view of this and the fact that En andE_t may be written as:
En = f f ' E a~ Eu ~ - - a u hu
a~? Ev a v hv
+
Ew hw
-
-
and
Et
=
=
=
XE)
-
1 [(aT? Ew hw au hu
+
Eu) eu hw -
+
(a~ Ew Ev ) +-ev hv hw
-I-
-
+(aT? E U ahu u +--arab?E~vv)-ew ?
(10)
a first-order representation of eqn. (7) can be obtained, where Eu, Ev and Ew are the orthogonal components of E. In particular, with rectangular coordinates we have
E = +
~-x ~
E y - E x) - ~-~ E x E y + E x E z -Ey
-~x
+ EyEz
- ~ Ex + - ~
~+ ] +
.. k_
(11)
The deformation of the interface is accompanied by a perturbation of the external field which, to the first order in small quantities, may be described by: _E = E H +6_E,
(12)
where the subscript H denotes the value of the variable corresponding to the equilibrium state. Retaining rectangular coordinates for simplicity, eqn. (12) may be introduced into eqn. (11) to give to the first order in small quantities, where the subscript H is now dropped:
306
F_ + ~ F ~+
-~x
Ey
~-~ (E~ - Ey
- ~ Ex Ey + Ex Ez + Ex ~ Ez + Ez ~ Ex [_+ - - ~ Ex Ey + Ey Ez + Ey 5 Ez + Ez 5 y
f_ +
+ ~.(E 1 z2 _ Ex2 _ E ; ) + E z 6 E z - E x S E x
-
Ey6Ey
~77
bx
EzEx
-
(13)
- -
from which
F__ = ~e (Ex Ez [__+ Ey Ez_~ + represents the total force of electrical origin acting at the hydrostatic interface, and
6F_ :
e
+
(E~ + Ey - E2x) - ~ E x E y
~
- E y + E x) - - ~
+ (EzSEz - ExSEx - EySEy
+ Ex6Ez + Ez6Ex
+Ey~Ez +Ez~Ey
~EzEx
-~EzEy -
[__ +
] +
)k-I
to the first order in small quantities. Equation (15) is important in that it describes the first-order perturbation induced in the Maxwell stress when the interface between t w o dielectric media undergoes a slight deformation, ~7(x,y) from its equilibrium position. It is a more general result than those obtained previously for the cases of "perpendicular" or "parallel" fields [2], where the appropriate components of the "hydrostatic" electric field were suppressed. References
I W.H. Panofsky and M. Phillips, Classical Electricity and Magnetism, Addison-Wesley, Reading, Massachusetts, 1955. 2 J.R. Melcher, Phys. Fluids, 5 (1962) 1130. 3 J.R. Melcher, I.E.E.E. Trans. Educ., E17 (1974) 100.
303
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
Short Communication
THE MAXWELL STRESS AT A D E F O R M A B L E S U R F A C E
J.M.H. PETERS*
Department of Applied Mathematics and Theoretical Physics, University of Liverpool, Liverpool (Gt. Britain) (Received October 26, 1977; in revised form December 9, 1977)
A dielectric under electric stress is subject to three electrical bulk forces. Two result from the translatory action of the electric field on free and polarisation charges which are present in the medium, and the third is associated with a very small volume change caused by the reorientation of dipoles in the direction of the external field [ 1 ]. With incompressible dielectrics, however, this latter force assumes the character of a pressure force and it may be included with the dynamic pressure when a total force balance is made [2]. For this reason, it is excluded from the present analysis. The t w o remaining forces associated with free and polarisation charge will be called the Coulomb and polarisation forces, respectively. Taken together, they may be written as the gradient of a dyadic (tensor) quantity when the external electric field is irrotational. Integration across an interface between two different dielectric media with the aid of Gauss's divergence theorem, then yields an expression known as the Maxwell stress. The Maxwell stress is important in problems on interfacial electrohydrodynamics since it gives the electrical contribution to the boundary conditions at an interface between two different fluid media. Previous work in this field has been mainly concerned with the analysis of the effects of electric fields directed either perpendicular to, or parallel to, a fluid--liquid interface and the evaluation of the Maxwell stress in these special cases has been greatly simplified. The present analysis considers, as an example, the more general case in which the electric field is directed obliquely to an initially plane interface in hydrostatic equilibrium. It is clear that the method suggested here may also be applied to other interfaces requiring curvilinear coordinates for their description, so that, for example, the first-order components of the Maxwell stress at a perturbed ellipsoidal interface may be evaluated systematically. If the volumic density of free charge present in the bulk of a dielectric of permittivity e be denoted by qv, then the s u m f of the Coulombic and polarisation volume forces is: 0
f = q v E _ - - ~1 _E2 V e, *Present address: The Polvtechnic. Livernnnl C'.t Rrlt~in
(1)
304
where E is the external electric field vector. With the aid of Gauss's law in differential form, namely [ 1 ] = V" ( e E ) ,
qv
(2)
and the fact that the electric field E is irrotational when magnetic effects are negligible, eqn. (1) may be written as 1 2 f = V-(eEE-~-E /).
