{"id":1796,"date":"2022-06-22T18:06:44","date_gmt":"2022-06-22T18:06:44","guid":{"rendered":"https:\/\/blog.praxilabs.com\/?p=1796"},"modified":"2025-08-22T22:14:59","modified_gmt":"2025-08-22T22:14:59","slug":"bernoulli-equation-and-applications","status":"publish","type":"post","link":"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/","title":{"rendered":"Learn All about Bernoulli Equation and Its Applications"},"content":{"rendered":"<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Bernoulli equation, or the incompressible steady flow energy equation, is considered one of the most well-known equations in physics ( fluid mechanics) and it explains the conservation of mechanical work-energy. The equation was published in 1738 by Daniel Bernoulli (a Swiss physicist) to help us understand fluid flow.<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\"><a href=\"https:\/\/en.wikipedia.org\/wiki\/Daniel_Bernoulli\" target=\"_blank\" rel=\"noopener\">Daniel Bernoulli<\/a> was born on February 8th, 1700, in the Netherlands. He studied mathematics and medicine under the guidance of his father Johann Bernoulli.<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">\u00a0In this article, we will discuss Bernoulli equation, principle, derivation, application and more.<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 12pt;\"><a class=\"maxbutton-3 maxbutton\" href=\"https:\/\/praxilabs.com\/\"><span class='mb-text'>Get started Praxilabs for FREE<\/span><\/a><\/span><\/strong><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-1798 aligncenter\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/images.png\" alt=\"\u00a0Bernoulli Equation\" width=\"377\" height=\"248\" \/><\/span><\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_83 counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\r\n<div class=\"ez-toc-title-container\">\r\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\r\n<span class=\"ez-toc-title-toggle\"><\/span><\/div>\r\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#_Bernoulli_Equation\" >\u00a0Bernoulli Equation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Bernoulli_Differential_Equations\" >Bernoulli Differential Equations<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Bernoulli_equation_for_V\" >Bernoulli equation for V<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#What_is_Bernoullis_Principle\" >What is Bernoulli&#8217;s Principle ?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Bernoullis_Principle_Formula\" >Bernoulli&#8217;s Principle Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Limitations_on_the_Use_of_Bernoulli_Theorem\" >Limitations on the Use of\u00a0 Bernoulli Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Bernoulli_Equation_Derivation\" >Bernoulli Equation Derivation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Bernoulli_Equation_Applications\" >Bernoulli Equation Applications<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Bernoulli_Theorem_application_in_fluid_mechanics\" >Bernoulli Theorem application in fluid mechanics<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Application_of_Bernoullis_equation_in_pumps\" >Application of Bernoulli&#8217;s equation in pumps<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Application_of_Bernoullis_Theorem_in_Aeroplanes\" >Application of Bernoulli&#8217;s Theorem in Aeroplanes \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Pressure_detection\" >Pressure detection<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Velocity_detection\" >Velocity detection<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Application_of_Bernoullis_equation_in_medicine\" >Application of Bernoulli&#8217;s equation in medicine<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Other_Applications_of_Bernoullis_Principle\" >Other Applications of Bernoulli&#8217;s Principle<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Bernoulli_Equation_Assumptions\" >Bernoulli Equation Assumptions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Example_of_Bernoulli_Differential_Equations\" >Example of Bernoulli Differential Equations<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Gamified_learning_experiences_with_Bernoullis_equation\" >Gamified learning experiences with Bernoulli&#8217;s equation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/06\/22\/bernoulli-equation-and-applications\/#Interactive_simulations_of_Bernoullis_equation\" >Interactive simulations of Bernoulli&#8217;s equation<\/a><\/li><\/ul><\/nav><\/div>\r\n<h2><span class=\"ez-toc-section\" id=\"_Bernoulli_Equation\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">\u00a0<\/span><b>Bernoulli Equation<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><strong>P<\/strong> is the static pressure (the pressure of the fluid).<\/span><\/li>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><strong>\u03c1<\/strong> is the density.<\/span><\/li>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><strong>V<\/strong> is the velocity.<\/span><\/li>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><strong>G<\/strong> is the gravitational acceleration.<\/span><\/li>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><strong>H<\/strong> is the elevation of the fluid.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Bernoulli&#8217;s theorem describes the relation between velocity, pressure, and elevation of a flowing fluid in a streamline.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">\u00a0It states that the dynamic pressure plus the static pressure in the flow, one half of the density (r) times the velocity (V) squared, is equal to a constant throughout the flow.\u00a0 The constant is called the total pressure of the flow (pt).<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Or simply, the Bernoulli equation states that the sum of the potential, kinetic, and flow energies of a fluid is constant in a streamline in steady flow.