{"id":3006,"date":"2024-08-21T17:40:43","date_gmt":"2024-08-21T12:10:43","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=3006"},"modified":"2025-07-16T16:49:43","modified_gmt":"2025-07-16T11:19:43","slug":"types-of-motion","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ce\/fluid-mechanics\/types-of-motion","title":{"rendered":"Types of Motion or Deformation of Fluid Elements"},"content":{"rendered":"<h2 style=\"text-align: justify;\">A fluid particle as it moves can undergo the following four types of displacement viz,<\/h2>\n<ol style=\"list-style-type: lower-roman; text-align: justify;\">\n<li>Linear Translation or Pure Translation<\/li>\n<li>Linear Deformation<\/li>\n<li>Rotation<\/li>\n<li>Angular Deformation<\/li>\n<\/ol>\n<ul style=\"text-align: justify;\">\n<li><strong>Linear Translation:<\/strong> The movement of a fluid element in such a way that it moves bodily from one position to another position.<\/li>\n<li><strong>Linear Deformation:<\/strong> The deformation of a fluid element in linear direction when the element moves. In this, the axes of the element in the deformed position are parallel with that of in undeformed position but, their length changes.<\/li>\n<li><strong>Rotation :<\/strong> The movement of a fluid element in such as way that both of its axes (horizontal as well as vertical) rotates in the same direction.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3007 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/deformation-3.jpg\" alt=\"Deformation\" width=\"585\" height=\"133\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/deformation-3.jpg 585w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/deformation-3-300x68.jpg 300w\" sizes=\"auto, (max-width: 585px) 100vw, 585px\" \/><\/p>\n<p style=\"text-align: justify;\">The rate of rotation or angular velocity of the fluid particle about any axis is defined as the average angular velocity of any two infinitesimal linear elements in the particle that are perpendicular to each other and to the axis of rotation.<br \/>\nConsider a fluid element PABC rotating d about z-axis at point P to new position PA\u2032B\u2032C\u2032.<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3008 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/fluid-elements.jpg\" alt=\"Fluid Elements\" width=\"295\" height=\"278\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3009 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/fluid-elements-1.jpg\" alt=\"Fluid Elements\" width=\"440\" height=\"539\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/fluid-elements-1.jpg 440w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/fluid-elements-1-245x300.jpg 245w\" sizes=\"auto, (max-width: 440px) 100vw, 440px\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3010 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/rotation-component.jpg\" alt=\"Rotation Component\" width=\"624\" height=\"477\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/rotation-component.jpg 624w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/rotation-component-300x229.jpg 300w\" sizes=\"auto, (max-width: 624px) 100vw, 624px\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Angular Deformation or Shear deformation:<\/strong> The average change in the angle contained by two adjacent sides. Let, \u2206\u03b8<sub>1<\/sub> and \u2206\u03b8<sub>2<\/sub> be the change in angle between two adjacent sides of fluid element.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3011 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/angular-deformation.jpg\" alt=\"Angular Deformation\" width=\"355\" height=\"159\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/angular-deformation.jpg 355w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/angular-deformation-300x134.jpg 300w\" sizes=\"auto, (max-width: 355px) 100vw, 355px\" \/><\/p>\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\/ce\/fluid-mechanics\/types-of-motion\/#VELOCITY-POTENTIAL\" >VELOCITY POTENTIAL<\/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\/ce\/fluid-mechanics\/types-of-motion\/#STREAM-FUNCTION\" >STREAM FUNCTION<\/a><\/li><\/ul><\/nav><\/div>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"VELOCITY-POTENTIAL\"><\/span>VELOCITY POTENTIAL<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul style=\"text-align: justify;\">\n<li>The velocity potential (\u03c6) is defined as a scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction.<\/li>\n<li>Mathematically, velocity potential is defined as :<br \/>\nIn cartesian coordinate system:<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">\u03c6 = f (x, y, z, t) for unsteady flow<br \/>\n\u03c6 = f(x, y, z) for steady flow<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3012 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/laplace-equation.jpg\" alt=\" Laplace Equation\" width=\"645\" height=\"482\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/laplace-equation.jpg 645w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/laplace-equation-300x224.jpg 300w\" sizes=\"auto, (max-width: 645px) 100vw, 645px\" \/><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3013 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/potential-flow-e1723784597192.jpg\" alt=\"Potential Flow\" width=\"819\" height=\"446\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/potential-flow-e1723784597192.jpg 819w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/potential-flow-e1723784597192-300x163.jpg 300w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/potential-flow-e1723784597192-768x418.jpg 768w\" sizes=\"auto, (max-width: 819px) 100vw, 819px\" \/><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"STREAM-FUNCTION\"><\/span>STREAM FUNCTION<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul style=\"text-align: justify;\">\n<li>The stream function (\u03c8) is defined as a scalar function of space and time, such that its partial derivative with respect to any direction gives the velocity component at right angle (in the counter clockwise) to that direction. This function is defined only in 2D i.e one plane.<\/li>\n<li>Mathematically, stream function may be defined as \u03c8 = f(x, y, t) for unsteady flow and \u03c8 = f(x, y) for steady flow.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3016 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/cauchy-riemann-.jpg\" alt=\"Cauchy Riemann \" width=\"673\" height=\"604\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/cauchy-riemann-.jpg 673w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/cauchy-riemann--300x269.jpg 300w\" sizes=\"auto, (max-width: 673px) 100vw, 673px\" \/><\/p>\n<p style=\"text-align: justify;\">If\u00a0 \u22072\u03c8 \u2260 0<br \/>\n\u2234 Rotational flow condition.<br \/>\nHence, \u03c8 exists in both rotational as well as irrotational flow.<\/p>\n<ul style=\"text-align: justify;\">\n<li>In terms of cylindrical polar coordinates<\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3017 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/irrotational-flow.jpg\" alt=\"Irrotational Flow\" width=\"267\" height=\"80\" \/><\/p>\n<p style=\"text-align: justify;\">where, V<sub>r<\/sub> and V<sub>\u03b8<\/sub> are the components of velocity in r and \u03b8 directions respectively.<\/p>\n<ul style=\"text-align: justify;\">\n<li>The Cauchy-Riemann equation in cylindrical polar coordinates may thus be expressed as<\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-3018 aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/cylindrical-polar.jpg\" alt=\"Cylindrical Polar\" width=\"268\" height=\"84\" \/><\/p>\n<p style=\"text-align: center;\"><a class=\"btn btn-danger\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/ce\/fluid-mechanics\/types-of-pressure\/\" target=\"_blank\" rel=\"noopener\">&lt;&lt; Previous<\/a> | <a class=\"btn btn-success\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/ce\/fluid-mechanics\/surface-tension\/\" target=\"_blank\" rel=\"noopener\"> Next &gt;&gt;<\/a><br \/>\n<strong> Must Read: <\/strong> <a href=\"https:\/\/study.madeeasy.in\/ce\/what-is-fluid-mechanics\/\" target=\"_blank\" rel=\"noopener\"><strong>What is Fluid Mechanics?<\/strong><\/a><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A fluid particle as it moves can undergo the following four types of displacement viz, Linear Translation or Pure Translation<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[814,2],"tags":[833,831,835,832,834],"class_list":["post-3006","post","type-post","status-publish","format-standard","hentry","category-fluid-mechanics","category-ce","tag-angular-deformation","tag-fluid-elements","tag-laplace-equation","tag-rotation-component","tag-shear-deformation"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/3006","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=3006"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/3006\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=3006"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=3006"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=3006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}