Title: Global self$-$similarity of dense granular flow in hopper: the role of hopper width

Abstract: The influence of hopper width on dense granular flow in a two$-$dimensional hopper is investigated through experiments and simulations. Though the flow rate remains stable for larger hopper widths, a slight reduction in hopper width results in a significant increase in flow rate for smaller hopper widths. Both Beverloo\('\)s and Janda\('\)s formula accurately capture the relationship between the flow rate and outlet size. Flow characteristics in the regions near the outlet exhibit local self$-$similarity, supporting Beverloo and Janda's principles. Moreover, global self$-$similarity is analysed, indicated by the transition in flow state from mass flow in regions far from the outlet to funnel flow near the outlet. The earlier occurrence of this transition favors to enhance the grain velocity and consequently increases the dense flow rate. An exponential scaling law is proposed to describe the dependencies of flow rate, grain velocity, and transition height between the mass flow pattern and the funnel flow pattern on silo width.