Restricted non-Newtonian constitutive examples

Let $M_3(\mathbb{R})={\mathbb{R}}^{3\times 3}$ be the real $3\times 3$ matrices. For $M=(M_{ij})\in M_3(\mathbb{R})$, define its trace by $\mathrm{tr}M={\sum }_{i=1}^3{M}_{ii}$, and let $I_3$ be the $3\times 3$ identity matrix, with diagonal entries $1$ and all other entries $0$.

Let $I\subseteq \mathbb{R}$ be a nonempty open interval containing $0$, let $B\subseteq {\mathbb{R}}^3$ be a bounded open reference body with $C^1$ boundary, and let $U$ be an open neighborhood of $\bar{B}$. A material motion is a map $\chi \in C^2(I\times U;{\mathbb{R}}^3)$ whose slices ${\chi }_t(a)=\chi (t,a)$ are orientation-preserving $C^2$ diffeomorphisms onto their images, with $\chi (0,a)=a$. Its swept open region is $\mathrm{\Omega }=\{(t,x):x\in {\chi }_t(U)\}$, its inverse label is $A(t,x)={\chi }_t^{-1}(x)$, and its Eulerian velocity is $u(t,\chi (t,a))={\mathrm{\partial }}_t\chi (t,a)$. Set $L_{ij}={\mathrm{\partial }}_j{u}_i$, $D=(L+L^{\mathsf{T}})\mathrm{/}2$, $\theta =\mathrm{tr}D=\mathrm{div}u$, and $D_t={\mathrm{\partial }}_t+u\cdot \mathrm{\nabla }$. A branch is incompressible when $\theta =0$, so $D$ is symmetric and trace-free. Write $S:T=\mathrm{tr}(S^{\mathsf{T}}T)$ for the Frobenius contraction and $\mathrm{\mid }S\mathrm{\mid }=(S:S{)}^{1\mathrm{/}2}$ for its norm. Define ${\mathrm{S}\mathrm{y}\mathrm{m}}_0(3)=\{D\in M_3(\mathbb{R}):D=D^{\mathsf{T}},\mathrm{tr}D=0\}$ and ${\mathrm{S}\mathrm{y}\mathrm{m}}^{+}(3)=\{C\in M_3(\mathbb{R}):C=C^{\mathsf{T}},z^{\mathsf{T}}Cz>0\text{ for every }z\mathrm{\ne }0\}$. A rigid observer change is $x^{\ast }=Q(t)x+c(t)$ with $Q\in C^2(I,\mathrm{S}\mathrm{O}(3))$, $c\in C^2(I,{\mathbb{R}}^3)$, and $\mathrm{S}\mathrm{O}(3)=\{Q:Q^{\mathsf{T}}Q=I_3,\det Q=1\}$. A second-order tensor transforms as $S^{\ast }=QS{Q}^{\mathsf{T}}$; a constitutive response is objective when it obeys this rule at corresponding events.

For the generalized Newtonian and Bingham branches, require $\theta =\mathrm{div}u=0$ throughout $\mathrm{\Omega }$, so $D\in {\mathrm{S}\mathrm{y}\mathrm{m}}_0(3)$. In these branches, $\tau$ is a symmetric deviatoric extra stress, meaning $\mathrm{tr}\tau =0$; the total tension-positive Cauchy stress may additionally contain an isotropic pressure. In the conformation branch, ${\tau }_p$ and ${\tau }_s$ denote the polymer and solvent stresses separately and the pressure is a Lagrange multiplier. Stress is measured in Pa, $D$ and shear rate in ${\mathrm{s}}^{-1}$, viscosity in $\mathrm{P}\mathrm{a}\mathrm{s}$, and a rate of stress power per volume in $\mathrm{W}{\mathrm{m}}^{-3}$.

For a generalized Newtonian branch, define

$$\dot{\gamma }=(2D:D{)}^{1\mathrm{/}2},\tau =2\eta (\dot{\gamma })D,$$

where $\eta :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is finite and continuous. Two permitted scalar choices are the Carreau form

$${\eta }_C(\dot{\gamma })={\eta }_{\mathrm{\infty }}+({\eta }_0-{\eta }_{\mathrm{\infty }})(1+({\lambda }_C\dot{\gamma }{)}^2{)}^{(n_C-1)\mathrm{/}2},$$

with ${\eta }_0\ge {\eta }_{\mathrm{\infty }}\ge 0$ in $\mathrm{P}\mathrm{a}\mathrm{s}$, ${\lambda }_C\ge 0$ in seconds, and dimensionless $n_C>0$, and the regularized power form

$${\eta }_R(\dot{\gamma })=m({\epsilon }^2+{\dot{\gamma }}^2{)}^{(n_R-1)\mathrm{/}2},$$

with $m>0$ in ${\mathrm{P}\mathrm{a}\mathrm{s}}^{n_R}$, $\epsilon >0$ in ${\mathrm{s}}^{-1}$, and dimensionless $n_R>0$. The unregularized power form ${\eta }_U=m{\dot{\gamma }}^{n_R-1}$ is defined only for $\dot{\gamma }>0$; its stress is extended by $\tau =0$ at $D=0$, so for $0<n_R<1$ the viscosity diverges at zero rate even though the extended stress can be continuous.

