The notion of objectivity is found in early Greek and Chinese philosophies. This concept was often implied in solid mechanics research, particularly in the theoretical foundations of nonlinear elasticity early in the 20th century. Oldroyd [7] used the idea effectively in his work in rheology. However, to my knowledge, Noll [8,9,10] was the first to quantify this concept for broader use in physics. Malvern [11] and Gurtin [12] have lucid descriptions of the underlying mathematics. See also, Speziale [13], Murdoch [14], Frewer [15] and Pucci et al. [16] for recent discussions of some basic issues. As cogently explained by Truesdell and Toupin [3], objectivity encompasses two concepts. One is simply a coordinate transformation involving just rotation and transformation. The other is concerned with the philosophical matter of comparing measurements made by different observers attached to these coordinate systems. At a fundamental level, a change of observer is more than just a change of coordinate systems.

As noted by [3], Noll’s formulation of MFI provided important restrictions to the construction of power law constitutive theories for viscoelastic materials. MFI also crystallizes the primitive concept that material responses to dynamic processes, such as stress or inter-constituent fluxes, are innate properties and thus insensitive to observer motions. I propose an additional reason. The most fundamental characteristic of non-Newtonian fluid behavior is irreversibility. Not only should constitutive equations that parameterize the intrinsic transport of mass, energy and momentum be insensitive to observer motions, they should unequivocally agree on irreversible behavior as posited by Grad [17].

Nevertheless, MFI is controversial. Müller [18] claimed that generally-accepted approximations for the momentum and heat fluxes arising from kinetic theory were incompatible with the principle of objectivity. Woods [19] further questioned the basic definitions. Murdoch [20] argued that Müller’s results were in fact objective and that other studies, such as [21], could be readily rendered into frame-indifferent forms. The utility of MFI as a constraint on constitutive equations was the topic of an incisive discussion reported in Physica A [22]. Here, several investigators questioned the general validity of objectivity in material where centrifugal and Coriolis accelerations are dominate. In a later study [23], Evans and Heyes showed that ensemble averages arising from molecular dynamics simulations and group theoretic considerations are frame dependent.

What should be made of this? As noted recently by Liu and Sampio [24], much of the debate about MFI arises from confusion between the coordinate frame indifference and material frame indifference. The former is concerned with transformations between coordinate systems moving with respect to each other, while the latter concerns intrinsic properties of fluids, which are independent of observers. Perhaps the position of Bird [22] is appropriate here. He agreed that MFI was an approximation, but it has proven useful in many studies. To this, I would add that MFI also plays an important, but rarely appreciated, role in irreversibility.

The rationale for objectivity is that intrinsic properties of substances are not affected by the motions of observers. Measurements by different observers vary because they may be made at different times and because the observers may be moving and rotating with respect to each other. However, relative motion and time delay effects cancel for objective measurements. To quantify this, consider observers α and β embedded in two coordinate systems that are translating and rotating with respect to each other: (1) x α ( X , t * ) = Q ( t ) · x β ( X , t ) + b ( t )

Here, Q accounts for the rotation of β with respect to α. It satisfies Q · Q † = I , where Q † is the inverse of Q and I is the unit matrix. In the context used here, Q is the rotation matrix. The X are material coordinates embedded in the observed specimen and, thus, are unaffected by the motion of the observers. Furthermore, b is the translation of β relative to α. Finally, the observer times are simply shifted relative to each other as given by t * = t − a , where a is an arbitrary time delay. I stress that objective transformations are more general than simple Galilean transformations in that they account for the instantaneous relative orientations of observers.

Obviously, relative motions of the observers will not affect measurements of scalar properties, such as temperature. It is not so simple with observations of vectors and tensors, as their components depend on observer coordinates. Objective vectors and tensors must depend only on the instantaneous orientations of the observers. That is, objective vectors v and tensors T must satisfy: (2) v * = Q · v T * = Q · T · Q †

Here, I have used * to denote the coordinates and objects of one of the coordinate systems. This is in distinction to Equation (1), where the superscripts denoted observers embedded in the coordinate systems. It is noted that in solid mechanics applications, certain tensors that are functions of both spatial and initial coordinates transform objectively as vectors in the spatial frame [11].

