# Numerical Study of Viscoplastic Flows Using a Multigrid Initialization Algorithm
## 2 Numerical Methodology  
### Governing Equations  
We consider the two-dimensional steady-state flow. Let \(\rho\) and \(\eta=\eta(\dot{\gamma})\) denote the density and viscosity of any generalized-Newtonian fluid. Incompressible steady Navier-Stokes equations are given by a pair of partial differential equations:  
\[\
abla.\mathbf{u}=0 \tag{4}\]  
\[\rho\mathbf{u}.\
abla\mathbf{u}=-\
abla p+\
abla.\tau+\mathbf{f} \tag{5}\]  
Equation (4) represents the conservation of mass, whereas Equation (5) represents the conservation of momentum. The velocity field is defined as \(\mathbf{u}(\mathbf{x},t)\), with components \(u\) and \(v\). The Cauchy stress tensor \(\sigma\) is introduced as the sum of isotropic and deviatoric parts, \(\sigma=-pI+\tau\). Here, the pressure \(p(\mathbf{x},t)\) is multiplied by the identity tensor, while the deviatoric part of the stress tensor is denoted as \(\tau(\mathbf{x},t)\). We have introduced \(\mathbf{f}\) to describe external body forces such as gravity acting on the fluid. Note that the mass conservation equation is simplified to (4) due to incompressibility, i.e., constant density within a fluid volume. Newtonian flow is characterized by a dynamic coefficient of viscosity \(\mu>0\), which is independent of the strain, yielding a linear relationship between rate-of-strain and stress in the rheological equation,  
\[\tau=\mu\dot{\gamma} \tag{6}\]  
For fluids where the dependency of the stress on the shear rate tensor is nonlinear, the apparent viscosity is a useful concept when considering rheological responses to strain. It is a generalization of the constant viscosity for a Newtonian flow, where we allow the viscosity to be a function of the magnitude of the shear rate tensor. Denoting the apparent viscosity by \(\eta\), we thus have:  
\[\eta=\frac{\tau}{\dot{\gamma}} \tag{7}\]  
When considering viscoplastic fluids, Equation (1) does not present any problems for yielded regions \(|\tau|>\tau_{y}\), but the corresponding apparent viscosity \(\eta=\mu+\tau_{y}/|\dot{\gamma}|\) has a singularity when the stress falls below the yield (and \(|\tau|\rightarrow\tau_{y}\)). To help address this issue, we employ the popular Papanastasiou regularization [8], which lies on an exponential relaxation according to:  
\[\frac{1}{\dot{\gamma}}\rightarrow\frac{1-e^{-m\dot{\gamma}}}{\dot{\gamma}} \tag{8}\]  
When \((m\dot{\gamma})>>1\), it is a suitable approximation that provides a continuous dependence between the shear and shear rate. While near the small shear rate magnitude limit, we have:  
\[\lim_{\dot{\gamma}\to 0}\frac{1-e^{-m\dot{\gamma}}}{\dot{\gamma}}=\lim_{ \dot{\gamma}\to 0}\left(1-\sum_{n=1}^{\infty}\frac{(-m\dot{\gamma})}{n!} \right)=m \tag{9}\]  
Substitution of Equation (8) into the apparent viscosity of the Bingham model yields the following regularized viscosity:  
\[\eta=\mu_{p}+\frac{\tau_{y}}{\dot{\gamma}}(1-e^{-m\dot{\gamma}}) \tag{10}\]The effect of \(m=1/\epsilon\) is related to the curves shown in Figure 1. For two-dimensional flow, the explicit form of the second invariant of the rate-of-strain tensor, \(\dot{\gamma}\) is given by:  
\[|\dot{\gamma}|=\left[2\left(\frac{\partial u}{\partial x}\right)^{2}+2\left( \frac{\partial v}{\partial y}\right)^{2}+\left(\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}\right)^{2}\right]^{1/2} \tag{11}\]  
Under these considerations, the conservation equation for a general flow variable \(\phi\) in the steady laminar case described above, can be written as:  
\[\frac{\partial}{\partial x}(\rho u\phi)+\frac{\partial}{\partial y}(\rho v\phi )=\frac{\partial}{\partial x}\left(\Gamma\frac{\partial\phi}{\partial x} \right)+\frac{\partial}{\partial y}\left(\Gamma\frac{\partial\phi}{\partial y }\right)+S_{\phi} \tag{12}\]  
where \(u\) and \(v\) are the \(x\) and \(y\) components of the velocity field and \(\Gamma\) is the diffusion coefficient. The terms on the left represent the net convection flow. The two terms on the right represent the net diffusion and the last term the source generation.