{"id":311538,"date":"2022-11-06T15:47:12","date_gmt":"2022-11-06T10:17:12","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/stokes-law-derivation\/"},"modified":"2025-02-10T17:35:53","modified_gmt":"2025-02-10T12:05:53","slug":"stokes-law-derivation","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/physics\/stokes-law-derivation\/","title":{"rendered":"Stokes Law Derivation"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">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\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><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><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/physics\/stokes-law-derivation\/#Stokes_law_derivation_for_laminar_flow\" title=\"Stokes law derivation for laminar flow\">Stokes law derivation for laminar flow<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/physics\/stokes-law-derivation\/#What_is_Stokes_Law\" title=\"What is Stokes Law?\">What is Stokes Law?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/physics\/stokes-law-derivation\/#Stokes_Law_Formula\" title=\"Stokes Law Formula\">Stokes Law Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/physics\/stokes-law-derivation\/#What_is_Stokes_Law_Derivation\" title=\"What is Stokes Law Derivation?\">What is Stokes Law Derivation?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/physics\/stokes-law-derivation\/#Terminal_Velocity_Formula_in_Physics\" title=\"Terminal Velocity Formula in Physics\">Terminal Velocity Formula in Physics<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/physics\/stokes-law-derivation\/#Assumptions_made_in_the_Stokes_Law\" title=\"Assumptions made in the Stokes Law:\">Assumptions made in the Stokes Law:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/physics\/stokes-law-derivation\/#Applications_of_Stoke_Law_in_Daily_Life\" title=\"Applications of Stoke Law in Daily Life\">Applications of Stoke Law in Daily Life<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Stokes_law_derivation_for_laminar_flow\"><\/span>Stokes law derivation for laminar flow<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"flex-1 overflow-hidden\">\n<div class=\"react-scroll-to-bottom--css-potab-79elbk h-full dark:bg-gray-800\">\n<div class=\"react-scroll-to-bottom--css-potab-1n7m0yu\">\n<div class=\"flex flex-col items-center text-sm h-full dark:bg-gray-800\">\n<div class=\"w-full border-b border-black\/10 dark:border-gray-900\/50 text-gray-800 dark:text-gray-100 group bg-gray-50 dark:bg-[#444654]\">\n<div class=\"text-base gap-4 md:gap-6 m-auto md:max-w-2xl lg:max-w-2xl xl:max-w-3xl p-4 md:py-6 flex lg:px-0\">\n<div class=\"relative flex w-[calc(100%-50px)] md:flex-col lg:w-[calc(100%-115px)]\">\n<div class=\"flex flex-grow flex-col gap-3\">\n<div class=\"min-h-[20px] flex flex-col items-start gap-4 whitespace-pre-wrap\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<p>Stokes&#8217; law is a mathematical equation used to describe the motion of a small sphere through a viscous fluid. It is named after George Gabriel Stokes, who derived the equation in the 19th century.<\/p>\n<p>The equation is given by:<\/p>\n<p>F = 6\u03c0\u03b7rv<\/p>\n<p>where:<\/p>\n<p>F is the force required to move the sphere through the fluid at a constant velocity (also known as the drag force) \u03b7 is the viscosity of the fluid r is the radius of the sphere v is the velocity of the sphere<\/p>\n<p>The equation can be derived by considering the balance of forces on the sphere as it moves through the fluid. The sphere experiences a viscous force from the fluid, which is proportional to the velocity of the sphere and the viscosity of the fluid. The sphere also experiences an opposing force due to its inertia, which is proportional to its mass and acceleration.<\/p>\n<p>Assuming the sphere is moving at a constant velocity, the viscous force and the inertial force must balance each other. This leads to the following equation:<\/p>\n<p>F = 6\u03c0\u03b7rv<\/p>\n<p>This equation is known as Stokes&#8217; law and can be used to predict the drag force experienced by a small sphere moving through a viscous fluid at a constant velocity.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-4 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"w-full h-48 flex-shrink-0\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-323068\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/11\/Stokes-Law-Derivation.png\" alt=\"Stokes Law Derivation\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/11\/Stokes-Law-Derivation.png?v=1672744482 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/11\/Stokes-Law-Derivation-300x212.png?v=1672744482 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"absolute bottom-0 left-0 w-full border-t md:border-t-0 dark:border-white\/20 md:border-transparent md:dark:border-transparent md:bg-vert-light-gradient bg-white dark:bg-gray-800 md:!bg-transparent dark:md:bg-vert-dark-gradient\">\n<form class=\"stretch mx-2 flex flex-row gap-3 pt-2 last:mb-2 md:last:mb-6 lg:mx-auto lg:max-w-3xl lg:pt-6\">\n<div class=\"relative flex h-full flex-1 md:flex-col\">\n<div class=\"ml-1 mt-1.5 md:w-full md:m-auto md:flex md:mb-2 gap-2 justify-center\"><\/div>\n<\/div>\n<\/form>\n<\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_is_Stokes_Law\"><\/span>What is Stokes Law?