Is the velocity of X as observed by α and β objective? Differentiation of Equation (1) gives: (3) d x α d t * = u α = d Q d t · x β + Q · d x β d t + d b d t = d Q d t · x β + Q · u β + d b d t

Clearly, velocity is not consistent with Equation (2), so such observations are not MFI. This is due to the relative rotation rate of the coordinate systems and any time dependence of the relative translation. A second differentiation of Equation (1) shows that acceleration observations also are not objective.

What about observations of gradients of objective vectors? The gradient of Equation (2) gives: (4) ∇ * v * = Q · ∇ v · ∇ * x = Q · ∇ v · Q †

Here, Equation (1) was solved for x β to get ∇ * x β = Q † . Apparently, the gradient of an objective vector is objective. However, the gradient operation applied to Equation (3) gives: (5) ∇ * u α = L α = Q · L β · Q † + d Q d t · Q † = Q · L β · Q † + Ω ∇ * u α † = L α † = Q † · L β † · Q + Q · d Q † d t = Q † · L β † · Q + Ω †

The velocity gradient is not MFI, because of the difference in coordinate rotations Ω. Differentiation of Q · Q † = I establishes that Ω + Ω † = 0 ; hence, the spin cancels for the symmetric part of the velocity gradient, as seen by adding the two equations of Equation (5). Thus: (6) L α + ( L α ) † = 2 D α = Q · [ L β + ( L β ) † ] · Q † = 2 Q · D β · Q †

In contrast, the fluid spin or vorticity is not objective, as the difference gives: (7) ∇ * u α − ( ∇ * u α ) † = 2 W α = Q · [ ∇ u β − ( ∇ u β ) † ] · Q † + 2 Ω = 2 Q · W β · Q † + 2 Ω

Expressions for d Q / d t and d Q † / d t are readily obtained from either Equation (5) or Equation (7). Two particularly useful in non-Newtonian fluid research are: (8) d Q d t = W α · Q − Q · W β = L α · Q − Q · L β d Q † d t = W β · Q † − Q † · W α = Q † · ( L α ) † − L β † · Q †

Viscoelastic non-Newtonian models require rates of change of dynamic quantities, such as stress or the heat flux, and even kinematic quantities, such as deformation rate or vorticity. Of course, these derivatives must also be objective; hence the question: Are material derivatives of objective quantities objective? Start with the derivative of the first equation of Equation (2) to get: (9) d v * d t * = Q · d v d t + d Q d t · v = Q · d v d t + W α · Q − Q · W β · v

Using Equation (8) and recognizing that u α = v * and u β = v , this reduces to: (10) d v * d t * − W α · v * = d j v * d t = Q · d v d t − W β · v = Q · d j v d t

The material derivative is not objective, but the operator denoted by the subscript j is. This is the Jaumann derivative [25]. An analogous calculation for an objective tensor leads to: (11) d j T * d t * = d T * d t * − W α · T * + T * · W α = Q · d T d t − W β · T + T · W β · Q †

Obviously, Equations (10) and (11) are not unique, as other combinations of terms in Equation (8) could be used. For example, straightforward calculations give: (12) d o v * d t * = d v * d t * − L α · v * = Q · d v d t − L β · v d o T * d t * = d T * d t * + L α † · T * + T * · L α = Q · d T d t + L β † · T + T · L β · Q †

This last operator is known as the convective or Oldroyd derivative [7].

It is stressed that although these and other co-rotation rate measures are objective, they are not equal. Hence, predictions of non-Newtonian fluid responses depend on which MFI derivative operator is used. Appendix D of [25] summarizes these derivatives and includes tables that compare notations.

