## 5 Interpretation of the Solution and Discussion  
Using the geometry of Figure 1, the torque is found as  
\[T=\frac{2\pi\tau R^{3}}{3\cos\beta}. \tag{44}\]  
It has been taken into account here that \(\sigma_{\theta\varphi}\) is independent of \(r\), as follows from (23). Moreover, \(\tau\) is the value of \(\sigma_{\theta\varphi}\) at \(\theta=\theta_{0}\). If the regime of sticking occurs then the value of \(\tau\) depends on \(\omega\) (or \(\mu\)). In particular, using (23), Equation (44) can be rewritten as  
\[t=\frac{3T}{2\pi R^{3}k_{0}}=\frac{C_{1}}{\cos^{3}\beta} \tag{45}\]  
where \(t\) is the dimensionless torque. It has been taken into account here that \(\theta_{0}=\pi/2-\beta\), as it is seen from Figures 1 and 2. The value of \(C_{1}\) is determined from (28) if there is no rigid region. If there is a rigid region then (3) should be used to transform (45) to  
\[t=\frac{\sin^{2}\theta_{c}}{\cos^{3}\beta} \tag{46}\]  
where \(\theta_{c}\) is determined from (32). If \(\alpha\) satisfies the inequality (41) and \(C_{1}\) attains the value of \(C_{m}\), then Equations (37) and (45) combine to give  
\[t_{m}=\frac{k_{s}}{k_{0}\cos\beta}. \tag{47}\]  
It is evident that the value of \(t_{m}\) is independent of \(\mu\) and is described by material properties and geometry. If \(\alpha\) satisfies the inequality (40) then \(C_{1}\to\infty\) as \(\omega\to\infty\). Therefore, it is seen from Equation (45) that \(t\to\infty\) as \(\omega\to\infty\). The dependence of \(t\) on \(\omega\) that follows from the discussion above is shown schematically in Figure 3 where curve \(1\) corresponds to a value of \(\alpha\) satisfying the inequality (41) and curve \(2\) to a value of \(\alpha\) satisfying the inequality (40). The qualitative behavior of these curves is independent of the function \(\Phi\big{(}\xi_{eq}/\xi_{0}\big{)}\) over any finite interval \(0\leq\xi_{eq}/\xi_{0}\leq\xi_{m}/\xi_{0}<\infty\). In general, for some materials, experimental observations confirm the general features of the dependence between the torque and \(\mu\) illustrated in Figure 3 by curve \(1\) (see, for example, Figure 8 in [26]). In particular, the torque is practically independent of \(\omega\) in the range \(\omega\geq\omega_{m}\) where \(\omega_{m}=2\xi_{0}/\mu_{m}\). The value of \(\mu_{m}\) is determined from (28) at \(C_{1}=C_{m}\). Then, using (37) one gets the following equation for \(\mu_{m}\):  
\[1=\mu_{m}\int\limits_{\partial_{0}}^{\pi/2}\frac{1}{\sin\theta}\Lambda\bigg{(} \frac{k_{s}}{k_{0}}\frac{\sin^{2}\theta_{0}}{\sin^{2}\theta}\bigg{)}d\theta.\]  
A conventional interpretation of the dependence between the torque and \(\omega\) illustrated in Figure 3 by curve \(1\) is that there is a transition region between weak and strong slip regions. The main result of the present paper provides another interpretation that the finite value of \(k_{s}\) exists and that the value of \(\alpha\) satisfies (41). In this case, experimental results similar to that illustrated in Figure 8 in [26] allow for the value of \(k_{s}/k_{0}\) to be determined using Equation (47). All other parameters of the material model can be determined using any available method, for example [31]. In some cases, rough surfaces are intentionally used for suppressing slip in rheometers [32]. This method has no effect of material flow in the vicinity of interfaces if the material satisfies the constitutive Equations (18) and (41). This is because the regime of flow at the interface is completely described by the constitutive equations. This feature of solutions is typical for several rigid plastic models [33]. It is, in general, assumed that the shear stress is approximately constant inside the sample if the angle \(\beta\) (Figure 1) is small enough [4; 11]. A consequent conclusion is that the shear strain rate is also approximately constant. This conclusion is not valid for the material models satisfying (18) at \(\alpha<0\)  
Figure 3: Schematic representation of the dependence of \(t\) on \(\omega\) for satisfying inequalities (41) and (40).