{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:11:41Z","timestamp":1760231501966,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,25]],"date-time":"2022-09-25T00:00:00Z","timestamp":1664064000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Axisymmetric granular flows in vertical cylindrical pipes under action of gravity are studied using mathematical particle\u2013particle models based on the Hertz\u2013Mindlin theory. By and large, in granular flows, the density field and the pressure are unknown scalar functions. A well-known relationship between these fields gives the pressure field a power law of the density. The aim of this paper was to study unsteady, axisymmetric, fully developed granular flow under gravity action in a vertical cylindrical pipe, under the assumptions that the density field is constant and the velocity on the pipe\u2019s wall is time-dependent. Using integral transforms method and appropriate initial-boundary conditions, the analytical solution for axial velocity is determined. The obtained analytical solution is used to determine the steady-state solution (the solution for large values of the time). The properties of the flow in some particular cases of the velocity on the pipe\u2019s surface are analyzed and the transient flow is compared with the stationary one.<\/jats:p>","DOI":"10.3390\/sym14102013","type":"journal-article","created":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T01:23:16Z","timestamp":1664414596000},"page":"2013","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Axial Symmetric Granular Flow Due to Gravity in a Circular Pipe"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9743-1442","authenticated-orcid":false,"given":"Numan","family":"Naeem","sequence":"first","affiliation":[{"name":"Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54000, Punjab, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9989-8452","authenticated-orcid":false,"given":"Dumitru","family":"Vieru","sequence":"additional","affiliation":[{"name":"Department of Theoretical Mechanics, Technical University \u201cGheorghe Asachi\u201d of Iasi, 70050 Iasi, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0901-4508","authenticated-orcid":false,"given":"Noor","family":"Muhammad","sequence":"additional","affiliation":[{"name":"Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54000, Punjab, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5816-0491","authenticated-orcid":false,"given":"Najma","family":"Ahmed","sequence":"additional","affiliation":[{"name":"Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54000, Punjab, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Nedderman, R.M. (1992). Statics and Kinematics of Granular Materials, Cambridge University Press.","DOI":"10.1017\/CBO9780511600043"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/BF01175958","article-title":"The dynamics of avalanches of granular materials from initiation to runout. Part I Analysis","volume":"86","author":"Savage","year":"1991","journal-title":"Acta Mech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2003","DOI":"10.1098\/rsta.2004.1428","article-title":"Discrete\u2013element modelling and smoothed particle hydrodynamics: Potential in the environmental sciences. Philosophical Transactions of the Royal Society of London","volume":"362","author":"Cleary","year":"2004","journal-title":"Ser. A Math. Phys. Eng. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1146\/annurev.fluid.40.111406.102142","article-title":"Flows of dense granular media","volume":"40","author":"Forterre","year":"2008","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"055024","DOI":"10.1088\/1361-6552\/ac8136","article-title":"Gravity-driven granular flows in pipes: Teaching experimental skills in the context of granular flows","volume":"57","author":"Williams","year":"2022","journal-title":"Phys. Educ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"A21","DOI":"10.1017\/jfm.2021.909","article-title":"Exact solutions for steady granular flow in vertical chutes and pipes","volume":"930","author":"Barker","year":"2022","journal-title":"J. Fluid Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1407","DOI":"10.1029\/2017JF004296","article-title":"Link between the dynamics of granular flows and the generated seismic signal: Insights from laboratory experiments","volume":"123","author":"Farin","year":"2018","journal-title":"J. Geophys. Res. Earth Surf."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ligneau, C., Sovilla, B., and Gaume, J. (2022). Numerical investigation of the effect of cohesion and ground friction on snow avalanches flow regimes. PLoS ONE, 17.","DOI":"10.1371\/journal.pone.0264033"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"103162","DOI":"10.1016\/j.jfluidstructs.2020.103162","article-title":"Numerical simulation of fast granular flow facing obstacles on steep terrains","volume":"99","author":"Yang","year":"2020","journal-title":"J. Fluids Struct."