Maxwells’ electric stress tensor

For ionic surfactant solution the body force tensor, Pb, is not isotropic - it is the Maxwell electric stress tensor, i.e. Pb = f6bEE - i6jE l2, where E = -V is the electric field (Landau and Lifshitz 1960). The density of the electric force plays the role of a spatial body force, f, in the Navier-Stokes equation of motion (3). In the lubrication approximation the pressure in the continuous phase depends on the vertical coordinate, z, only through its osmotic part generated from the electric potential and the pressure in the middle plane (or the pressure, pn, corresponding to the case of zero potential) ... 

We can now identify the first term in (A.4) with Maxwell s stress tensor, which acts on any dielectric in an electric field. The magnitude of this force Pe is given by... 

Now, find the force acting on the conducting drop. The momentum fiux density in an electric field is defined by Maxwell s stress tensor [77]... 

The electric stress tensor reduces to what is known as the Maxwell form for the vacuum where e = 1, Equation 2.38 also can be written in the form... 

In what follows, the functions negative Helmholtz energy. Therefo on purely dimensional grounds, a Sijj, must represent surface charge d coefficient of the displacement vec represent a stress tensor. In fact electrostatic field (e.g. for p = 0 Maxwell s electric stress tensor, the next Section that Maxwell s str a part of in our colloid model,... 

We have just described the linearized theory of capillarity. In the electrostatic analogy the field M(r ) is identified with a 2D electrostatic potential ( capillary potential ) and Il(r ) with a charge density ( capillary charge ) Equation 2.8 reduces to the Poisson equation of electrostatics and Equation 2.9 relates the tensor Tn, which has the form of Maxwell s stress tensor, with the electric force exerted on the capillary charge n(rn) (also the usual boundary conditions imposed on the interface have a close electrostatic analogy [34,35]). 

At the same time, the coulomb attraction acts between the charges on the particle surface and the counterions within the electrical double layer, which is obtained by integrating the Maxell stress tensor over an arbitrary surface surrounding the particle. The Maxwell stress tensor is given by... 

The effective electrical tension, transmembrane potential, Pja, is defined by the Maxwell stress tensor [59, 89, 92]... 

The coupling between the hydrodynamics and the electric field therefore arises through the Maxwell stress tensor. The total electric force density comprises the sum of the Coulombic force arising from the presence of free charges... 

Equation [19] expresses the condition of equilibrium as a balance between the electrostatic pressure on ions, where the first term is simply the Maxwell electric field stress tensor,and the osmotic pressure with respect to bulk, given by the second term. Only in special cases can the integral in Eq. [17] be evaluated analytically. Several analytical approximations to the planar PB equation for asymmetric electrolytes have been suggested. " We now present three examples possessing exact analytical solutions beginning with the classic Gouy-Chapman solution. 

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