There is considerable empirical evidence suggesting that the constitutive behavior of non-Newtonian fluids should include some measure of fluid vorticity. Consequently, considerable effort has been spent to develop objective measures using MFI operators, such as the above. See, for example, [26] for an application to Rivlin–Ericksen fluids, [27] for an application to a general polymeric fluid, [28] for a straightforward application to viscoelastic flows and [29,30] for applications to objective measures of vortices in geophysical fluid settings. Useful theoretical templates for constructing objective measures of skew-symmetric tensors are given by [31,32,33,34].

Typically, these measures depend on a “co-rotation rate” of the principle axes of the deformation tensor. In this case, care should be exercised in applying these measures whenever two or more of the axes are the same. Moreover, delineating these axes from experimental or computational experiments is not always straightforward. Nevertheless, as the cited studies show, inclusion of an objective spin in constitutive models has produced some remarkable successes.

Application of MFI to multiphase materials and mixtures raises additional issues. Massoudi [35] has discussed these in the context of an application to lift forces in multiphase flows. His approach is followed here.

Instead of a single test specimen with material coordinates X , consider two specimens with material coordinates X 1 and X 2 . The specimens may be different constituents or different phases of the same constituent. A fundamental tenant of mixture theory is that the two specimens occupy the same geometric point, i.e., x 1 = x 2 . See [36,37] for thorough discussions of the tenants of mixture theory. Whether a mixture or multiphase material, Equation (1) is generalized to: (13) x 1 α ( X 1 , t † ) = Q ( t ) · x 1 β ( X 1 , t ) + b ( t ) x 2 α ( X 2 , t † ) = Q ( t ) · x 2 β ( X 2 , t ) + b ( t )

The velocities of the specimens are: (14) d x 1 α d τ = u 1 α = d Q d t · x 1 β + Q · u 1 β + d b d t d x 2 α d τ = u 2 α = d Q d t · x 2 β + Q · u 2 β + d b d t

Then: (15) u 1 α − u 2 α = u 12 = Q · ( u 1 β − u 2 β )

Here, u 12 = − u 21 is the “diffusion” velocity. For those not familiar with mixture theory, it is noted that the material derivatives of the two substances are not the same, since their velocities are different (see Massoudi [35] and Rajagopal and Tao [37] for a careful analysis and discussion of this matter). Nevertheless, the constituent velocity differences are objective! Moreover, as shown by [35], the relative acceleration and spin differences are also objective. This is a key result for developing constitutive equations for mixtures of solids and fluids.

The purpose of this section is to use two recent results in non-Newtonian fluid mechanics to show how MFI impacts irreversibility. The following subsection focuses on a constitutive model for a mixture of two viscous fluids at different temperatures. Although the constitutive equations are linear, there is considerable interaction between the temperature and velocity fields of the two fluids. Subsection 4.3 deals with a nonlinear constitutive model for a granular substance. Here, the nonlinear cross-coupling between the vector and tensor processes is fundamentally nonlinear. In both cases, objective specification of the vector and tensor processes is a key aspect to demonstrating irreversibility.

A distinctive characteristic of many non-Newtonian fluids is the robust interaction between different dynamic fields, such as stress and heat flux. An illustration of the interaction of these fields and the consequent impact on irreversibility was reported by [44]. The vector processes are restricted to just sensible heat flux associated with two temperature gradients and heat generated by the friction of the fluids moving past each other. Vectorial mechanisms, such as Dufour and Soret processes, are certainly important in many applications, but are not considered here. Tensor processes include the viscous stresses of each of the constituents in addition to viscous interactions between the constituents. Consequently, the constitutive equations for the stresses include the asymmetric effects depicted in Equation (21). Characterization of asymmetric stresses uses the spin differences of the two fluids in accordance with [35].

