Frame-indifference

Campbell, G.A., Sweeney, P.A., Dontula, N., and Wang, Ch., Frame Indifference Fluid Flow in Single Screw Pumps and Extruders, Int. Polym. Process., 11, 199 (1996)... 

Since historically the dissipation is evaluated using the local velocity at the boundary and the shear stress is evaluated as the product of the viscosity and the shear rate at the boundary, it follows that if the velocity is not frame indifferent then the dissipation will not be frame indifferent. As discussed previously in this chapter, rotation of the barrel at the same angular velocity as the screw are the conditions that produce the same theoretical flow rate as the rotating screw. Because the flow rate is the same and the dissipation is different, it follows that the temperature increase for barrel and screw rotation is different. This section will demonstrate this difference from both experimental data and a theoretical analysis. 

The viscous energy dissipation is calculated using velocities in the laboratory (Eulerian) reference frame. As previously stated here and by Malvern [48], velocities are not frame indifferent. The Eulerian velocities and provided by... 

Since flow is frame indifferent, then the transformed flow rate calculation for rate can be calculated using the above Eq. A7.51. The flow rate is obtained by integrating the z-direction velocity of the area as follows ... 

In consideration of the principle of material frame indifference the fundamental laws for the multi-component mixture in the Eulerian description are formulated in the following local forms. 

The frame indifference principle states that constitutive equations must exhibit coordinate indifference, that is, the properties of a material must be independent of the reference frame. 

The principle of material frame indifference (objectivity) states that variables, for which constitutive equations are needed, cannot depend on the coordinate frame (i.e.. Euclidean space, plus time) in which the variables are expressed. 

However, just after this paper was published, Acrivos pointed out in a personal communication with Drew and Lahey that a fundamental error had been made in the derivation. Therefore, in a corrigendum by Drew and Lahey [34] the conclusions regarding objectivity were modified to apply only approximately to a fluid with very small vorticity. In practice, the sum of the added mass and lift forces is thus not necessarily frame-indifferent when applied to reactor simulations. 

The rate of deformation and the pressure are frame-indifferent (e.g., see [6], p 141 [54], p 400 [28, 31, 34]) so we can simply re-write the divergence operator and the stress terms into the rotating reference frame notation. The transformed momentum equation delds ... 

In more complicated material models we modify oruse further constitutive principles determinism is enlarged for densities (mass concentrations) in mixtures (cf. Sects. 2.4,3.5,4.5), and the definition of fluid used in this principle is in fact the result of constitutive principle of symmetry (see Rem. 30 in Chap. 3). Another constitutive principle is the objectivity (frame indifference) principle. Here it is trivially satisfied because motion is neglected and all quantities are objective (see Sects. 3.2,3.5). In nonuniform systems the influence of neighborhood is described in the principle of local action (cf. Sect. 3.5). In mixtures, the property of mixture invariance [32] may also be used as a constitutive principle [33]. 

Many quantities used in the following considerations are called objective ot frame-indifferent, if they are invariant in the change of frame (3.25), (3.26) as follows (because this change contains rotations and/or inversions of corresponding Cartesian systems as a very special case (cf. Fig. 3.1), the following definition is motivated by (b), (c) of Rem. 4) ... 

Objective or frame-indifferent scalar a, vector a and (second order) tensor A transform by the change of frame on scalar a, vector a and tensor A as follows ... 

Applying the change of frame (3.25), (3.26) to the above definitions of Sect. 3.1 and, using these precepts, we can decide about objectivity or nonobjectivity (frame indifference or not) of the following quantities (more detailed proofs of some of them are written in the footnote-sized script below) the remainder from the next sections may be proved analogously. 

Deformation gradient F (3.10) is not a frame-indifferent (objective) tensor, because it transforms as... 

Transformation properties of some objects (mostly derivatives useful in the following chapters) formed from scalar a, vector a, tensor A which are objective (frame indifferent) (3.30)-(3.32), will be discussed now. We must realize that these objective conditions must be valid at any x, t transforming by (3.25), (3.26) to X, t (the same event seen from different frames passing at the same particle X) therefore ... 

As a primitive we assign to each particle X of this body the (mass) density p— positive and (assuming) objective (frame indifferent) scalar. Mass of the body or its arbitrary part with material volume V is then... 

At the end of these Sects. 3.3 and 3.4 we note that energy balance and entropy inequality motivated by procedures like those in Chap. 1 together with generalization of frame indifference (plausible objectivity is postulated not only for motion (Sect. 3.2) but also, e.g., for power of surface and body forces or heating) permit to deduce balances in Sect. 3.3 (i.e., for mass, linear and angular momentum), internal energy, entropy and their objectivity, etc. For details see, e.g., [1, 22, 42, 43] and other works on modern thermomechanics [7, 8, 18, 20, 41]. 

Further reduction of constitutive equations (3.119) may be achieved by the constitutive principle of frame indifference or the principle of objectivity, the material properties and therefore also constitutive equations must be independent of the choice of frame. This principle is a generalization of common experience with mate-... 

In a special choice Q = 1 and arbitrary b, c, c, Q, these transformations must give the same values of responses in (3.119) and (3.120) (because T is the same in both frames) and this is possible (change from (3.119) to (3.120) is valid for any values of independent variables) only ifresponses are independent of variables x, t, v and W. This means that two observers with a shift in origins of time and space and with different velocities of translation and rotation must obtain the same responses. Therefore, the constitutive equations (3.119) must be reduced by the principle of frame indifference (or objectivity) to the form... 

Murdoch, A.I. On material frame-indifference. Proc. R. Soc. Lond. A380, 417 26 (1982)... 

Silhavy, M. Mass, internal energy, and Cauchy s equations in frame-indifferent thermodynamics. Arch. Ration. Mech. Anal. 107(1), 1-22 (1989)... 

Muschik W Objectivity and frame indifference. Arch. Mech. 50, 541-547 (1998)... 

Similarly as for the single substance in Sect. 3.4 we postulate that scalars <, q, Q are objective (frame indifferent) as well as ba, T . 

Moreover, because q and (arbitrary) n are objective (see end of Sect. 3.2), then from (4.78) follows (similarly as below (3.101)) that the heat flux q in mixture is objective (frame indifferent) vector cf. also Rem. 21 in Chap. 3. 

We also note that mixture properties (4.90)-(4.92) are objective (frame indifferent). For later applications, it is useful to define the total stress T by... 

Now we restrict such constitutive equations—responses (4.120) as functions of (4.126)—by the principle of objectivity (or (material) frame indifference), cf. Sect. 3.5 constitutive equations cannot depend explicitly on (non-objective) x and t... 

Note that the force vector represented in the form of (2.5) is a fundamental hypothesis of Newtonian mechanics (i.e., the frame indifference of a force vector see Sect. 2.2.2). The third law gives the interacting forces for a two-body problem, and this will not be treated here. 

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