Maximum ceasing angle of inclination andflux formula for granular orifice flow in water
Authors: Xie Wen-Tao , Li Ruo-Ru , Peng Zheng , Jiang Yi-Min ,   School of Physics and Electronics, Central South University, Changsha 410083, China,   Powder Metallurgy Research Institute, Central South University, Changsha 410083, China
Published: December 2020

In previous work [<i>Acta Phys. Sin</i>. <b>60</b> 054501 (2011)], we found that for inclined Granular Orifice Flow (GOF) in air, regardless of the orifice size, the flow rate <i>Q</i> had a good linear relationship with the cosine of the inclination <inline-formula><tex-math id="M3">\begin{document}$\cos \theta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M3.png"/></alternatives></inline-formula>, i.e. <inline-formula><tex-math id="M4">\begin{document}$\dfrac{Q}{{{Q_0}}} = 1 - \dfrac{{\cos \theta - 1}}{{\cos {\theta _{\rm c}} - 1}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M4.png"/></alternatives></inline-formula>, where <i>Q</i><sub>0</sub> is flow rate at <inline-formula><tex-math id="M5">\begin{document}$\theta ={0^ \circ }$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M5.png"/></alternatives></inline-formula>, and <inline-formula><tex-math id="M6">\begin{document}${\theta _{\rm c}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M6.png"/></alternatives></inline-formula> is the critical angle of flow ceasing obtained by linear extrapolation. Moreover, <inline-formula><tex-math id="M7">\begin{document}${\theta _{\rm c}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M7.png"/></alternatives></inline-formula> increased linearly with ratio between grain and orifice diameter <i>d</i>/<i>D</i>, and at the limit of <i>d</i>/<i>D</i> going to zero (that is, <i>D</i> going to infinity), the angle of repose of the sample <inline-formula><tex-math id="M8">\begin{document}${\theta _{\rm r}}~( = 180^ \circ - \theta _{\rm c\infty})$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M8.png"/></alternatives></inline-formula> was obtained. Since the flow of GOF is very stable, we believe that the linear extrapolation of the above-mentioned inclined GOF provides a novel method for accurately measuring the angle of repose of granular materials. This method has been proved to be effective in a wider orifice size range by another work [<i>Acta Phys. Sin</i>. <b>65</b> 084502 (2016)]; and three angles, namely the repose angle measured by GOF, the free accumulation angle of a sandpile and the internal friction angle of the granular material measured by Coulomb yielding, are confirmed to be consistent. In this work, we extend this method to underwater, measuring the mass flow rate of a granular sample (glass beads) which completely immersed in water and driven by gravity, discharged from an inclined orifice for various inclination angles and orifice diameters. It is found that similar to the case in air, regardless of the orifice size, the flow rate increase linearly with the cosine of the inclination; the critical angle of flow ceasing increases linearly with ratio between grain and orifice diameter; at the limit of infinite orifice, this critical angle is consistent with the repose angle of the underwater sample within the experimental error range. In addition, all measurements can be well fitted by using the Beverloo formula <inline-formula><tex-math id="M9">\begin{document}$Q = {C_0}\rho {g^{1/2}}{(D - kd)^{5/2}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200217_M9.png"/></alternatives></inline-formula>, where the parameters <i>C</i><sub>0</sub> and <i>k</i> are only related to the cosine of the inclination, and are linear and inversely squared, respectively. Compared with the results of GOF in air reported by previous work, it is found that the difference mainly comes from the influence of buoyancy and fluid drag forces on the parameter <i>C</i><sub>0</sub>. These results show that both the method of measuring angle of repose with the inclined GOF and the Beverloo formula have certain universality. The behavior of GOF is qualitatively the same whether the interstitial fluid is water or air.