<\/span><\/p>\n<p><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">Note:<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The streamline in steady flow is the path traced by a single particle within the fluid.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The following figure shows the streamline in steady flow:<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"> <img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1812 size-full aligncenter\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/stream-line-flow.png\" alt=\"streamline in steady flow\" width=\"385\" height=\"145\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/stream-line-flow.png 385w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/stream-line-flow-300x113.png 300w\" sizes=\"auto, (max-width: 385px) 100vw, 385px\" \/><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">To apply Bernoulli\u2019s equation:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The flow must be steady (Velocity, density, and pressure must not change at any point in the streamline).<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The flow must be incompressible: even when the pressure varies, the density must remain constant along the streamline.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Friction by viscous forces must be negligible or minimal.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Note:<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">\u00a0Incompressible flows are gasses and gasses with a low Mach number and constant fluid density, regardless of the pressure flow. By using incompressible flow, we will have the simplest form of Bernoulli\u2019s equation.<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Bernoulli_Differential_Equations\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Bernoulli Differential Equations<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">A <\/span><b>Bernoulli differential equation<\/b><span style=\"font-weight: 400;\"> is a specific type of ordinary differential equation that can be expressed in the form:<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\"><br \/>\n<\/span> <b>y\u2032+P(x)y=Q(x)yn<\/b><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Where:<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">n is a real number (except for 0 or 1), because it is notable for its non-linear nature and if n=0 or n=1 then the equation is linear.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">\u00a0<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Bernoulli_equation_for_V\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Bernoulli equation for V<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">In Bernoulli\u2019s equation, we can find the velocity of the fluid<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3110\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/media_708_7087b8ec-bbef-4a0a-83f6-e37810e403ac_image.png\" alt=\"Bernoulli equation for V\" width=\"208\" height=\"80\" \/><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Where:<\/span><\/p>\n<ul>\n<li><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><strong>\u03c1<\/strong> is the density of the fluid.<\/span><\/li>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><strong>P<\/strong> is the measured pressure.<\/span><\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"What_is_Bernoullis_Principle\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>What is Bernoulli&#8217;s Principle ?<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Bernoulli\u2019s principle states that an increase in the velocity (the speed of a fluid) occurs simultaneously and must be accompanied by a decrease in the potential energy of the fluid (or the static pressure).<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Bernoulli\u2019s principle can be derived from the conservation of energy principle. In case of steady flow, the sum of energy forms in a fluid will remain the same at all points of that streamline. While all the energy remains constant, an increase in the fluid velocity will imply that there is an increase in the kinetic energy or dynamic pressure. This happens with a decrease in the potential energy (the static pressure and internal energy).<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Bernoullis_Principle_Formula\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Bernoulli&#8217;s Principle Formula<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Bernoulli\u2019s equation formula is considered a relation between pressure, potential energy, and kinetic energy of a fluid.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The formula given is as follows:<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1797 aligncenter\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/maxresdefault-e1655921095200.jpg\" alt=\"\u00a0Bernoulli Equation\" width=\"398\" height=\"157\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/maxresdefault-e1655921095200.jpg 908w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/maxresdefault-e1655921095200-300x118.jpg 300w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/maxresdefault-e1655921095200-768x303.jpg 768w\" sizes=\"auto, (max-width: 398px) 100vw, 398px\" \/><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The Bernoulli equation can be considered to be the conservation of energy principle for the flowing fluids. Bernoulli effect is the lowering of fluid pressure in cases where the velocity is increased. In the case of high velocity flow through the constriction, the kinetic energy must increase at the expense of pressure energy.