Power-law fluid

A power-law fluid has shear stress $\tau =K|\dot{\gamma }{|}^{n-1}\dot{\gamma }$. It is shear thinning for $n<1$, Newtonian for $n=1$, and shear thickening for $n>1$.

For a thin, shear-free coating on a horizontal cylinder, lubrication theory balances tangential gravity against the transverse shear stress gradient. With angle measured counterclockwise from the rightward horizontal and wall speed $U=\Omega a>0$, the flux is $Uh-\mathrm{sgn}(\cos \theta )\frac{n}{2n+1}(\rho g|\cos \theta |/k{)}^{1/n}{h}^{(2n+1)/n}$. On the uphill side this function has a finite maximum. The tightest restriction is at $\cos \theta =1$, where differentiation gives $h_{\ast }=[k{U}^n/(\rho g){]}^{1/(n+1)}$ and the displayed global steady-flux bound. This is the leading gravity-driven model without higher-order pressure-gradient or capillary corrections.

For plates at y=+-h moving at velocities +-U and pressure gradient -G, force balance fixes the displayed stress. The power-law fluid law gives $u_y=(Gh/k{)}^{1/n}\mathrm{sgn}(\alpha -y/h)|\alpha -y/h{|}^{1/n}$. Integrating from the lower no-slip wall yields $u=-U+hn(Gh/k{)}^{1/n}[|\alpha +1{|}^{1+1/n}-|\alpha -y/h{|}^{1+1/n}]/(n+1)$. The upper wall fixes alpha uniquely for k,n,G,U>0. Absolute values are essential when the shear reverses inside the gap.

Shear thinning is the decrease of effective viscosity with increasing shear rate.