# Numerical Study of Viscoplastic Flows Using a Multigrid Initialization Algorithm
## 3 Numerical Results
### Influence of the Stress Growth Parameter: Pipe Flow  
The results of Equation (10) are plotted in Figure 1 with dashed lines, along with the the actual Bingham constitutive equation, \(\tau=(\mu+\tau_{y}/|\dot{\gamma}|)\dot{\gamma}\). It is clear that with an increasing exponent \(m\), we can achieve quick stress growth at very low shear rates, which is consistent with the behavior of the material in its unyielded region. In fact, regularized models yield results that do not contain truly unyielded or solid zones and hence the solution may, in some cases, be sensitive to the chosen stress growth exponent. Previous studies, including [10], found that different values of 100, 200, and 400 for \(m\) do not influence the location of the yield surface significantly, for the particular case of cavity flow. However, very large values of the exponent parameter may be needed if the region where the stress close to the yield stress is relatively large. Figure 8: Streamline contours in Bingham flow for \(Bn=2\). Figure 10: Streamline contours in Bingham flow for \(Bn=100\). Figure 9: Streamline contours in Bingham flow for \(Bn=10\). Regarding the viscoplastic flows though pipes, two distinct unyielded regions can possibly occur. First, in the layer adjacent to no-slip boundary where the velocity of the fluid is set to zero, it is called the stagnant zone. Then, a second unyielded region with non-zero fluid velocity is referred to as the plug zone, see Figure 11. One result of importance in a range of applications is the location of the plug zone, where \(|\dot{\gamma}|\) tends to zero, as previously stated. In Figure 12, we present a comparison of the velocity profiles for different values of \(m\): 100, 200, 400, and 800. As \(m\) is increased, the location of the unyielded region would be slightly affected. The difference especially vanished between \(m=400\) and \(m=800\) for both cases. Figure 12 shows velocity profiles, which deviate slightly from the reference result considered here in red dotted lines (\(m=800\)) and plug zones which get flatter as \(m\) is increased. Indeed, beyond a certain value of stress growth exponent, yield lines (\(\tau=\tau_{y}\)) are less sensitive to its variation. This verifies the assumption that the yield areas calculated using a regularization approach will converge to the true yield areas as \(m\to\infty\). It should be pointed out that, unlike the velocity field, there is no theoretical proof that yield surfaces converge to the exact solution [11]. In Figure 13, we compare the simulated results of the apparent viscosity profiles along the circular cross-section of the pipe calculated with \(m=100\) and \(m=800\). It is shown that using a high value of the exponent \(m\) leads to fast growing apparent viscosity as the shear rates are decreasing, meaning that the larger the value of \(m\), the larger the region of the solid-like behavior of the Bingham model is reproduced. Figure 13 also shows that, with regularized models, the unyielded material (plug zone) is no longer a rigid solid but a highly viscous fluid that approximates the ideal viscoplastic behavior. The degree of approximation depends on the adequate choice of the stress growth parameter in order to satisfy the [PERSON] criterion. Figure 11: Numerical results: the developing velocity profile of a Bingham fluid entering the pipe, \(Re=10^{2}\) and \(Bn=10\). Figure 12: Velocity profiles at \(x=0.5L\) calculated with \(m=100\) (solid lines), \(m=200\) (dashed lines), \(m=400\) (dotted lines), and \(m=800\) ( red dotted lines). For \(Re=10^{2}\), (**a**) \(Bn=1\), and (**b**) \(Bn=10\). An important issue when using the Papanastasiou regularization is that the degree of non linearity increases sharply with higher values of \(m\) that ensure better approximations. Therefore, as a matter of practicality, the choice of exponent \(m\) has to be reasonably reduced depending on the problem under study. In this work, we limited our study to the comparisons showing the influence of \(m\) on the quality of the results.