Maxwell’s stress tensor

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 retarded dispersion energy between macroscopic particles was treated by Liftshitz [28]. He considered half-spaces. Going half the way from the microscopic to the macroscopic approach, Lifshitz expanded the local fluctuations within the half-spaces in terms of plane waves and coupled them to the outgoing (reflected) radiation field. Then, satisfying the boundary conditions for the radiation field across the surfaces of the half-spaces under consideration, he found their force of attraction from Maxwell s stress tensor in the interspace. 

The calculation of the force of attraction from Maxwell s stress tensor, i.e. the omission of the Poynting vector in the total energy balance, is equivalent to calculating the dispersion energy merely from the real part of the free energy gain according to Eq. (3.72). 

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]). 

A frequently used example of Oldroyd-type constitutive equations is the Oldroyd-B model. The Oldroyd-B model can be thought of as a description of the constitutive behaviour of a fluid made by the dissolution of a (UCM) fluid in a Newtonian solvent . Here, the parameter A, called the retardation time is de.fined as A = A (r s/(ri + s), where 7]s is the viscosity of the solvent. Hence the extra stress tensor in the Oldroyd-B model is made up of Maxwell and solvent contributions. The Oldroyd-B constitutive equation is written as... 

For the Maxwell field, the energy-momentum tensor Tfi(A) derived from Noether s theorem is unsymmetric, and not gauge invariant, in contrast to the symmetric stress tensor derived directly from Maxwell s equations [318], Consider the symmetric tensor 0 = T + AT, where... 

Note first that if the fluid is at a state of equilibrium with no flow, then the time derivative d is equal to zero, and the velocity gradient Vv is also zero. This implies from the above equations that = G8. Hence cr, i = <7 2 = ct t, = G at equilibrium, and aj = 0, for i j. Thus, although the diagonal stress components are not zero at equilibrium, they are all equal to each other, and the nondiagonal components are all equal to zero. Hence, the stress tensor is isotropic, but nonzero at equilibrium. (If one redefines the stress tensor as H = a — G8, then S " = 0 at equilibrium. The upper-convected Maxwell equation can then be rewritten in terms of Z .)... 

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,... 

Here, C denotes the configuration tensor and D describes the rate-of-strain tensor of the material continuum. The dimensionless anisotropy factor, cc, characterizes the anisotropic character of the particle mobility. It is easy to show that a attains values between zero and one. The limiting case a = 0 corresponds to the isotropic motion and ultimately leads to an upper converted Maxwell material. In order to derive a deformation-dependent constitutive equation, a Hookean law connecting the tensor of external stresses S, the configuration tensor C, and the shear modulus G was suggested ... 

Just as there are various possible finite strain tensors, there are various time derivatives that can be used in place of the ordinary derivative of stress in Eq. 10.21 to satisfy the continuum mechanics requirements for a model to be able to describe large, rapid deformations in arbitrary coordinate systems. The derivative that yields a differential model equivalent to Lodge s Eq. 10.6 is the upper convected time derivative (defined in Eq. 11.19), and the resulting model is called the upper-convected Maxwell model. Other possibilities include the lower-convected derivative and the corotational derivative. Furthermore, a weighted-sum of two of these derivatives can be used to formulate a differential constitutive equation for polymeric liquids. In particular, the Gordon-Schowalter convected derivative [7] is defined in this manner. 

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