Balance of moments

As the name implies, the cup-and-bob viscometer consists of two concentric cylinders, the outer cup and the inner bob, with the test fluid in the annular gap (see Fig. 3-2). One cylinder (preferably the cup) is rotated at a fixed angular velocity ( 2). The force is transmitted to the sample, causing it to deform, and is then transferred by the fluid to the other cylinder (i.e., the bob). This force results in a torque (I) that can be measured by a torsion spring, for example. Thus, the known quantities are the radii of the inner bob (R ) and the outer cup (Ra), the length of surface in contact with the sample (L), and the measured angular velocity ( 2) and torque (I). From these quantities, we must determine the corresponding shear stress and shear rate to find the fluid viscosity. The shear stress is determined by a balance of moments on a cylindrical surface within the sample (at a distance r from the center), and the torsion spring ... 

Therefore this work concerns the formulation of a proposal for the thermochemistry of an immiscible mixture of reacting materials with microstructure in presence of diffusion a new form of the integral balance of moment of momentum appears in the theory, in which the presence of the microstructure is taken into account. Moreover, the density fields can no longer be regarded as determined by the deformation fields because chemical reactions are present,... 

To formulate another main principle—the balance of moment of momentum—we introduce for some part of body (or body itself) with material volume V in actual configuration of the considered frame the moment of momentum or angular moment related to the point y as follows ... 

To obtain a simple form of the balance of moment of momentum, we confine its formulation to inertial frame with angular moment (3.88) having point y fixed here (although we use here the inertial frame fixed with distant stars, resulting formulations are valid in any inertial frame as will be shown at the end of this section). Again, the main reason for that is the nonobjectivity of x, y, v in (3.88), cf. (3.25), (3.38) generalization of this balance in the arbitrary frame will be discussed below but we note that the main local result—symmetry of stress tensor (3.93) below—is valid in the arbitrary frame. 

Therefore, the balance of moment of momentum or balance of angular momentum related to the fixed point y in aetual configuration in the inertial frame (fixed with distant stars) for (arbitrary part of) body with material volume V and its surface 9V is postulated as... 

Inserting it in (3.91) and using in this inertial reference configuration the balance of momentum (3.76) multiplied by (x — y) A from the left, we obtain the local balance of moment of momentum as... 

Using here the Reynolds theorem (3.24) we can, e.g., write the balance of moment of momentum related to even a nonfixed point y in an arbitrary (even noninertial) frame for a fixed volume V in actual configuration as... 

Balance of moment of momentum (3.93) expressed through the symmetry of a stress tensor (at least for mechanically nonpolar materials, cf. Rem. 17) is valid in any frame, even noninertial. Finally we can see that because (3.48) is valid for transformations between any inertial frames, the balances of angular moment related to fixed y (3.89)-(3.91) are valid in any inertial frame and not only in those fixed with distant stars. 

Summary. The first three balance equations are formulated in this section. The balances are necessary conditions to be fulfilled not only in thermodynamics but generally (in continuum mechanics). The balance of mass was formulated locally in several alternatives—(3.62), (3.63), or (3.65). The most important consequence of the balance of momentum is the Cauchy theorem (3.72), which introduces the stress tensor. The local form of this balance is then expressed by (3.76) or (3.77). The most relevant outcome of the balance of moment of momentum is the symmetry of the stress tensor (3.93). Note that in this section also an important class of quantities— the specific quantities—was introduced by (3.66) note particularly their derivative properties (3.67) and (3.68). 

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. 

This thermokinetic process (with validity of mass balance) generates an admissible thermodynamic process we can obtain the values of independent variables of constitutive equations (3.127), (3.146) in the whole body at any time (see (3.126), (3.112), (3.15)) and therefore also fields of responses (3.115) (with /). Further the balance of moment of momentum (3.93) is satisfied because of symmetric tensor T... 

The balance of moment ofmomentumfor constituent a in the inertial frame relative to the fixed point y is postulated in any fixed volume V with fixed surface 9 V in the mixture as (we use outer product from Rem. 16 in Chap. 3)... 

According to general procedure discussed in Sect. 4.1 (summing of Lh.s. of (4.65) and compensation of interactions), we postulate the balance of moment of momentum for the mixture in inertial frame relatively to the fixed place y for any fixed volume V with the surface 9 V in the mixture as... 

But the left-hand side of this equation is zero because of the local momentum balance for each constituent (4.56) and for mixture (4.63). So, the local balance of moment of momentum for the mixture has been obtained... 

Summary. The balance of momentum postulated for individual constituents leads to the Cauchy s theorem for partial stress tensors (4.53) and the local form of this balance is given by (4.56) or (4.57). The balance of momentum for mixture as a whole is given by (4.63) or (4.64). The balance of moment of momentum postulated for individual constituents gives the symmetry of the partial stress tensor—see (4.70). Analogical balance for mixture as a whole gives symmetry of sum of these tensors, cf. (4.75). Note that in mixture conceptually new quantities entered these balances— especially partial quantities and the interaction forces between constituents. 

The electric current in the electromagnetic coil is chosen such that the resulting mechanical moment (M ) of the magnetic forces keeps the beam exactly horizontal. Then the balance of moments is... 

Transitioning from the stress state of a particle to the stress field of the continuum, the interaction of the Cauchy stress tensor components of neighboring points needs to be investigated. They have to satisfy the conditions of local equilibrium to be established with the aid of an arbitrary infinitesimal volume element. Such an element with faces in parallel to the planes of the Cartesian coordinate system is subjected to the volume force and on the faces to the components of the Cauchy stress tensor with additional increments in the form of the first element of Taylor expansions on one of the respective opposing faces. The balance of moments proves the symmetry of the stress tensor,... 

For electric fields, certainly for the special cases generally considered, the outcome is rather similar in that the equation for the balance of forces integrates to give the pressure, and one can recast that for balance of moments in the same way as above. However, a general treatment does not appear to be available. 

What is less clear is that the equilibrium equation (2.143) actually arises from a balance of moments, as we shall now demonstrate. Consider an infinitesimal, rigid rotation co for which... 

Since a is an arbitrary constant vector, the properties of the scalar triple product (see equation (1-12)) then show that we have the balance of moments... 

Equation (6.78) represents a balance of forces, while, similar to the static theory of nematics, equations (6.79) and (6.80) are equivalent to a balance of moments see also Remark (i) below. Notice that the a-equations in (6.79) are coupled to the c-equations in (6.80) via the multiplier p. Recall that the Lagrange multipliers 7, p and T are scalar valued functions while is a vector function. 

As in the nematic liquid crystal case, it is clear that (6,78) represents a balance of forces at equilibrium. This can be seen by applying arguments that parallel those in Remark (i) on page 40. Similarly, it can be shown that the equilibrium equations (6.79) and (6.80) actually arise from a balance of moments, as has been demonstrated by Stewart and McKay [267], by means of an appropriate extension to the Ericksen identity (B.6) (cf. [181]). For the present, however, we restrict our attention to the derivation of the couple stress tensor Uj and its associated couple stress vector Analogous to the discussion for the nematic case in Remark (i) on page 40 when the balance of moments was discussed, consider an arbitrary, infinitesimal, rigid body rotation a for which... 

---