External body moment

Here x represents the position vector, K is the external body moment per unit volume, and t is any surface moment per unit area. In this equation, the velocity v is subject to the constraint... 

If we suppose that the external body moment K per unit mass is related to the generalised body force G via the relationship pK = n x G, that is,... 

In many instances of practical interest when the effect of gravity is neglected, an external body force F and external body moment K can arise from the presence of a magnetic field. The body moment due to an external magnetic field is assumed to take the form [74]... 

The vector Lagrange multiplier j3 has a physical interpretation, as pointed out by Leslie [177]. If we consider planar layers subject to an external body moment, for example via an external field, or, as we shall see below in Section 6.3, perhaps subject to fiow, then equation (6.89) shows that the smectic layers are subject to the moments li = liji/j where i/ is the unit outward surface normal. If the smectic layers are constrained to be planar, for example by boundary plates parallel to the xy-plane with the smectic layer normal a = (0,0,1) coincident with the z axis, then, by (6.90),... 

Following the form of equation (4.100), the external body moment may be written as... 

The thermokinetic process (3.133)-(3.135) in thermoelastic material (3.125) fulfilling mass balance generates the admissible thermodynamic process. Indeed, for chosen values of F, T and g = (Gradr)F at X (or place x) and t (see (3.13)) the fields of responses (3.115) follow by (3.125) the symmetric response T fulfils the balance of moment of momentum. Mass balance is satisfied by (3.135) and the balance of momentum (3.78) and energy (3.107) are satisfied by the appropriate choice of external body force b(Y, t) (or/and inertial force i(Y, r)) and volume heating Q(V,x) because (3.116) are controlled Ifom the outside. 

The equations (Eq. 23) representing balance of forces and moments constitute six equations for four unknowns, two components of the unit vector n and the scalars p and y-However, this apparent overdeterminacy does not materialize if the external body force and moment meet a certain requirement [5], To see this combine the two equations as follows... 

We shall always assume isothermal conditions and therefore ignore thermal effects. In these circumstances, as in any classically based continuum theory, conservation laws for mass, linear momentum and angular momentum must hold. The balance law for linear momentum, given below, is basically similar to that for an isotropic fluid, except that the resulting stress tensor (to be derived later) need not be symmetric. The balance law for angular momentum is also suitably augmented to include explicit external body and surface moments. 

Whenever the weight of a body is significant in comparison to the external forces, the weight, or body force, must be considered in both the force and moment balances. 

Therefore, for the internal (Neel) relaxation the parameter, r m plays the same role as the fluid viscosity r in the mechanism of the external (Brownian) diffusion. Note that the density of the anisotropy energy K is not included in x. This means that xD can be considered as the internal relaxation time of the magnetic moment only for magnetically isotropic particles (where K = a = 0). The sum of the rotations—thus allowing for both the diffusion of the magnetic moment with respect to the particle and for the diffusion of the particle body relative to the liquid matrix—determines the angle ft of spontaneous rotation of the vector p at the time moment t ... 

Raman scattering depends on the time correlation function of the many-body polarizability of the liquid, collective dipole moment. In the case of Raman scattering, an external electric field (from a laser) generates an induced collective dipole in the liquid ... 

The dipole moment m(f) = RgAfexp(+ icof), induced in a dielectric body by an externally applied electric field, e(t) = RgiSexp(+ itor), may... 

Applying external forces to an elastic body we change the relative position of its different parts which results in a change in body size and shape, i.e. under stressed conditions an elastic body undergoes deformation. As the particles of a body are shifted with respect to each other, the body develops elastic forces, namely stresses, opposing the deformation. In the course of deformation these forces increase and at a certain instant of time they can even counter-balance the effect of the external stress. At this moment the deformation process comes to an end, and the body is in a state of elastic equilibrium. As the stress is removed gradually, the elastic body returns to its initial state however, the abrupt disappearance of the outside force causes the particles inside the body to oscillate. To describe these oscillations, it is necessary to quantify the relationships between the forces arising at each point of the deformed... 

Gases exert pressure, because their particles bombard the surface of a material they come in contact with. At this moment, billions of gas particles, which make up the air around you, are striking your body every second. Under normal circumstances, your internal pressure keeps these collisions from having any noticeable effect on you. Sometimes the difference between the external air pressure and your internal pressure is evidenced by a discomfort in your eardrums. When your ears pop, your body attempts to compensate for the difference in pressure. 

Let the system consist of a molecule with (Rayleigh) polarizability tti, and a metallic body with the polarizability a2- Let the external field be Eq. The moments of the induced dipoles, in the molecule /jli and in the metal /X2, are given by the product of the polarizability and the total field operating on each moiety. Here one assumes the particles to be much smaller than the wavelength and, for simplicity, the tensorial properties are dropped. Thus,... 

M(t) instantaneous dipole moment of a dielectric body in the absence of an external field... 

When a metal body is exposed to an electric field, free electrons are displaced by electric forces until the field in the body vanishes. In an ideal dielectric (dc conductivity is zero), there exists only bound charges (electrons, ions) that can be displaced from their equilibrium positions until the field force and the oppositely acting elastic force are equal. This phenomenon is called displacement polarization (electronic or ionic polarization). A dipole moment is induced in every atom or between ion pairs. The molecular dipoles can only be rotated by an electric field. Usually, their dipole moments are randomly oriented. In an external held, however, an orientation parallel to the field direction is preferred so that a dipole moment is induced. This process is called orientational polarization. 

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