# Title: Creeping Couette flow of Generalised Newtonian fluids # # Description: # # Following \cite{vola2004}, we solve for the 2D creeping flow between # two coaxial cylinders. The inner cylinder rotates at a constant # speed. The outer cylinder is fixed. The viscosity is a function of # the second principal invariant of the shear strain rate tensor: # $$|D|=\sqrt{\sum_{i,j}D_{ij}D_{ij}}$$ # where $D_{ij}=(\partial_iu_j+\partial_ju_i)/2$. # # We use a general Herschel-Bulkley formulation of the form: # $$\mu(|D|)={\tau_y\over 2|D|}+\mu|D|^{N-1},$$ where $\tau_y$ is the # yield stress. The solutions obtained for the stationary tangential # velocity profiles for Newtonian, Power law ($N=0.5$), # Herschel-Bulkley ($\mu=0.0672$, $\tau_y=0.12$, $N=0.5$) and Bingham # ($\mu=1$, $\tau_y=10$, $N=1$) fluids are illustrated on Figure # \ref{prof}, together with the analytical solutions given by # \cite{bird87}. # # The Bingham fluid case is a particularly severe test of the # diffusion solver, as the outer part of the fluid ring ($r>0.35$) # behaves likes a rigid body attached to the outer boundary. # # \begin{figure}[htbp] # \caption{\label{prof}Tangential velocity as a function of radial position for # various Generalised Newtonian fluids.} # \begin{center} # \includegraphics[width=\hsize]{prof.eps} # \end{center} # \end{figure} # # Author: St\'ephane Popinet # Command: sh couette.sh couette.gfs # Version: 1.0.0 # Required files: couette.sh profile prof-0.ref prof-1.ref prof-2.ref prof-3.ref # Running time: 32 minutes # Generated files: prof.eps # 1 0 GfsSimulation GfsBox GfsGEdge {} { Time { iend = 100 dtmax = 1e-2 } Refine 6 Solid (ellipse (0,0,0.25,0.25)) { flip = 1 scale = 1.9999 } Solid (ellipse (0,0,0.25,0.25)) ApproxProjectionParams { tolerance = 1e-6 } AdvectionParams { scheme = none } SourceViscosity {} { double mu, ty, N, mumax = 1000.; double m; switch (MODEL) { case 0: /* Newtonian */ mu = 1.; ty = 0.; N = 1.; break; case 1: /* Power-law (shear-thinning) */ mu = 0.08; ty = 0.; N = 0.5; break; case 2: /* Herschel-Bulkley */ mu = 0.0672; ty = 0.12; N = 0.5; break; case 3: /* Bingham */ mu = 1.; ty = 10.; N = 1.; break; } if (D2 > 0.) m = ty/(2.*D2) + mu*exp ((N - 1.)*log (D2)); else { if (ty > 0. || N < 1.) m = mumax; else m = N == 1. ? mu : 0.; } return MIN (m, mumax); } { # Crank-Nicholson does not converge for these cases, we need backward Euler # (beta = 0.5 -> Crank-Nicholson, beta = 1 -> backward Euler) beta = 1 } GfsSurfaceBc U Dirichlet (x*x + y*y > 0.140625 ? 0. : - ay) GfsSurfaceBc V Dirichlet (x*x + y*y > 0.140625 ? 0. : ax) EventStop { istep = 1 } U 1e-4 DU OutputScalarNorm { istep = 1 } du-MODEL { v = DU } OutputLocation { start = end } { awk '{if ($1 != "#") print$2,\$8;}' > prof-MODEL } profile } GfsBox {}