Electric stress tensor

Here t is any vector tangent to the interface and it follows that the potentials on either side of the interface differ by at most a constant. If no work is done in transferring charge across the interface, the constant is zero. 

The study of interactions between macroscopic bodies requires the force per unit area, i.e. the stress that an external field exerts on a surface. 

We will see first the expression of the stress tensor in a homogeneous fluid, then illustrate its application to a sphere immersed in an uncharged dielectric. 

The derivation of the stress in a fluid dielectric containing free charge is not straightforward, in part because of the difficulty in establishing how the presence of the electric field contributes to the pressure and thereby alters the stress. The derivation below is based on that of Landau and Lifshitz (1960) [2]. 

First the force due to the electric field acting on an isolated dipole is derived. Consider a pair of charges, Q and -Q, at relative position d. The electrical force on the pair is 

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Electrostatic. In many practical situations, both membrane and solute have net negative charges. Hence, as the solute approaches a pore in the membrane it experiences an electrostatic repulsion. A quantitative theoretical description of this interaction requires solution of the non-linear Poisson-Boltzmann equation for the interacting solute and membrane followed by calculation of the resulting force by integrating the electric stress tensor on the solute surface. Due to the complexity of the geometry... 

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) ... 

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 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... 

From the field stress tensor components, we may write the electric and magnetic field components as... 

The state of strain in a body is fully described by a second-rank tensor, a strain tensor , and the state of stress by a stress tensor, again of second rank. Therefore the relationships between the stress and strain tensors, i.e. the Young modulus or the compliance, are fourth-rank tensors. The relationship between the electric field and electric displacement, i.e. the permittivity, is a second-rank tensor. In general, a vector (formally regarded as a first-rank tensor) has three components, a second-rank tensor has nine components, a third-rank tensor has 27 components and a fourth-rank tensor has 81 components. 

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 solntion to Equations 4.75 and 4.76 yields the mechanical displacements and the electrical potential in piezoelectric medium. The above mechanical and electrical eqnations are conpled by matrix K ,j,that is represented in terms of the piezoelectric stress tensor e. As e —> 0, K ,j, —> 0 and the two sets (Equations 4.75 and 4.76) then represent pnre mechanical finite element and electrostatic field models, respectively. The sets of eqnation represented by Eqnation 4.75 and 4.76 can be solved using various commercially available packages such as ANS YS [67], PZFLEX, ABAQUS, and so forth (Section 4.7.6), all of which offer excellent postprocessing capabilities. 

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]... 

Consider a homogeneously stressed material and imagine a small surface area 55 somewhere within it (see fig. A.l). Let the direction of the vector 55 represent the normal to the area and let the length of 55 represent the magnitude of the area. The material on the side of 55 towards which the vector 55 points exerts a force on the material on the opposite side of 55. Let this force be 5f. Because the stress is homogeneous, 5F must be proportional to 55, but it must also depend on the direction of the normal to 55. Thus the two vectors 5F and 55 must be related by a second-rank tensor [ujfi, called the stress tensor, so that 5F, = 55,. (The symbol ct is conventionally used both for electrical... 

To close the system of equations for the fluid motion the tangential stress boundary condition and the force balance equation are used. The boundary condition for the balance of the surface excess linear momentum, see equations (8) and (9), takes into account the influence of the surface tension gradient, surface viscosity, and the electric part of the bulk pressure stress tensor. In the lubrication approximation the tangential stress boundary condition at the interface, using Eqs. (17) and (18), is simplified to... 

Particle trajectory is the result of the interaction of the particle with the electric field and the flow field. To simulate the particle trajectories, there are two approaches. The first approach is the Lagrangian tracking method, which neglects the finite size of the particles and treats them as point particles and solves the field variables without the presence of the particles [8]. In this case, only the effect of the field variables on the particle is considered. The second approach is the stress tensor approach, which includes the size effect of the particle. In this approach, the field variables are solved with the presence of the finitesized particle, and the particle translates as a result of the interaction of the particle with the electric and flow field [8]. In each incremental movement of the particle, the field variables need to be resolved. The former approach is very simple and works good to some extent, and the latter approach is accurate yet computationally expensive. 

To evaluate the resultant electromechanical force, one needs to utilize the fact that the tangential component of the electrical field at the surface of the liquid droplet vanishes, and accordingly, one may express the components of E on a plane with direction cosines n, as , = Utilizing the Cauchy s theorem relating the traction vector with the stress tensor components, one can write... 

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... 

Hoburg and Melcher [6] demonstrated electrohydrodynamic instabilities in macroscale systems at an oil-oil interface with a discrete conductivity change at the interface under the influence of an applied electric field. In the presence of an applied electric field, charge accumulates at the fluid-fluid interface, and the electrical force on the interface is balanced by the fluid interfacial stress tensor. At a critical field strength, the electrical force exceeds the... 

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