{"id":22208,"date":"2025-06-16T14:10:24","date_gmt":"2025-06-16T14:10:24","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=22208"},"modified":"2025-06-16T14:10:28","modified_gmt":"2025-06-16T14:10:28","slug":"state-bernoullis-theorem-for-steady-flow-of-an-incompressible-fluid","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/state-bernoullis-theorem-for-steady-flow-of-an-incompressible-fluid\/","title":{"rendered":"State Bernoulli\u2019s theorem for steady flow of an incompressible fluid"},"content":{"rendered":"\n<p>State Bernoulli\u2019s theorem for steady flow of an incompressible fluid. Derive an expression for Bernoulli\u2019s equation from first principles (Starting from Euler\u2019s Equation) and state the assumptions made for such a derivation.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p><strong>Correct Answer:<\/strong><\/p>\n\n\n\n<p><strong>Bernoulli\u2019s Theorem:<\/strong><br>For a steady, incompressible, and non-viscous fluid flowing along a streamline, the total mechanical energy (pressure energy, kinetic energy, and potential energy) per unit volume remains constant.<\/p>\n\n\n\n<p><strong>Mathematically:<\/strong> P+12\u03c1v2+\u03c1gh=constantP + \\frac{1}{2} \\rho v^2 + \\rho g h = \\text{constant}<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>PP = Pressure energy per unit volume<\/li>\n\n\n\n<li>\u03c1\\rho = Fluid density<\/li>\n\n\n\n<li>vv = Fluid velocity<\/li>\n\n\n\n<li>gg = Acceleration due to gravity<\/li>\n\n\n\n<li>hh = Height above a reference level<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Derivation of Bernoulli\u2019s Equation from Euler\u2019s Equation:<\/strong><\/p>\n\n\n\n<p><strong>Euler\u2019s Equation for Inviscid Flow Along a Streamline:<\/strong> dP\u03c1+g\u2009dz+v\u2009dv=0\\frac{dP}{\\rho} + g\\,dz + v\\,dv = 0<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>dP\u03c1\\frac{dP}{\\rho} = Change in pressure head<\/li>\n\n\n\n<li>g\u2009dzg\\,dz = Change in gravitational potential head<\/li>\n\n\n\n<li>v\u2009dvv\\,dv = Change in kinetic energy head<\/li>\n<\/ul>\n\n\n\n<p><strong>Integrating each term:<\/strong> \u222bdP\u03c1+\u222bg\u2009dz+\u222bv\u2009dv=0\\int \\frac{dP}{\\rho} + \\int g\\,dz + \\int v\\,dv = 0 \u21d2P\u03c1+gz+v22=constant\\Rightarrow \\frac{P}{\\rho} + g z + \\frac{v^2}{2} = \\text{constant}<\/p>\n\n\n\n<p>Multiply through by \u03c1\\rho to get Bernoulli\u2019s Equation: P+12\u03c1v2+\u03c1gz=constantP + \\frac{1}{2} \\rho v^2 + \\rho g z = \\text{constant}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Assumptions Made:<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Steady Flow<\/strong> \u2013 Properties of the fluid (velocity, pressure) at a point do not change with time.<\/li>\n\n\n\n<li><strong>Incompressible Fluid<\/strong> \u2013 Density \u03c1\\rho is constant.<\/li>\n\n\n\n<li><strong>Non-viscous Fluid<\/strong> \u2013 No energy is lost due to internal friction.<\/li>\n\n\n\n<li><strong>Flow along a streamline<\/strong> \u2013 The equation applies to fluid particles moving along the same path.<\/li>\n\n\n\n<li><strong>No external work done<\/strong> \u2013 No energy is added or extracted (e.g., by pumps or turbines).<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Explanation (300 words):<\/strong><\/p>\n\n\n\n<p>Bernoulli\u2019s theorem is a fundamental principle in fluid dynamics that relates the pressure, velocity, and elevation in a moving fluid. It states that for a steady, incompressible, and frictionless flow along a streamline, the total mechanical energy per unit volume remains constant. This total energy comprises pressure energy (PP), kinetic energy per unit volume (12\u03c1v2\\frac{1}{2} \\rho v^2), and potential energy per unit volume (\u03c1gh\\rho g h).<\/p>\n\n\n\n<p>To derive Bernoulli\u2019s equation from first principles, we start with Euler\u2019s equation for the motion of a fluid particle along a streamline. Euler\u2019s equation is derived from Newton\u2019s Second Law, applied to fluid motion, and expresses the balance of forces on a fluid element due to pressure, gravity, and acceleration. It can be written as: dP\u03c1+g\u2009dz+v\u2009dv=0\\frac{dP}{\\rho} + g\\,dz + v\\,dv = 0<\/p>\n\n\n\n<p>Each term represents a different form of energy per unit mass. When we integrate this equation, we obtain: P\u03c1+gz+v22=constant\\frac{P}{\\rho} + g z + \\frac{v^2}{2} = \\text{constant}<\/p>\n\n\n\n<p>Multiplying by the fluid density \u03c1\\rho transforms the equation into Bernoulli\u2019s form: P+12\u03c1v2+\u03c1gz=constantP + \\frac{1}{2} \\rho v^2 + \\rho g z = \\text{constant}<\/p>\n\n\n\n<p>This implies that if the speed of the fluid increases, its pressure must decrease if the height remains constant, and vice versa. Bernoulli\u2019s equation is widely used in applications such as airplane wing design, venturimeters, and pipe flow analysis.<\/p>\n\n\n\n<p>However, its validity relies on several assumptions, such as inviscid flow, no heat or work interactions, and application only along a streamline. If these conditions are not met, corrections or other more complex fluid dynamic equations must be used.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>State Bernoulli\u2019s theorem for steady flow of an incompressible fluid. Derive an expression for Bernoulli\u2019s equation from first principles (Starting from Euler\u2019s Equation) and state the assumptions made for such a derivation. The correct answer and explanation is: Correct Answer: Bernoulli\u2019s Theorem:For a steady, incompressible, and non-viscous fluid flowing along a streamline, the total mechanical [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-22208","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22208","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=22208"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22208\/revisions"}],"predecessor-version":[{"id":22209,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22208\/revisions\/22209"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=22208"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=22208"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=22208"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}