The appropriate vector objective variables are the two temperature gradients, ∇ θ 1 and ∇ θ 2 , along with the diffusion velocity u 12 . Applying the restricted principle of equipresence noted earlier, the constitutive equations are given as: (23) q i = − ϕ i k i i ∇ θ i + ∑ j = 1 2 ϕ j k i j ∇ θ j + h i j u i j m i = − ∑ j = 1 2 ϕ i ϕ j l i j u i j + ∑ γ = 1 2 r i j γ ∇ θ γ

Here, q i and m i are the heat flux and internal momentum source of substance/phase i, respectively; ϕ i is the volume fraction of that constituent; and k i j , h i j , l i j and r i j γ are phenomenological coefficients. These equations were formulated so that if one of the constituents vanishes, the system reduces to just Fourier’s law of heat conduction for a single constituent.

Tensor processes are simply generalizations of the Navier–Stokes law to include all objective tensor quantities. These are the deformation of each phase D i and the spin difference W i . Consistent with Equation (18), deformation is divided into trace and deviator components.

The tensor constitutive equation is: (24) T i = ϕ i λ i i D i + ∑ j = 1 2 ϕ j λ i j D j I + 2 μ i i D d i + ∑ j = 1 2 ϕ j μ i j D d j + ∑ j = 1 2 ν i j ϕ j W i − W j

Note that if one of the volume fractions vanishes, then Equation (24) reduces to the Navier–Stokes equations for a single constituent.

The irreversibility condition is given by: (25) σ = − ∑ i = 1 2 C i C i q i · ∇ θ i + m i · v i − T i : ∇ v i ≥ 0

Here, C i = θ i − 1 . This is sometimes called the coldness of constituent i. Examination of Equation (25) indicates a dilemma: all of the dependent variables are objective, except v i . This obstacle was overcome in [44] by imposing the following constraints on l i j and r i j γ : (26) C 1 l 12 = − C 2 l 21 = L C 1 r 121 = − C 2 r 211 = R 1 C 1 r 122 = − C 2 r 212 = R 2

Then, Equation (25) reduces to: (27) C 1 2 ϕ 1 k 11 ∇ θ 1 · ∇ θ 1 + C 2 2 ϕ 2 k 22 ∇ θ 2 · ∇ θ 2 + ϕ 1 ϕ 2 C 1 2 k 12 + C 2 2 k 21 ∇ θ 1 · ∇ θ 2 + ϕ 1 ϕ 2 u d · C 1 2 h 12 ∇ θ 1 − C 2 2 h 21 ∇ θ 2 + ϕ 1 ϕ 2 u d · L u d + R 1 ∇ θ 1 + R 2 ∇ θ 2 + ϕ 1 λ 11 C 1 D 1 2 + ϕ 2 λ 22 C 2 D 2 2 + ϕ 1 ϕ 2 D 1 D 2 C 1 λ 12 + C 2 λ 21 + 2 ϕ 1 μ 11 C 1 D d 1 2 + ϕ 2 μ 22 C 2 D d 2 2 + 2 ϕ 1 ϕ 2 D d 1 : D d 2 C 1 μ 12 + C 2 μ 21 + ϕ 1 ϕ 2 ν 12 C 1 W 1 2 + ν 21 C 2 W 2 2 − C 1 ν 12 + C 2 ν 21 W 1 : W 2 ≥ 0

The irreversibility constraints arising from Equation (27) can be divided into two groups. The first are essentially those that would arise from single fluid analysis: (28) k i i , λ i i , μ i i , ≥ 0

The second group are Onsager-type constraints involving the magnitudes of the interaction coefficients. These are: (29) 4 λ 11 λ 22 C 1 C 2 ≥ ϕ 1 ϕ 2 ( C 1 λ 21 + C 1 λ 22 ) 2 4 μ 11 μ 22 C 1 C 1 ≥ ϕ 1 ϕ 2 ( C 1 μ 12 + C 2 λ 21 ) 2 ν 12 C 1 = ν 21 C 2 and that the roots of the matrix: C 1 2 ϕ 1 k 11 ϕ 1 ϕ 2 ( C 1 2 k 12 + C 2 2 k 21 ) / 2 ϕ 1 ϕ 2 ( R 1 + C 1 2 h 12 ) / 2 ϕ 1 ϕ 2 ( C 1 2 k 12 + C 2 2 k 21 ) / 2 C 2 2 ϕ 1 k 22 ϕ 1 ϕ 2 ( R 2 − C 2 2 h 21 ) / 2 ϕ 1 ϕ 2 ( R 1 + C 1 2 h 12 ) / 2 ϕ 1 ϕ 2 ( R 2 − C 2 2 h 21 ) / 2 ϕ 1 ϕ 2 L be nonnegative. A special case of the latter condition is when the diffusion velocity u 12 = 0 . Then: (30) 4 C 1 2 C 2 2 k 11 k 22 ≥ ϕ 1 ϕ 2 ( C 1 2 k 12 + C 2 2 k 21 ) 2