(3)
The term in brackets, which is here most conveniently expressed in dyadic notation - - / b e i n g the unit dyad -- is the Maxwell stress dyad. The total-force _F acting on a thin disc which straddles the interface is the integral of eqn. (3) over the volume of the disc, namely:
F = f ;'(¢_EE_-2_E2Ddv V
--
f
d,,
(4)
$
where h is the unit normal to the interface between the dielectric media -labelled (1) and (2), respectively -- which points from region (1) to region (2). When the disc is made infinitely thin, the surface integral in eqn. (4) reduces to:
where the notation [H ~ represents the difference H (2) - H (1) and the quantity is evaluated at the interface. If_Et denotes the tangential component of the electric field -- it is a vector lying in the plane of the interface -- and E n denotes the magnitude of the normal c o m p o n e n t of the field, then_E may be written as: _E = Et + En_ .
(6)
Introducing this into eqn. (5) gives: F
=
e
~
h_ + E n E t
.
(7)
If the interface is deformable and n denotes its normal displacement from equilibrium, then when 77 is small, the unit normal h to the interface may be approximated to the first order in small quantities, by:
£w --
3u
hu
3v
hv
hw '
305
where u, v and w are orthogonal coordinates with u and v in the plane of the initially unperturbed interface and w directed normally to it, hu, hv and hw are the corresponding scale factors and eu, ev and ~w are unit vectors. In view of this and the fact that En andE_t may be written as:
En = f f ' E a~ Eu ~ - - a u hu
a~? Ev a v hv
+
Ew hw
-
-
and
Et
=
=
=
XE)
-
1 [(aT? Ew hw au hu
+
Eu) eu hw -
+
(a~ Ew Ev ) +-ev hv hw
-I-
-
+(aT? E U ahu u +--arab?E~vv)-ew ?
(10)
a first-order representation of eqn. (7) can be obtained, where Eu, Ev and Ew are the orthogonal components of E. In particular, with rectangular coordinates we have
E = +
~-x ~
E y - E x) - ~-~ E x E y + E x E z -Ey
-~x
+ EyEz
- ~ Ex + - ~
~+ ] +
.. k_
(11)
The deformation of the interface is accompanied by a perturbation of the external field which, to the first order in small quantities, may be described by: _E = E H +6_E,
(12)
where the subscript H denotes the value of the variable corresponding to the equilibrium state. Retaining rectangular coordinates for simplicity, eqn. (12) may be introduced into eqn. (11) to give to the first order in small quantities, where the subscript H is now dropped:
306
F_ + ~ F ~+
-~x
Ey
~-~ (E~ - Ey
- ~ Ex Ey + Ex Ez + Ex ~ Ez + Ez ~ Ex [_+ - - ~ Ex Ey + Ey Ez + Ey 5 Ez + Ez 5 y
f_ +
+ ~.(E 1 z2 _ Ex2 _ E ; ) + E z 6 E z - E x S E x
-
Ey6Ey
~77
bx
EzEx
-
(13)
- -
from which
F__ = ~e (Ex Ez [__+ Ey Ez_~ + represents the total force of electrical origin acting at the hydrostatic interface, and
6F_ :
e
+
(E~ + Ey - E2x) - ~ E x E y
~
- E y + E x) - - ~
+ (EzSEz - ExSEx - EySEy
+ Ex6Ez + Ez6Ex
+Ey~Ez +Ez~Ey
~EzEx
-~EzEy -
[__ +
] +
)k-I
to the first order in small quantities. Equation (15) is important in that it describes the first-order perturbation induced in the Maxwell stress when the interface between t w o dielectric media undergoes a slight deformation, ~7(x,y) from its equilibrium position. It is a more general result than those obtained previously for the cases of "perpendicular" or "parallel" fields [2], where the appropriate components of the "hydrostatic" electric field were suppressed. References
I W.H. Panofsky and M. Phillips, Classical Electricity and Magnetism, Addison-Wesley, Reading, Massachusetts, 1955. 2 J.R. Melcher, Phys. Fluids, 5 (1962) 1130. 3 J.R. Melcher, I.E.E.E. Trans. Educ., E17 (1974) 100.