For the Bingham branch, let ${\mu }_p\ge 0$ in $\mathrm{P}\mathrm{a}\mathrm{s}$ and ${\tau }_y\ge 0$ in $\mathrm{P}\mathrm{a}$. Adopt $\tau \in \mathcal{B}(D)$, where the set-valued graph on ${\mathrm{S}\mathrm{y}\mathrm{m}}_0(3)$ is

$$\mathcal{B}(D)=\{\begin{array}{ll}\{2{\mu }_{p}D+{\tau }_{y}D\mathrm{/}\mathrm{\mid }D\mathrm{\mid }\},& D\mathrm{\ne }0,\\ \{S\in {\mathrm{S}\mathrm{y}\mathrm{m}}_0(3):\mathrm{\mid }S\mathrm{\mid }\le {\tau }_y\},& D=0,\end{array}$$

where ${\mathrm{S}\mathrm{y}\mathrm{m}}_0(3)$ is the space of trace-free symmetric $3\times 3$ real matrices. Here ${\tau }_y$ is, by definition, the radius of the Frobenius stress ball at rest; other invariant normalizations require a corresponding parameter conversion. The graph is single-valued for $D\mathrm{\ne }0$ and set-valued at rest.

For an isothermal conformation branch, let $K=[t_0,t_1]\subset I$ be a compact interval with $t_0<t_1$. A pathline on $K$ is a continuously differentiable curve $x:K\to {\mathbb{R}}^3$ satisfying $(t,x(t))\in \mathrm{\Omega }$ and $x^{\mathrm{\prime }}(t)=u(t,x(t))$ for every $t\in K$; derivatives at the endpoints are one-sided. Assume $\mathrm{div}u=0$ on $\mathrm{\Omega }$ and take such a pathline. Let $C\in C^1(K;{\mathrm{S}\mathrm{y}\mathrm{m}}^{+}(3))$ be a dimensionless matrix-valued function. Along this pathline, the material derivative of $C$ is its ordinary derivative, $D_{t}C(t,x(t)):=\frac{d}{dt}C(t)$. Set $L(t)=\mathrm{\nabla }u(t,x(t))$ and $D(t)=(L(t)+L(t{)}^{\mathsf{T}})\mathrm{/}2$. Let $G>0$ be a modulus in Pa, $\lambda >0$ a relaxation time in seconds, ${\eta }_s\ge 0$ a solvent viscosity in $\mathrm{P}\mathrm{a}\mathrm{s}$, and $T_0>0$ a fixed temperature in K. Here an external thermostat is an idealized heat reservoir that maintains $T_0$ and accounts for heat released by this branch; it is an additional model assumption. Adopt

$$\frac{dC}{dt}=LC+C{L}^{\mathsf{T}}-(C-I_3)\mathrm{/}\lambda ,{\tau }_p=G(C-I_3),{\tau }_s=2{\eta }_{s}D,w(C)=\frac{G}{2}(\mathrm{tr}C-\log \det C-3)\mathrm{.}$$

The polymer and solvent stresses ${\tau }_p,{\tau }_s$ are measured in Pa, and $w$ is a stored-energy density in $\mathrm{J}{\mathrm{m}}^{-3}$. The determinant is positive because $C$ is symmetric positive definite. These are selected model equations on the stated smooth incompressible regime; they assert no empirical applicability to a particular material, fitted parameter values, PDE solution existence, or universal validity.

pathline and material-derivative definitions

J. N. Reddy, An Introduction to Continuum Mechanics, 2nd ed., Chapter 6, equation (6.5.15) and §§6.6.1–6.6.2.3, PDF pp. 25–27 (printed pp. 245–247)

Chiang C. Mei, MIT 1.63J/2.21J Lecture Notes on Fluid Dynamics, §1.6.2, equations (1.6.13)–(1.6.18), complete 4-page PDF

John W. Barrett and Sébastien Boyaval, Existence and Approximation of a (Regularized) Oldroyd-B Model, arXiv:0907.4066v2, §1.1 and §2.3, PDF pp. 2–11

S. Boyaval, T. Lelièvre, and C. Mangoubi, Free-energy-dissipative schemes for the Oldroyd-B model, ESAIM: M2AN 43 (2009), §§2.1.1–2.1.2, PDF pp. 4–6

David Tong, Fluid Mechanics, Part I, §§1.1.1–1.1.3, PDF pp. 6–9