Generalized Newtonian fluid (GNF) model for incompressible non-Newtonian fluids. $$ \begin{equation} \boldsymbol{T} = -p \boldsymbol{1} + 2 \eta \boldsymbol{D} = -p \boldsymbol{1} + \boldsymbol{\tau} \tag{5.67} \end{equation} $$
Stress deviator $$ \begin{equation} \boldsymbol{\tau} = 2 \eta \boldsymbol{D} \tag{5.68} \end{equation} $$
Viscosity function \( \eta \) $$ \begin{equation} \eta = \eta(\dot{\gamma}) \tag{5.69} \end{equation} $$
Shear measure \( \dot{\gamma} \) $$ \begin{equation} \dot{\gamma} = \sqrt{2D_{ij}D_{ij}} = 2\sqrt{-II_D} \tag{5.70} \end{equation} $$
Shear measure reduce to the strain rate for simple shear flow $$ \begin{equation} \dot{\gamma} = \partd{v_1}{x_2} \tag{5.71} \end{equation} $$
Power-law fluid. Viscosity function $$ \begin{equation} \eta = K \dot{\gamma}^{n-1} \tag{5.72} \end{equation} $$
Power-law index \( n \). Consistency parameter $$ \begin{equation} K = K_0 \exp (-A \, (\theta - \theta_0)) \tag{5.73} \end{equation} $$
Pros and cons for the power-law model: cannot fit \( \eta \) for extremal values of \( \dot{\gamma} \), however it is convenient for analytical solutions.
Most real fluids are shear thinning (\( n < 1 \)), where \( \eta \downarrow \) as \( \dot{\gamma} \uparrow \).
Carreau-Yasuda model $$ \begin{equation} \frac{\eta -\eta_\infty}{\eta_0 -\eta_\infty} = (1+(\lambda\dot{\gamma})^2)^{n-1} \tag{5.74} \end{equation} $$
Some GNFs which fits experiments well. Casson $$ \begin{align} \tau^{\frac{1}{m}} &= \tau_0^{\frac{1}{m}} + (\eta_\infty \dot{\gamma})^{\frac{1}{m}} \tag{5.75}\\ \eta &= \eta_\infty \left [ 1 + \left (\frac{\tau_0}{\eta_\infty \dot{\gamma}} \right )^{\frac{1}{m}} \right ]^m \tag{5.76} \end{align} $$
Viscoplastic model. Bingham: \( m=1 \), Casson: \( m=2 \) The Casson-model was originally suggested for pigment/oil mixtures, but it has alos been used for blood for small \( \dot{\gamma} \). The Casson-model has Newtonian behavior as \( \dot{\gamma}\quad \uparrow \).
Cauchy equations for GNFs. Generalized Newtonian fluids $$ \begin{equation} \boldsymbol{T} = -p \boldsymbol{1} + \boldsymbol{\tau} \tag{5.77} \end{equation} $$ Cauchy's equations $$ \begin{equation} \rho \partd{\boldsymbol{v}}{t} + \rho (\boldsymbol{v} \cdot \nabla) \boldsymbol{v} = \nabla \cdot \boldsymbol{T} + \rho \boldsymbol{b} \ \tag{5.78} \end{equation} $$
Cauchy equations for generalized Newtonian fluids $$ \begin{equation} \rho \partd{\boldsymbol{v}}{t} + \rho (\boldsymbol{v} \cdot \nabla) \boldsymbol{v} = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \rho \boldsymbol{b} \ \tag{5.79} \end{equation} $$
Component form $$ \begin{equation} \rho ( \partial_t v_i + v_k v_{i,k} ) = - p_{,i} + \tau_{ik,k} + \rho b_i \ \tag{5.80} \end{equation} $$
Cauchy equation in cylinder coordinates for GNF $$ \begin{align} \rho \left (\partd{v_r}{t} \right . & \left .+ v_r \partd{v_r}{r} + \frac{v_\theta}{r} \, \partd{v_r}{\theta} + v_z \partd{v_r}{z} - \frac{v_\theta^2}{r} \right ) \nonumber \\ & = -\partd{p}{r} + \frac{1}{r} \, \partd{}{r} (r \tau_{rr} ) + \frac{1}{r} \, \partd{\tau_{r\theta}}{\theta} + \partd{\tau_{rz}}{z} - \frac{\tau_{\theta\theta}}{r} + \rho b_r \tag{5.81} \end{align} $$ $$ \begin{align} \rho \left (\partd{v_\theta}{t} \right . & \left .+ v_r \partd{v_\theta}{r} + \frac{v_\theta}{r} \, \partd{v_\theta}{\theta} + v_z \partd{v_\theta}{z} + \frac{v_r v_\theta }{r} \right ) \nonumber \\ & = -\frac{1}{r}\partd{p}{\theta} + \frac{1}{r^2} \, \partd{}{r} (r^2 \tau_{r\theta} ) + \frac{1}{r} \, \partd{\tau_{\theta\theta}}{\theta} + \partd{\tau_{\theta z}}{z} + \rho b_\theta \tag{5.82} \end{align} $$ $$ \begin{align} \rho \left (\partd{v_z}{t} \right . & \left .+ v_r \partd{v_z}{r} + \frac{v_\theta}{r} \, \partd{v_z}{\theta} + v_z \partd{v_z}{z} \right ) \nonumber \\ & = -\partd{p}{z} + \frac{1}{r} \, \partd{}{r} (r \tau_{zr} ) + \frac{1}{r} \, \partd{\tau_{\theta z}}{\theta} + \partd{\tau_{zz}}{z} + \rho b_z \tag{5.83} \end{align} $$
Simplifications: \( \boldsymbol{v} \) independent of \( z \) and \( \theta \). Deviatoric stresses are independent of \( z \) and \( \theta \).n Symmetry \( \Rightarrow \tau_{rz}=\tau_{zr} \).