<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Limitations_on_the_Use_of_Bernoulli_Theorem\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Limitations on the Use of\u00a0 Bernoulli Theorem <\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The Bernoulli equation is applicable to steady flow as we explained before, but in some cases we can&#8217;t use the Bernoulli equation.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The Bernoulli theorem is not applicable in a flow section like machines (e.g.,\u00a0 fan or turbine or pump), because these machines can damage the streamlines and make energy interactions with the fluid. So in these cases, the energy equation should be used instead of Bernoulli equation.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The equation also should not be used for significant temperature change flow sections like cooling or heating sections, because the gas density is inversely proportional to the temperature.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Every flow involves some friction and if the frictional effects are not be negligible, the Bernoulli theorem is not applicable in this case.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">So, the Bernoulli equation should not be used in the flow of:<\/span><\/p>\n<ul>\n<li><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">Sudden expansion.<\/span><\/span><\/li>\n<li><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">Heating section.<\/span><\/span><\/li>\n<li><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">Valve.<\/span><\/span><\/li>\n<li><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">Fan.<\/span><\/span><\/li>\n<li><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">Narrow long tube.<\/span><\/span><\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Bernoulli_Equation_Derivation\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Bernoulli Equation Derivation<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">We can drive B<\/span><span style=\"font-weight: 400;\">ernoulli differential equation as follows:<\/span><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">We will consider that we have a pipe with a varying height and diameter through which an incompressible fluid is flowing. The relationship between the areas of cross-sections A, pressure p the speed <span style=\"font-weight: 400;\">of the flow v, and the height h at two different points 1 and 2 is described in the following figure:<\/span><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1801 size-full\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/cub_bernoulli_lesson01_figure1web.jpg\" alt=\" a pipe with a varying height and diameter -Bernoulli Equation Derivation\" width=\"451\" height=\"288\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/cub_bernoulli_lesson01_figure1web.jpg 451w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/cub_bernoulli_lesson01_figure1web-300x192.jpg 300w\" sizes=\"auto, (max-width: 451px) 100vw, 451px\" \/><\/span><\/p>\n<h2><\/h2>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Assumptions:<\/b><\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The incompressible fluid density remains constant at both points.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The fluid energy is conserved because\u00a0 there are no viscous forces in the fluid.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">So, the work done on the fluid is given as:<\/span><\/p>\n<ul>\n<li><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">dW = F1dx1 \u2013 F2dx2<\/span><\/strong><\/span><\/li>\n<li><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">dW = p1A1dx1 \u2013 p2A2dx2<\/span><\/strong><\/span><\/li>\n<li><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">dW = p1dv \u2013 p2dv = (p1 \u2013 p2)dv<\/span><\/strong><\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">We know that the work done on the fluid was due to the conservation of change in both the kinetic energy and gravitational potential energy<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">\u00a0The change in kinetic energy of the fluid is given as:<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1802 size-full\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/Capture.png\" alt=\"\" width=\"401\" height=\"77\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/Capture.png 401w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/Capture-300x58.png 300w\" sizes=\"auto, (max-width: 401px) 100vw, 401px\" \/><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The change in potential energy is given as:<\/span><\/p>\n<ul>\n<li><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">dU = m2gy2 \u2013 m1gy1 = \u03c1dvg(y2 \u2013 y1)<\/span><\/strong><\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Then, the energy equation is given as:<\/span><\/p>\n<p style=\"text-align: left;\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0dW = dK + dU<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1803\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/1.png\" alt=\"bernoulli equation\" width=\"403\" height=\"91\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/1.png 403w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/1-300x68.png 300w\" sizes=\"auto, (max-width: 403px) 100vw, 403px\" \/><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">And finally by rearranging the above equation, we get<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1797 aligncenter\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/maxresdefault.jpg\" alt=\"\u00a0Bernoulli Equation\" width=\"426\" height=\"240\" \/><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><strong>For more understanding, watch the following video about Bernoulli equation derivation<\/strong><\/span><\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/DW4rItB20h4\" width=\"560\" height=\"314\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>PraxiLabs virtual science labs enable you to conduct various laboratory experiments in physics, chemistry, and biology online anytime and anywhere. <\/b><a href=\"https:\/\/praxilabs.com\/en\/sign-up\"><b>Create your free account<\/b><\/a><b> and try our virtual labs now.