D. Alvaro-Berlanga, R. Planet, A. Fernandez-Nieves
{"title":"Torricelli's experiment and conservation of momentum","authors":"D. Alvaro-Berlanga, R. Planet, A. Fernandez-Nieves","doi":"10.1119/5.0145991","DOIUrl":null,"url":null,"abstract":"Torricelli's law states that the speed of a fluid as it flows out of an orifice of a large reservoir open to the atmosphere is equal to 2gh, where g is the acceleration of gravity and h is the distance between the free surface of the fluid in the reservoir and the orifice. Obtaining the flow rate from this speed is not at all trivial, despite how simple the situation seems, as the fluid jet issued at the orifice does not have a constant cross section and the motion of the fluid near the orifice is not really known. Here, we use Torricelli's experiment as a practical way to illustrate how to properly apply the momentum balance equation to solve hydrodynamic problems. We compare the horizontal component of the force exerted by the container on the fluid using both momentum conservation and the integral of the stress tensor; this results in a contradiction that we use to review, after considering simple experimental results, the assumptions made during the calculations, to finally resolve the discrepancy and rationalize the intricacies of this every-day situation.","PeriodicalId":0,"journal":{"name":"","volume":"259 1‐4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1119/5.0145991","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Torricelli's law states that the speed of a fluid as it flows out of an orifice of a large reservoir open to the atmosphere is equal to 2gh, where g is the acceleration of gravity and h is the distance between the free surface of the fluid in the reservoir and the orifice. Obtaining the flow rate from this speed is not at all trivial, despite how simple the situation seems, as the fluid jet issued at the orifice does not have a constant cross section and the motion of the fluid near the orifice is not really known. Here, we use Torricelli's experiment as a practical way to illustrate how to properly apply the momentum balance equation to solve hydrodynamic problems. We compare the horizontal component of the force exerted by the container on the fluid using both momentum conservation and the integral of the stress tensor; this results in a contradiction that we use to review, after considering simple experimental results, the assumptions made during the calculations, to finally resolve the discrepancy and rationalize the intricacies of this every-day situation.
托里拆利定律指出,流体从向大气开放的大型储液器孔口流出时的速度等于 2gh,其中 g 是重力加速度,h 是储液器中流体自由表面与孔口之间的距离。尽管情况看起来非常简单,但要从这个速度中获得流速却并非易事,因为在孔口喷射的流体截面并不恒定,而且孔口附近流体的运动情况也不为人所知。在此,我们以托里切利的实验为实例,说明如何正确应用动量平衡方程来解决流体力学问题。我们同时使用动量守恒和应力张量积分来比较容器对流体施加的力的水平分量;这导致了一个矛盾,在考虑简单的实验结果后,我们利用这个矛盾来回顾计算过程中所做的假设,最终解决了差异,并合理解释了这种日常情况的复杂性。