Generalized Newtonian, Bingham and conformation model equations

Use SI units and an oriented Euclidean three-space with Cartesian coordinates. Here $C^r$ means that coordinate partial derivatives through total order $r$ exist and are continuous. Let $I\subseteq \mathbb{R}$ be a nonempty open interval containing $0$, let $B\subseteq {\mathbb{R}}^3$ be a nonempty bounded open reference body with $C^1$ boundary, and let $\mathcal{U}$ be an open neighborhood of $\bar{B}$. A smooth motion is a map $\chi \in C^2(I\times \mathcal{U};{\mathbb{R}}^3)$ such that each ${\chi }_t(a):=\chi (t,a)$ is an orientation-preserving $C^2$ diffeomorphism onto its image; when labels are initially the current positions, ${\chi }_0(a)=a$. Set ${\mathrm{\Omega }}_t={\chi }_t(B)$. For $x\in {\chi }_t(\mathcal{U})$ its inverse label map is $A(t,x)={\chi }_t^{-1}(x)$, and its Eulerian velocity is $u(t,x)={\mathrm{\partial }}_t\chi (t,A(t,x))$. Position and material-label coordinates have units $\mathrm{m}$, time has units $\mathrm{s}$, and velocity has units $\mathrm{m}{\mathrm{s}}^{-1}$. The velocity gradient is $L=\mathrm{\nabla }u$, the rate-of-strain tensor is $D=(L+L^{\mathsf{T}})\mathrm{/}2$, the spin is $W=(L-L^{\mathsf{T}})\mathrm{/}2$, and the dilatation is $\theta =\mathrm{tr}D=\mathrm{tr}L$; all have units ${\mathrm{s}}^{-1}$. The material derivative is $D_t={\mathrm{\partial }}_t+u\cdot \mathrm{\nabla }$, acting componentwise on tensor fields, with units ${\mathrm{s}}^{-1}$ on dimensionless fields. Matrix contraction is $S:T=\mathrm{tr}(S^{\mathsf{T}}T)$, with Frobenius norm $\mathrm{\mid }S\mathrm{\mid }=(S:S{)}^{1\mathrm{/}2}$. These definitions are the mathematical motion and kinematics used below.

Let $M_3(\mathbb{R})$ denote the real $3\times 3$ matrices and $I_3$ the identity matrix. Write ${\mathrm{S}\mathrm{y}\mathrm{m}}_0(3)=\{D\in M_3(\mathbb{R}):D^{\mathsf{T}}=D,\mathrm{tr}D=0\}$. The generalized Newtonian branch is incompressible, so $\theta =0$ and $D\in {\mathrm{S}\mathrm{y}\mathrm{m}}_0(3)$. Define the shear-rate scalar $\dot{\gamma }=(2D:D{)}^{1\mathrm{/}2}$, with units ${\mathrm{s}}^{-1}$, and let $\eta (\dot{\gamma })$ be a finite nonnegative viscosity with units $\mathrm{P}\mathrm{a}\mathrm{s}$. The deviatoric stress is $\tau =2\eta (\dot{\gamma })D$, in $\mathrm{P}\mathrm{a}$; no pressure term is included.

For $a>0$ and $r\in \mathbb{R}$, use the positive-base real power $a^r:=\exp (r{\log }_{\mathrm{n}\mathrm{a}\mathrm{t}}a)$, where ${\log }_{\mathrm{n}\mathrm{a}\mathrm{t}}$ denotes the natural logarithm. The formulas apply this power to positive numerical values in the fixed SI units, with the resulting unit exponents carried by the listed parameter units. This is the convention in the viscosity formulas below; the unregularized form is evaluated only when $\dot{\gamma }>0$. The integral logarithm $\log$ used in the conformation branch is defined separately there.

Two named viscosity choices are the Carreau form

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

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

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

with $m>0$ in $\mathrm{P}\mathrm{a}{\mathrm{s}}^{\mathrm{n}}$, $\epsilon >0$ in ${\mathrm{s}}^{-1}$, and dimensionless $n>0$. For comparison, the unregularized power form ${\eta }_U=m{\dot{\gamma }}^{n-1}$ is defined only for $\dot{\gamma }>0$; its stress is extended by $\tau =0$ at $D=0$. When $0<n<1$, this viscosity diverges as $\dot{\gamma }\downarrow 0$, although the extended stress may remain continuous.