<\/b><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 12pt; font-family: tahoma, arial, helvetica, sans-serif;\"><strong><a class=\"maxbutton-3 maxbutton\" href=\"https:\/\/praxilabs.com\/en\/pricing\"><span class='mb-text'>Book FREE Live Demo Now<\/span><\/a>\u00a0<\/strong><\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Bernoulli_Equation_Applications\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Bernoulli Equation Applications<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Bernoulli&#8217;s equation has many applications. For example, we can use it to explain how planes generate lift, to calculate the velocity of fluid flow, or why ships have to run away from each other as they pass, and other applications that we find in our daily lives. We will go over in detail some of the most popular applications of bernoulli&#8217;s theorem.<\/span><\/p>\n<ol>\n<li>\n<h3><span class=\"ez-toc-section\" id=\"Bernoulli_Theorem_application_in_fluid_mechanics\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Bernoulli Theorem application in fluid mechanics<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/li>\n<\/ol>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The Bernoulli equation is applied to all incompressible fluid flow problems. The Bernoulli equation can be applied to devices such as the orifice meter, Venturi meter, and <a href=\"https:\/\/en.wikipedia.org\/wiki\/Pitot_tube\" target=\"_blank\" rel=\"noopener\">Pitot tube<\/a> and its applications for measuring flow in open channels and inside tubes.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Venturi meter is a device that is based on Bernoulli&#8217;s principle and used to determine the rate of flow through a pipe. It works by measuring the pressure drop across a section near the pipe. For an incompressible fluid, reducing the diameter will increase the fluid flow velocity.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Note: Bernoulli&#8217;s principle states that there must be a pressure drop in the region of the reduced diameter. This phenomenon is known as the venturi effect.<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1806 aligncenter\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/download.jpg\" alt=\"Venturi meter\" width=\"318\" height=\"159\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/download.jpg 318w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/download-300x150.jpg 300w\" sizes=\"auto, (max-width: 318px) 100vw, 318px\" \/><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><strong>Venturi meter<\/strong><\/span><\/p>\n<ol start=\"2\">\n<li>\n<h3><span class=\"ez-toc-section\" id=\"Application_of_Bernoullis_equation_in_pumps\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b> Application of Bernoulli&#8217;s equation in pumps<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/li>\n<\/ol>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1805 aligncenter\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/Bernoul-1.gif\" alt=\"Application of Bernoulli's equation in pumps\" width=\"431\" height=\"192\" \/><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">As we mentioned before,the<\/span><span style=\"font-weight: 400;\"> speed of a fluid flow can be measured by devices like Venturi meter, which is used\u00a0 to reduce the diameter of the flow and placed into a pipeline.<\/span><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">The maximum possible discharge rate for a tank with an opening or a spout at the base can be calculated directly by Bernoulli&#8217;s equation, and it has been found to be proportional to the square root of the height of the liquid in the tank (this is <a href=\"https:\/\/www.tec-science.com\/mechanics\/gases-and-liquids\/discharge-outflow-liquid-speed-torricellis-law\/\" target=\"_blank\" rel=\"noopener\">Torricelli&#8217;s law<\/a>) and this shows that Torricelli&#8217;s law is compatible with Bernoulli&#8217;s principle. Viscosity reduces the rate of discharge.<\/span><\/span><\/p>\n<ol start=\"3\">\n<li>\n<h3><span class=\"ez-toc-section\" id=\"Application_of_Bernoullis_Theorem_in_Aeroplanes\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Application of Bernoulli&#8217;s Theorem in Aeroplanes <\/b><b>\u00a0<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/li>\n<\/ol>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1807 aligncenter\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/txz8_vhwl_220322-scaled.jpg\" alt=\"Application of Bernoulli's equation in Aeroplanes \u00a0\" width=\"575\" height=\"309\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/txz8_vhwl_220322-scaled.jpg 2560w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/txz8_vhwl_220322-300x161.jpg 300w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/txz8_vhwl_220322-1024x550.jpg 1024w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/txz8_vhwl_220322-768x413.jpg 768w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/txz8_vhwl_220322-1536x825.jpg 1536w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/txz8_vhwl_220322-2048x1101.jpg 2048w\" sizes=\"auto, (max-width: 575px) 100vw, 575px\" \/><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">How aircraft wings generate lift can be explained by Bernoulli&#8217;s equation for fluids. In the case of flight, the fluid flowing above the wing of the aircraft moves faster than the flowing under the wing, and according to Bernoulli&#8217;s principle this causes the creation of a region of low pressure above the surface of the air and a region of high pressure under the plane and this pressure difference is what generates altitude.