Note that the interaction irreversibility constraints impose both temperature and volume fraction dependences on the phenomenological coefficients. Both functions are obtained from solutions to the dynamic equations. That they also appear in the irreversibility constraint indicates a coupling between the solution and the irreversibility condition. This connection is rarely, if ever, explored.

Yang et al. [45] proposed a general nonlinear constitutive model for granular materials that includes both dissipation and heat conduction. Furthermore, see Massoudi and Kirwan [46] for additional analysis and discussion of this model. For an incompressible fluid, the constitutive equations reduce to: (31) T d = β 3 D d + β 4 ( ∇ ϕ ⊗ ∇ ϕ ) d q = [ a 1 I + a 3 D d + a 5 D d · D d ] · ∇ θ + [ a 2 I + a 4 D d + a 6 D d · D d ] · ∇ ϕ

Here, ϕ is the volume fraction; θ is the temperature; D d is the deviator of the velocity gradient; and the subscript on ( ∇ ϕ ⊗ ∇ ϕ ) d indicates the deviator of the tensor product. I imposed this restriction so that this model only applies to the deviator component of T . Furthermore, the phenomenological coefficients β j and a j are functions of ϕ and perhaps geometric invariants of D d . Note also that D d : D d was neglected in the equation for T in Equation (31). As detailed by [45], the variables ∇ θ , ∇ ϕ and D d are objective. Note also that Equation (31) reduces to the Navier–Stokes equation for viscous stress when only β 3 ≠ 0 and Fourier’s heat conduction law when only a 1 ≠ 0 .

Application of the Clausius–Duhem inequality, Equation (16), to Equation (31) requires: (32) q · ∇ θ + T d : D d = σ ≥ 0

Using Equation (31) in Equation (32) produces the inequality: (33) β 3 ( D d ) 2 + β 4 ( ∇ ϕ ⊗ ∇ ϕ ) d : D d + a 1 ( ∇ θ ) 2 + a 2 ∇ ρ · ∇ θ + a 3 ( D d · ∇ θ ) · ∇ θ + a 5 D d · ( D d · ∇ θ ) + a 4 ( D d · ∇ ϕ ) · ∇ θ + a 6 ( D d · D d · ∇ ϕ ) · ∇ θ ≥ 0 .

The constraints that arise from Equation (33) are: (34) β 3 , a 1 ≥ 0 β 3 ( D d ) 2 + a 1 ( ∇ θ ) 2 ≥ − [ β 4 ( ∇ ϕ ⊗ ∇ ϕ ) d : D d + a 2 ∇ ρ · ∇ θ + a 3 ( D d · ∇ θ ) · ∇ θ + a 5 D d · ( D d · ∇ θ ) + a 4 ( D d · ∇ ϕ ) · ∇ θ + a 6 ( D d · D d · ∇ ϕ ) ] · ∇ θ .

The first two conditions are recognized as the irreversibility conditions for Navier–Stokes fluids and Fourier’s law for heat conduction. As in the previous example, the second condition restricts the interaction coefficients and the solution properties. In view of Equation (29) and the fundamental nonlinearity of the constitutive model, it is not surprising that the dependent dynamic variables appear in the irreversibility constraint. I am unaware of any study that evokes this condition on solutions for the dynamic variables.