{"id":227,"date":"2022-10-26T02:45:00","date_gmt":"2022-10-26T01:45:00","guid":{"rendered":"https:\/\/ramyrashad.com\/?p=227"},"modified":"2023-08-17T15:38:17","modified_gmt":"2023-08-17T14:38:17","slug":"fourier-navier-stokes-equations","status":"publish","type":"post","link":"https:\/\/ramyrashad.com\/index.php\/2022\/10\/26\/fourier-navier-stokes-equations\/","title":{"rendered":"Fourier-Navier-Stokes equations"},"content":{"rendered":"\n<p><strong>Research Overview<\/strong><\/p>\n\n\n\n<p>In this paper, we present a comprehensive description of thermodynamics in a differential geometric setting for a general non-relativistic continuum mechanical system. In order to deal with manifestly coordinate-invariant equations valid on general Riemannian manifolds, we make use of a formulation based on exterior calculus. We show how to complement this covariant formulation, already well known in continuum mechanics, with a consistent description of thermodynamics valid in general mechanical field theories, generalizing the procedure carried through on Euclidean spaces using vector calculus.<\/p>\n\n\n\n<p>We specialize the equations to Fourier\u2013Navier\u2013Stokes fluids in order to shed light on the entropy creation mechanisms in Newtonian fluids on manifolds.<\/p>\n\n\n\n<p>1st Law of Thermodynamics (Total energy balance)<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-20.png\" alt=\"\" class=\"wp-image-537\" width=\"467\" height=\"60\" srcset=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-20.png 925w, https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-20-300x39.png 300w, https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-20-768x100.png 768w\" sizes=\"(max-width: 467px) 100vw, 467px\" \/><\/figure><\/div>\n\n\n<p>Kinetic Energy and Internal Energy balances<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-layout-1 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-21.png\" alt=\"\" class=\"wp-image-538\" width=\"281\" height=\"58\" srcset=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-21.png 377w, https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-21-300x62.png 300w\" sizes=\"(max-width: 281px) 100vw, 281px\" \/><\/figure><\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-22.png\" alt=\"\" class=\"wp-image-539\" width=\"325\" height=\"53\" srcset=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-22.png 442w, https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-22-300x50.png 300w\" sizes=\"(max-width: 325px) 100vw, 325px\" \/><\/figure><\/div><\/div>\n<\/div>\n\n\n\n<p>2nd Law of Thermodynamics<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-23.png\" alt=\"\" class=\"wp-image-540\" width=\"257\" height=\"89\" srcset=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-23.png 538w, https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-23-300x104.png 300w\" sizes=\"(max-width: 257px) 100vw, 257px\" \/><\/figure><\/div>\n\n\n<p><strong>Publication<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>F. Califano, R. Rashad, S. Stramigioli&nbsp;(2022)&nbsp;A differential geometric description of thermodynamics in continuum mechanics with application to Fourier-Navier-Stokes fluids,&nbsp;Physics of Fluids&nbsp;34(10),&nbsp;AIP Publishing LLC,&nbsp;<a href=\"https:\/\/doi.org\/10.1063\/5.0119517\" target=\"_blank\" rel=\"noreferrer noopener\">doi:10.1063\/5.0119517<\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Research Overview In this paper, we present a comprehensive description of thermodynamics in a differential geometric setting for a general non-relativistic continuum mechanical system. In order to deal [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":542,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"ub_ctt_via":"","_mi_skip_tracking":false,"footnotes":""},"categories":[10],"tags":[],"aioseo_notices":[],"featured_image_src":"https:\/\/ramyrashad.com\/wp-content\/uploads\/2022\/10\/FNS.png","author_info":{"display_name":"Ramy","author_link":"https:\/\/ramyrashad.com\/index.php\/author\/ramy-abdelmonemgmail-com\/"},"_links":{"self":[{"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/posts\/227"}],"collection":[{"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/comments?post=227"}],"version-history":[{"count":6,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/posts\/227\/revisions"}],"predecessor-version":[{"id":543,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/posts\/227\/revisions\/543"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/media\/542"}],"wp:attachment":[{"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/media?parent=227"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/categories?post=227"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/tags?post=227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}