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">That is, we can explain the reason for the shape of the wings and their upward curvature (according to Bernoulli&#8217;s theory) that the air passes at a higher speed on the upper surface than the lower one. The difference in air velocity is calculated using Bernoulli&#8217;s principle of difference in pressure.<\/span><\/p>\n<ol start=\"4\">\n<li>\n<h3><span class=\"ez-toc-section\" id=\"Pressure_detection\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Pressure detection<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/li>\n<\/ol>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">In some cases of fluid flows, we know the velocities at two points of the streamline and pressure at only one point. The unknown is the pressure at the other point of the fluid.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">So in this case\u00a0 (if the Bernoulli equation applies to it ) we can use Bernoulli\u2019s Equation to find the unknown pressure.\u00a0<\/span><\/p>\n<ol start=\"5\">\n<li>\n<h3><span class=\"ez-toc-section\" id=\"Velocity_detection\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Velocity detection<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/li>\n<\/ol>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">In cases where the elevation and\u00a0 pressure at two points of the streamline are known and velocity at one point is\u00a0 also known, and we want to find the unknown velocity, Bernoulli\u2019s Equation is applied to calculate and find the required velocity.\u00a0<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>For example: the flow through a siphon<\/b><span style=\"font-weight: 400;\">\u00a0<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">In this case, we need\u00a0 to find the velocity with which the fluid leaves the siphon. By applying Bernoulli\u2019s Equation between the reservoir surface and the siphon exit point where the fluid leaves the tube, pressure at both points is the same and equal to the atmospheric pressure, and the velocity at the reservoir is negligible because the reservoir is large. We can calculate the velocity at the exit point by using the values of elevation at the two points.<\/span><\/p>\n<ol start=\"6\">\n<li>\n<h3><span class=\"ez-toc-section\" id=\"Application_of_Bernoullis_equation_in_medicine\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b> Application of Bernoulli&#8217;s equation in medicine<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/li>\n<\/ol>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">In echocardiography, Bernoulli&#8217;s principle can be applied when interpreting blood flow to describe a localized pressure decrease produced by high flow rate near blockages.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">\u00a0For clinical medicine, the equation is used for an easy estimation of pressure gradients from velocity.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Bernoulli&#8217;s equation can also be used in\u00a0 the venturi mask.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">\u00a0The venturi mask is a medical oxygen device that delivers oxygen concentration to patients on controlled oxygen therapy. The mask has a tube that is connected to a nozzle which connects to a supply of pure oxygen. The tube that is directly connected to the mask has a small window which allows room air to flow into the mask.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The venturi mask can control the amount of oxygen that flows in, along with pure oxygen delivered from its connected nozzle.\u00a0<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Venturi masks<\/b><span style=\"font-weight: 400;\"> use the Bernoulli principle, <\/span><span style=\"font-weight: 400;\">As oxygen flows into the tube, it creates a decrease in pressure due to oxygen passing through a narrow opening of the tube. The air is allowed to flow into the mask by the drop of pressure, mixing with the pure oxygen from the nozzle, which is the consequence of Bernoulli&#8217;s principle.<\/span><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-1804 aligncenter\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/multi-flow-venturi-mask-500x500-1.jpg\" alt=\"6. Application of Bernoulli's equation in medicine- venturi mask\" width=\"221\" height=\"221\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/multi-flow-venturi-mask-500x500-1.jpg 500w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/multi-flow-venturi-mask-500x500-1-300x300.jpg 300w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2022\/06\/multi-flow-venturi-mask-500x500-1-150x150.jpg 150w\" sizes=\"auto, (max-width: 221px) 100vw, 221px\" \/><\/span><\/p>\n<p>&nbsp;<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Other_Applications_of_Bernoullis_Principle\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Other Applications of Bernoulli&#8217;s Principle<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Bernoulli&#8217;s theory is used to study the unstable potential flow used in the theory of ocean surface waves and acoustics.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">When we stand at a railway station and a train comes in, we tend to fall towards the train. We can explain this using Bernoulli&#8217;s principle. As the train passes, the speed of the air between us and the train increases. Hence, from the equation, we can say that the pressure is decreasing. So the pressure from the back pushes us towards the train and that depends on the Bernoulli effect.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The Bernoulli equation also explains how a Bunsen burner works. When the gas valve is opened, the gas flows into the barrel at a high velocity. According to Bernoulli&#8217;s theory, this high velocity creates a low pressure area in the barrel (which draws air through the regulator) allowing the gas to burn more efficiently.