Cauchy's equations in cylindrical coordinates reduce to: $$ \begin{align} 0 &= -\partd{p}{r} + \frac{1}{r} \, \partd{}{r} (r \tau_{rr} ) - \frac{\tau_{\theta\theta}}{r} \quad \mathrm{and} \quad 0 = -\frac{1}{r}\partd{p}{\theta} \notag \tag{5.84}\\ 0 &= -\partd{p}{z} + \frac{1}{r} \, \partd{}{r} (r \tau_{zr} ) \notag \tag{5.85} \end{align} $$ Constant streamwise pressure gradient (i.e., \( \partial_z p = c \)) due to $$ \begin{equation} \frac{\partial^2 p}{\partial \theta \partial z} = \frac{\partial^2 p}{\partial r \partial z} = \frac{\partial^2 p}{\partial z^2} = 0 \tag{5.86} \end{equation} $$
From simplified Cauchy equation in z-direction $$ \begin{equation} 0 = -\partd{p}{z} + \frac{1}{r} \, \partd{}{r} (r \tau_{zr} ) \tag{5.87} \end{equation} $$ we get $$ \begin{equation} \partd{}{r} (r \tau_{zr} ) = r \partd{p}{z} \tag{5.88} \end{equation} $$
By integration $$ \begin{equation} r \tau_{zr} = \frac{r^2}{2} \partd{p}{z} + C_1 \tag{5.89} \end{equation} $$
As \( \tau_{rz}(r=0) = 0 \Rightarrow C_1 = 0 \). Equilibrium equation for stationary pipeflow $$ \begin{equation} \tau_{zr} = \frac{r}{2} \partd{p}{z} \tag{5.90} \end{equation} $$
May be applied to all GNFs
After subst (\( \partial_z p < 0 \)) $$ \begin{equation} \partd{v_z}{r} = -\left (\frac{1}{2K} \partd{p}{z} \, r\right )^{\frac{1}{n}} \tag{5.93} \end{equation} $$
Integrate and impose BC $$ \begin{equation} v_z = \left ( \partd{p}{z} \frac{a}{2K} \right )^{\frac{1}{n}} \frac{a}{\frac{1}{n}+1} \left ( \left ( 1 - \frac{r}{a} \right )^{\frac{1}{n} +1} \right ) \tag{5.94} \end{equation} $$
Power law and Newtonian fluid for stationary pipeflow.
Velocity profile for a power law fluid $$ \begin{equation} v_z = \left ( \partd{p}{z} \frac{a}{2K} \right )^{\frac{1}{n}} \frac{a}{\frac{1}{n}+1} \left ( 1 - \left ( \frac{r}{a} \right )^{\frac{1}{n} +1} \right ) \tag{5.95} \end{equation} $$
Newtonian fluid (\( K=\mu \) and \( n=1 \)) in \( \eta = K \dot{\gamma}^{n-1} \) From power law velocity profile $$ \begin{align} v_z &= v_0 \left ( 1 - \left (\frac{r}{a} \right )^2 \right ) \tag{5.96}\\ v_0 &= -\frac{d^2}{16 \mu} \, \partd{p}{z} \tag{5.97} \end{align} $$ \( \Rightarrow \) power law velocity profile reduces to Newtonian expression
Velocity profiles for GNFs
Velocity profiles for Bingham fluids are obtained with similar procedure Based on equilibrium relation for stationary pipeflows Velocity profiles