For the Bingham branch use the Frobenius norm just defined, ${\mu }_p\ge 0$ in $\mathrm{P}\mathrm{a}\mathrm{s}$, and yield parameter ${\tau }_y\ge 0$ in $\mathrm{P}\mathrm{a}$. Its set-valued deviatoric stress graph on ${\mathrm{S}\mathrm{y}\mathrm{m}}_0(3)$ is

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

The value at rest is therefore a whole stress ball when ${\tau }_y>0$. The chosen Frobenius normalization fixes the numerical convention for ${\tau }_y$; a source using a differently normalized second invariant requires a corresponding parameter conversion.

For the conformation branch, set $\mathrm{S}\mathrm{y}\mathrm{m}(3)=\{M\in M_3(\mathbb{R}):M=M^{\mathsf{T}}\}$ and ${\mathrm{S}\mathrm{y}\mathrm{m}}^{+}(3)=\{M\in \mathrm{S}\mathrm{y}\mathrm{m}(3):v^{\mathsf{T}}Mv>0\text{ for every }v\in {\mathbb{R}}^3\setminus \{0\}\}\mathrm{.}$ On the swept open set $\mathrm{\Omega }=\{(t,x)\in I\times {\mathbb{R}}^3:x\in {\chi }_t(\mathcal{U})\}$ from the smooth-motion definition, let the dimensionless conformation field be $C\in C^1(\mathrm{\Omega };\mathrm{S}\mathrm{y}\mathrm{m}(3))$. Thus $C$ is a $C^1$ matrix-valued field, and $D_{t}C={\mathrm{\partial }}_{t}C+u\cdot \mathrm{\nabla }C$ is defined componentwise. The ambient state space for the evolution equation is $\mathrm{S}\mathrm{y}\mathrm{m}(3)$; the stored-energy function below is defined only on the admissible subset ${\mathrm{S}\mathrm{y}\mathrm{m}}^{+}(3)$, and this definition does not assume that a solution remains in that subset along a trajectory.

For each $C\in {\mathrm{S}\mathrm{y}\mathrm{m}}^{+}(3)$, the finite-dimensional real spectral theorem supplies an orthonormal eigenbasis $v_1,v_2,v_3$ with $C{v}_i=c_i{v}_i$. Positive definiteness gives $c_i=v_i^{\mathsf{T}}C{v}_i>0$ for each unit eigenvector $v_i$, and determinant multiplicativity then gives $\det C=c_1{c}_2{c}_3>0$. Define $\log :(0,\mathrm{\infty })\to \mathbb{R}$ by $\log r={\int }_1^{r}ds\mathrm{/}s$. Let $G>0$ be a modulus in $\mathrm{P}\mathrm{a}$, $\lambda >0$ a relaxation time in $\mathrm{s}$, and ${\eta }_s\ge 0$ a solvent viscosity in $\mathrm{P}\mathrm{a}\mathrm{s}$. On an incompressible motion, meaning $\mathrm{div}u=\mathrm{tr}L=0$, adopt the conformation equation and stress and storage definitions

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

Here both sides of the conformation equation have units ${\mathrm{s}}^{-1}$; ${\tau }_p,{\tau }_s$ are polymer and solvent stresses in $\mathrm{P}\mathrm{a}$, and $w$ is a stored-energy density in $\mathrm{J}{\mathrm{m}}^{-3}$. The equation is defined for symmetric matrix fields, while $w$ is evaluated only where $C$ is positive definite. These are mathematical constitutive definitions and adopted model equations. They assert neither empirical validity for a particular material nor existence of a solution to a fluid partial differential equation.

smooth-motion definition and Eulerian kinematics.

orthonormal eigenbasis for a real symmetric matrix.

determinants multiply under matrix products.

positive-base real powers are defined by $a^r=\exp (r\log a)$.

Reddy, An Introduction to Continuum Mechanics, Chapter 6.

MIT OpenCourseWare, Lecture Notes on Fluid Dynamics, §1.6.2.

Barrett and Boyaval, Oldroyd-B model, §1.1.

Boyaval, Lelièvre, and Mangoubi, Oldroyd-B free-energy estimate, §§2.1.1–2.1.2.