\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Many flow meters are based on Bernoulli&#8217;s principle to determine the velocity of a flowing fluid. The most famous of these devices is the Pitot-Static Tube, which is used in aircraft to measure flight speed.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Bernoulli&#8217;s principle also applies to swinging a cricket ball. During the match, players constantly polish one side of the ball. After some time, one side is completely rough and the other is still smooth. Hence, when the ball is thrown in the air, the velocity on one side of the ball is faster than the other, due to this difference in smoothness. And this leads to a difference in pressure between the two sides. This causes the ball to spin (&#8220;swing&#8221;) as it travels through the air.<\/span><\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Bernoulli_Equation_Assumptions\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Bernoulli Equation Assumptions<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">For Bernoulli\u2019s equation to be applied, the following assumptions must be met:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The flow must be steady. (Velocity, pressure and density cannot change at any point).<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The flow must be incompressible \u2013 even when the pressure varies, the density must remain constant along the streamline.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Friction by viscous forces must be minimal.<\/span><\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Example_of_Bernoulli_Differential_Equations\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Example of Bernoulli Differential Equations<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Try to solve the Bernoulli differential equation: y\u2032 + y = xy2<\/b><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Solution:<\/b><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Rewrite the equation in standard form:<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">y\u2032+y = xy2<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Divide through by y2:<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">\\frac{y\u2019}{y^2} + \\frac{y}{y^2} = x<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Simplified:<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">y^{-2}y\u2019 + y^{-1} = x<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Substitute v = y{-1}:<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Derivative:<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">\\frac{dv}{dx} = -y^{-2} \\frac{dy}{dx}<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Rewrite the original equation:<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">-\\frac{dv}{dx} + v = x<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Rearrange:<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">\\frac{dv}{dx} \u2013 v = -x<\/span><\/strong><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Find the integrating factor:<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">\\mu(x) = e^{\\int -1 \\, dx} = e^{-x}<\/span><\/strong><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Multiply through by the integrating factor:<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">e^{-x} \\frac{dv}{dx} \u2013 e^{-x} v = -xe^{-x}<\/span><\/strong><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Simplify:<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">\\frac{d}{dx} \\left( e^{-x} v \\right) = -xe^{-x}<\/span><\/strong><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Integrate both sides:<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">e^{-x} v = \\int -xe^{-x} \\, dx + C<\/span><\/strong><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Using integration by parts:<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">\\int -xe^{-x} \\, dx = -xe^{-x} \u2013 \\int e^{-x} \\, dx = -xe^{-x} + e^{-x}<\/span><\/strong><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Thus:<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">e^{-x} v = -xe^{-x} + e^{-x} + C<\/span><\/strong><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Simplify:<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">v = -x + 1 + Ce^{x}<\/span><\/strong><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Back-substitute v = y{-1}:<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">y^{-1} = -x + 1 + Ce^{x}<\/span><\/strong><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Solve for y:<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">y = \\frac{1}{-x + 1 + Ce^{x}}<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Source: <a href=\"https:\/\/www.geeksforgeeks.org\/how-to-solve-bernoulli-differential-equation\/#steps-to-solve-a-bernoulli-differential-equation\" target=\"_blank\" rel=\"noopener\">How to Solve Bernoulli Differential Equation<\/a><\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Gamified_learning_experiences_with_Bernoullis_equation\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Gamified learning experiences with Bernoulli&#8217;s equation<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Gamification is the method that depends on using game-design elements and game mechanics in non-game contexts. Some researchers suggest that it could also be used in web-based education as a tool to increase student motivation and engagement.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">For example, using gamified learning elements in teaching Bernoulli\u2019s equation can motivate and encourage learners to participate in the learning process and thus improve their learning outcomes and also enhance their engagement and motivation.