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Stokes Law is a law that describes the motion of objects through a fluid. It is named after Sir George Stokes, who first derived it in 1851. The law is only valid for objects that are small compared to the size of the fluid, and for fluids that are viscous.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Stokes_Law_Formula\"><\/span>Stokes Law Formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Stokes law formula is used to calculate the force required to move a object through a fluid. The formula is:<\/p>\n<p>F = 6 x pi x r x v<\/p>\n<p>where:<\/p>\n<p>F = force required (N)<\/p>\n<p>r = radius of the object (m)<\/p>\n<p>v = velocity of the object (m\/s)<\/p>\n<p>pi = 3.14<\/p>\n<h2><span class=\"ez-toc-section\" id=\"What_is_Stokes_Law_Derivation\"><\/span>What is Stokes Law Derivation?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Here is a step-by-step derivation of Stokes&#8217; law:<\/p>\n<ol>\n<li>Consider a small sphere of radius r moving through a viscous fluid at a constant velocity v.<\/li>\n<li>The sphere experiences a viscous force Fv from the fluid, which is proportional to the velocity of the sphere and the viscosity of the fluid. The proportionality constant is given by 6\u03c0\u03b7, where \u03b7 is the viscosity of the fluid. Thus, we can write:<\/li>\n<\/ol>\n<p>Fv = 6\u03c0\u03b7rv<\/p>\n<ol start=\"3\">\n<li>The sphere also experiences an opposing force Fi due to its inertia, which is proportional to its mass m and acceleration a. We can <a href=\"https:\/\/www.britannica.com\/science\/Newtons-laws-of-motion\/Newtons-second-law-F-ma\">write<\/a>:<\/li>\n<\/ol>\n<p>Fi = ma<\/p>\n<ol start=\"4\">\n<li>The viscous force and the inertial force must balance each other in order for the sphere to move at a constant velocity. We can write this as:<\/li>\n<\/ol>\n<p>Fv = Fi 6\u03c0\u03b7rv = ma<\/p>\n<ol start=\"5\">\n<li>Solving for the velocity, we get:<\/li>\n<\/ol>\n<p>v = (ma)\/(6\u03c0\u03b7r)<\/p>\n<ol start=\"6\">\n<li>Substituting Fv for the viscous force gives us the final form of Stokes&#8217; law:<\/li>\n<\/ol>\n<p>F = 6\u03c0\u03b7rv<\/p>\n<p>This equation can be used to predict the drag force experienced by a small sphere moving through a viscous fluid at a constant velocity.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Terminal_Velocity_Formula_in_Physics\"><\/span>Terminal Velocity Formula in Physics<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"w-full border-b border-black\/10 dark:border-gray-900\/50 text-gray-800 dark:text-gray-100 group bg-gray-50 dark:bg-[#444654]\">\n<div class=\"text-base gap-4 md:gap-6 m-auto md:max-w-2xl lg:max-w-2xl xl:max-w-3xl p-4 md:py-6 flex lg:px-0\">\n<div class=\"relative flex w-[calc(100%-50px)] md:flex-col lg:w-[calc(100%-115px)]\">\n<div class=\"flex flex-grow flex-col gap-3\">\n<div class=\"min-h-[20px] flex flex-col items-start gap-4 whitespace-pre-wrap\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<p>The terminal velocity of an object is the maximum velocity it can reach as it falls through a fluid, such as air or water. It is the velocity at which the force of gravity pulling the object downward is balanced by the drag force of the fluid opposing the motion.<\/p>\n<p>The terminal velocity of an object can be calculated using the following formula:<\/p>\n<p>v = \u221a((2<em>m<\/em>g)\/(\u03c1<em>A<\/em>Cd))<\/p>\n<p>where:<\/p>\n<p>v is the terminal velocity m is the mass of the object g is the acceleration due to gravity \u03c1 is the density of the fluid A is the cross-sectional area of the object facing the fluid Cd is the coefficient of drag of the object<\/p>\n<p>The coefficient of drag is a measure of how resistant the object is to moving through the fluid. It is a function of the shape and surface properties of the object and the viscosity of the fluid.<\/p>\n<p>The terminal velocity of an object depends on its mass, size, and shape, as well as the density and viscosity of the fluid it is moving through. For example, a heavy object with a large cross-sectional area and a low coefficient of drag will have a lower terminal velocity than a lighter object with a small cross-sectional area and a high coefficient of drag.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-4 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2><span class=\"ez-toc-section\" id=\"Assumptions_made_in_the_Stokes_Law\"><\/span>Assumptions made in the Stokes Law:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>&nbsp;<\/p>\n<div class=\"w-full border-b border-black\/10 dark:border-gray-900\/50 text-gray-800 dark:text-gray-100 group bg-gray-50 dark:bg-[#444654]\">\n<div class=\"text-base gap-4 md:gap-6 m-auto md:max-w-2xl lg:max-w-2xl xl:max-w-3xl p-4 md:py-6 flex lg:px-0\">\n<div class=\"relative flex w-[calc(100%-50px)] md:flex-col lg:w-[calc(100%-115px)]\">\n<div class=\"flex flex-grow flex-col gap-3\">\n<div class=\"min-h-[20px] flex flex-col items-start gap-4 whitespace-pre-wrap\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<p>There are several assumptions that are made when using Stokes&#8217; law to calculate the drag force experienced by a small sphere moving through a viscous fluid at a constant velocity:<\/p>\n<ol>\n<li>The sphere is small compared to the dimensions of the fluid. This allows us to ignore any viscous effects at the surface of the sphere.