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"03014","DOI":"10.1051\/epjconf\/202124903014","article-title":"The effect of boundary roughness on dense granular flows. EDP Sciences","volume":"249","author":"Fazelpour","year":"2021","journal-title":"EPJ Web Conf."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/s10035-004-0191-9","article-title":"Steady-state granular flow in a 3D cylindrical hopper with flat bottom: Macroscopic analysis","volume":"7","author":"Zhu","year":"2005","journal-title":"Granul. Matter"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"198","DOI":"10.2991\/jnmp.1999.6.2.6","article-title":"New Mathematical Models for Particle Flow Dynamics","volume":"6","author":"Blackmore","year":"1999","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1017\/jfm.2011.335","article-title":"Granular column collapse as a continuum: Validity of a two-dimensional Navier_Stokes model with a rheology","volume":"686","author":"Legree","year":"2011","journal-title":"J. Fluid Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4222","DOI":"10.1016\/j.apt.2020.08.028","article-title":"DEM study of monodisperse granular flow in a pipe","volume":"31","author":"Shabir","year":"2020","journal-title":"Adv. Powder Thech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.nucengdes.2017.01.008","article-title":"DEM study of granular discharge rate through a vertical pipe with a bend outlet in small absorber system","volume":"314","author":"Li","year":"2017","journal-title":"Nucl. Eng. Des."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/s40571-015-0075-2","article-title":"Tapped granular column dynamics: Simulations, experiments and modeling","volume":"3","author":"Rosato","year":"2016","journal-title":"Comput. Part. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"A22","DOI":"10.1017\/jfm.2020.973","article-title":"Coupling rheology and segregation in granular flows","volume":"909","author":"Barker","year":"2021","journal-title":"J. Fluid Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1017\/jfm.2017.612","article-title":"Compressibility regularizes the \u03bc(I)-rheology for dense granular flows","volume":"830","author":"Heyman","year":"2017","journal-title":"J. Fluid Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"088002","DOI":"10.1103\/PhysRevLett.125.088002","article-title":"Power-law scaling in granular rheology across flow geometries","volume":"125","author":"Kim","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/S0167-6636(96)00034-8","article-title":"Contact force displacement laws and the mechanical behavior of random packs of identical spheres","volume":"24","author":"Elata","year":"1996","journal-title":"Mech. Mater."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.compgeo.2014.12.011","article-title":"A novel three-dimensional contact model for granulates incorporating rolling and twisting resistances","volume":"65","author":"Jiang","year":"2015","journal-title":"Comput. Geotech."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Davies, B. (2002). Integral Transforms and Their Applications, Springer Science & Business Media.","DOI":"10.1007\/978-1-4684-9283-5"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Debnath, L., and Bhatta, D. (2016). Integral Transforms and Their Applications, Chapman and Hall\/CRC.","DOI":"10.1201\/9781420010916"},{"key":"ref_24","unstructured":"Watson, G.N. (1922). A Treatise on the Theory of Bessel Functions, Cambridge University Press."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Grigoletto, E.C., and de Oliveira, E.C. (2018). A Note on the Inverse Laplace Transform. Cad. IME-S\u00e9rie Matem\u00e1tica, 39\u201346.","DOI":"10.12957\/cadmat.2018.34026"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1337","DOI":"10.1134\/S0965542514090073","article-title":"Studies on the zeros of Bessel functions and methods for their computation","volume":"54","author":"Kerimov","year":"2014","journal-title":"Comput. Math. Math. Phys."},{"key":"ref_27","unstructured":"Roberts, G.E., and Kaufman, H. (1966). Table of Laplace Transforms, W.B. Saunders Co."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2013\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:39:07Z","timestamp":1760143147000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2013"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,25]]},"references-count":27,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14102013"],"URL":"https:\/\/doi.org\/10.3390\/sym14102013","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,9,25]]}}}