<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">In an attempt to verify those theories, researchers <\/span><span style=\"font-weight: 400;\">conducted a study<\/span><span style=\"font-weight: 400;\"> \u201cGamifying Learning Experiences: Practical Implications and Outcomes\u201d they\u00a0 designed and developed a gamification plugin for a well-known e-learning platform. They conducted an experiment using this plugin in a university course, collecting quantitative and qualitative data in the process.<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Their findings suggest that some common beliefs about the benefits obtained when using games in education can be challenged. Students who completed the gamified experience\u00a0 scored better in practical assignments and achieved\u00a0 higher overall scores, but their findings also suggest that these students performed poorly on written assignments and participated less in class activities, although their initial motivation was higher.<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\"><a href=\"https:\/\/www.researchgate.net\/publication\/256194365_Gamifying_Learning_Experiences_Practical_Implications_and_Outcomes\" target=\"_blank\" rel=\"noopener\">Study<\/a><\/span><b><\/b><\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Interactive_simulations_of_Bernoullis_equation\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Interactive simulations of Bernoulli&#8217;s equation<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Virtual lab simulations can enrich the experimentation process, help students to understand Bernoulli&#8217;s equation.<\/span><\/p>\n<p data-pm-slice=\"1 1 []\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Educational virtual labs for fluid mechanics such as the<span style=\"font-weight: 400;\">\u00a0interactive simulations of Bernoulli\u2019s equation simulate the water flowing through a pipe. You can adjust the outlet height, outlet pipe diameter, inlet pressure, and inlet velocity using the sliders.<\/span><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">Virtual labs for <\/span>Online experiments with fluid flow phenomena <span style=\"font-weight: 400;\">provide a wide range of interactive features:<\/span><\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">\u00a0Interactive 3D virtual labs provide students with tools that allow for more focus, attention, and engagement. Students stay interested, alert, and absorb information more easily.\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Virtual labs eliminate potential safety hazards associated with handling hazardous materials or conducting experiments that require specialized equipment. Students can learn and experiment in a safe environment, reducing the chances of accidents or mishaps.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">By using virtual labs, you can save time and effort, as they eliminate the need to move between different laboratories. Students can access experiments and learning resources without the need for setup or cleanup, allowing them to focus more on the core learning objectives.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Virtual labs provide a virtual learning and teaching environment that aims to develop practical skills and learning outcomes.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">\u00a0Students can conduct experiments with 24\/7 unlimited accessibility and repeat virtual experiments as many times as needed until they grasp and understand all the information.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Since virtual simulations are accessible via the Internet, students can perform numerous experiments without being confined to specific locations or times, unlike when using real laboratories.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b><a href=\"https:\/\/praxilabs.com\/\">PraxiLabs simulation lab <\/a>enable you to conduct various laboratory experiments in physics, chemistry, and biology online anytime and anywhere. <\/b><a href=\"https:\/\/praxilabs.com\/en\/sign-up\"><b>Create your free account<\/b><\/a><b> and try PraxiLabs virtual labs now.<\/b><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 12pt;\"><a class=\"maxbutton-3 maxbutton\" href=\"https:\/\/praxilabs.com\/en\/virtual-physics-lab\"><span class='mb-text'>Join our virtual physics lab and start your virtual journey<\/span><\/a><\/span><\/strong><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bernoulli equation, or the incompressible steady flow energy equation, is considered one of the most well-known equations in physics ( fluid mechanics) and it explains the conservation of mechanical work-energy. The equation was published in 1738 by Daniel Bernoulli (a Swiss physicist) to help us understand fluid flow. Daniel Bernoulli was born on February 8th, &hellip;<\/p>\n","protected":false},"author":8,"featured_media":4439,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[4],"tags":[],"class_list":["post-1796","post","type-post","status-publish","format-standard","has-post-thumbnail","","category-physics"],"modified_by":"Muhamed Elmesery","_links":{"self":[{"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/posts\/1796","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/comments?post=1796"}],"version-history":[{"count":17,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/posts\/1796\/revisions"}],"predecessor-version":[{"id":4129,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/posts\/1796\/revisions\/4129"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/media\/4439"}],"wp:attachment":[{"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/media?parent=1796"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/categories?post=1796"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/tags?post=1796"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}