{"id":4732,"date":"2025-04-26T11:14:58","date_gmt":"2025-04-26T05:44:58","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=4732"},"modified":"2025-07-22T14:43:15","modified_gmt":"2025-07-22T09:13:15","slug":"velocity-potential-function","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/me\/fluid-mechanics-me\/velocity-potential-function","title":{"rendered":"Velocity Potential Function"},"content":{"rendered":"<h2>Function of\u00a0Velocity Potential<\/h2>\n<p style=\"text-align: justify;\">It is defined as a scalar function of space and time such that its negative derivative along any direction gives the component of <a href=\"https:\/\/study.madeeasy.in\/ce\/open-channel-flow\/velocity-distribution\/\" target=\"_blank\" rel=\"noopener\">velocity<\/a> along that direction.<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-4733 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/04\/derivative.jpg\" alt=\"Derivative\" width=\"236\" height=\"39\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Remember:<\/strong><\/p>\n<ul style=\"text-align: justify;\">\n<li>Negative sign implies flow always occurs in the direction of decreasing potential.<\/li>\n<li>Velocity potential function exists only for irrotational flow and potential flow.<\/li>\n<li>If velocity potential function satisfies the Laplace equation, then it also satisfies continuity equation and hence the flow is possible.<\/li>\n<\/ul>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_1 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.madeeasy.in\/study\/me\/fluid-mechanics-me\/velocity-potential-function\/#Flow-Through-Pipes\" >Flow Through Pipes<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.madeeasy.in\/study\/me\/fluid-mechanics-me\/velocity-potential-function\/#Darcy-Weisbach-equation\" >Darcy Weisbach equation.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.madeeasy.in\/study\/me\/fluid-mechanics-me\/velocity-potential-function\/#Laminar-Flow-Through-Circular-Pipe-Hagen-Poiseuille-Flow\" >Laminar Flow Through Circular Pipe (Hagen-Poiseuille Flow)<\/a><\/li><\/ul><\/nav><\/div>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Flow-Through-Pipes\"><\/span>Flow Through Pipes<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/study.madeeasy.in\/me\/introduction-to-friction\/\" target=\"_blank\" rel=\"noopener\">Friction<\/a> Loss\/Darcy Weisbach Equation<\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Darcy-Weisbach-equation\"><\/span>Darcy Weisbach equation.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4735 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/04\/darcy-weisbach.jpg\" alt=\"Darcy Weisbach equation \" width=\"637\" height=\"300\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/04\/darcy-weisbach.jpg 637w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/04\/darcy-weisbach-300x141.jpg 300w\" sizes=\"auto, (max-width: 637px) 100vw, 637px\" \/><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Laminar-Flow-Through-Circular-Pipe-Hagen-Poiseuille-Flow\"><\/span>Laminar Flow Through Circular Pipe (Hagen-Poiseuille Flow)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong> Shear stress (\u03c4) distribution<\/strong><\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-4736 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/04\/shear-stress.jpg\" alt=\"Shear stress\" width=\"202\" height=\"105\" \/><\/p>\n<p style=\"text-align: justify;\">The negative sign on x \u00a0\u2202p\/\u2202x indicates decrease in pressure in the direction of flow. The pressure must decrease because pressure force is the only means available to compensate for resistance to the flow, the potential and kinetic energy remain constant.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Function of\u00a0Velocity Potential It is defined as a scalar function of space and time such that its negative derivative along<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1413,10],"tags":[1419,1418],"class_list":["post-4732","post","type-post","status-publish","format-standard","hentry","category-fluid-mechanics-me","category-me","tag-darcy-weisbach-equation","tag-laminar-flow-through-circular-pipe"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/4732","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/comments?post=4732"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/4732\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=4732"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=4732"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=4732"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}