<\/li>\n<li>The velocity of the sphere is constant. This means that the sphere is moving at a steady state, without accelerating or decelerating.<\/li>\n<li>The fluid is Newtonian, meaning that the shear stress is directly proportional to the shear rate. This allows us to use the viscosity of the fluid to describe its resistance to flow.<\/li>\n<li>The flow is laminar, meaning that the fluid flows in smooth, parallel layers. This allows us to use the simple form of the drag force equation given above. If the flow is turbulent, a more complex equation must be used to take into account the additional forces that are present.<\/li>\n<li>The temperature and pressure of the fluid are constant. This means that the viscosity of the fluid is constant, which simplifies the calculation of the drag force.<\/li>\n<li>The sphere is a perfect sphere with a smooth surface. If the surface of the sphere is rough or irregular, the drag force may be different from what is predicted by Stokes&#8217; law.<\/li>\n<\/ol>\n<p>It&#8217;s important to note that these assumptions are idealizations that are used to simplify the calculation of the drag force. In real-world situations, it may be necessary to take into account additional factors that can affect the drag force.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-4 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2><span class=\"ez-toc-section\" id=\"Applications_of_Stoke_Law_in_Daily_Life\"><\/span>Applications of Stoke Law in Daily Life<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"w-full border-b border-black\/10 dark:border-gray-900\/50 text-gray-800 dark:text-gray-100 group dark:bg-gray-800\">\n<div class=\"text-base gap-4 md:gap-6 m-auto md:max-w-2xl lg:max-w-2xl xl:max-w-3xl p-4 md:py-6 flex lg:px-0\">\n<div class=\"relative flex w-[calc(100%-50px)] md:flex-col lg:w-[calc(100%-115px)]\">\n<div class=\"flex flex-grow flex-col gap-3\">\n<div class=\"min-h-[20px] flex flex-col items-start gap-4 whitespace-pre-wrap\">10 Applications of Stoke Law in Daily Life<\/div>\n<\/div>\n<div class=\"text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-4 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"w-full border-b border-black\/10 dark:border-gray-900\/50 text-gray-800 dark:text-gray-100 group bg-gray-50 dark:bg-[#444654]\">\n<div class=\"text-base gap-4 md:gap-6 m-auto md:max-w-2xl lg:max-w-2xl xl:max-w-3xl p-4 md:py-6 flex lg:px-0\">\n<div class=\"w-[30px] flex flex-col relative items-end\">\n<div class=\"relative h-[30px] w-[30px] p-1 rounded-sm text-white flex items-center justify-center\"><\/div>\n<\/div>\n<div class=\"relative flex w-[calc(100%-50px)] md:flex-col lg:w-[calc(100%-115px)]\">\n<div class=\"flex flex-grow flex-col gap-3\">\n<div class=\"min-h-[20px] flex flex-col items-start gap-4 whitespace-pre-wrap\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<ol>\n<li>Understanding the drag force experienced by a small object moving through a fluid is important in a variety of engineering applications, such as the design of aircraft, boats, and automobiles.<\/li>\n<li>Stokes&#8217; law is often used to calculate the settling velocity of small particles in a fluid, which is important in fields such as chemical engineering and environmental science.<\/li>\n<li>The motion of small particles through fluids is also important in the pharmaceutical and food industries, where it is used to design equipment for the separation and purification of materials.<\/li>\n<li>Stokes&#8217; law can be used to calculate the drag force experienced by a small object moving through air, which is important in the design of sports equipment such as golf balls, baseballs, and arrows.<\/li>\n<li>The drag force experienced by a small object moving through water is important in the design of underwater vehicles and in the study of ocean currents.<\/li>\n<li>Stokes&#8217; law is also used to calculate the drag force experienced by a small object moving through a fluid in the human body, such as blood or mucus. This is important in the study of blood flow and the design of medical devices.<\/li>\n<li>The motion of small particles through fluids is important in the study of atmospheric science, as it can help to understand the transport of pollutants and other particles through the atmosphere.<\/li>\n<li>Stokes&#8217; law is used in the design of inkjet printers, where it is used to calculate the drag force experienced by small droplets of ink moving through air.<\/li>\n<li>The drag force experienced by small objects moving through fluids is also important in the study of meteorology, as it can help to understand the motion of raindrops and snowflakes through the atmosphere.<\/li>\n<li>Stokes&#8217; law is used in the design of consumer products such as vacuum cleaners and air filters, where it is used to calculate the drag force experienced by small particles moving through a fluid.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-4 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Stokes law derivation for laminar flow Stokes&#8217; law is a mathematical equation used to describe the motion of a small [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"stokes law","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"stokes law derivation","custom_permalink":"physics\/stokes-law-derivation\/"},"categories":[4],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - 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