{"id":10253,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10253"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"conservation-of-energy-and-mass-in-fluid-flow","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/conservation-of-energy-and-mass-in-fluid-flow\/","title":{"rendered":"Conservation Of Energy And Mass In Fluid Flow"},"content":{"rendered":"<div class=\"article-watermark-wrapper\">\n<div style=\"position: relative; z-index: 1;\">\n<p style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 9pt; color: #444444;\">KAPDEC&reg; | Elite STEM Learning Platform | <a href=\"https:\/\/kapdec.com\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"color: #444444; text-decoration: underline;\">https:\/\/kapdec.com<\/a><\/p>\n<hr \/>\n<h2><strong>Unit: <\/strong><strong>Fluids: Pressure and Forces<\/strong><\/h2>\n<p><strong>Chapter: Conservation of Energy and mass in fluid flow<\/strong><\/p>\n<p><em>Reference: AP Physics Algebra, Fluids, Conservation of Energy and mass in fluid flow, Streamline Motion, Equation of Continuity Critical Velocity and Reynolds\u2019s Number, <\/em><em>Bernoulli\u2019s principle, <\/em><em>Applications of Bernoulli\u2019s Theorem<\/em><\/p>\n<p><strong>After studying this chapter, you should be able to,<\/strong><\/p>\n<ul>\n<li><strong>Understand the concept of Buoyancy<\/strong><\/li>\n<li><strong>to draw free body diagram of the object<\/strong><\/li>\n<\/ul>\n<p><strong>Streamline Motion <\/strong><\/p>\n<ul>\n<li>The path followed by fluid particles is called the line of flow. If every particle passing through a given point of the path follows the same line of flow as that of preceding particles, the flow is said to be streamlined.<\/li>\n<li>A streamline can be represented as the curve or path whose tangent at any point gives the direction of the liquid velocity at that point. In steady flow, the streamlines coincide with the line of flow.<\/li>\n<li>When the velocity of flow is less than the critical velocity of a given liquid flowing through a tube, the motion is streamlined.<\/li>\n<li>If the velocity of flow exceeds the critical velocity, the mixing of streamlines takes place and the flow path becomes zig-zag. Such a motion is said to be turbulent.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<p><a name=\"_Hlk134279660\"><strong>Equation of Continuity<\/strong><\/a><\/p>\n<p>\u00a0<\/p>\n<p>Let <em>A<\/em>1 and <em>A<\/em>2 denote the areas of cross-section of the tube where the fluid is entering and leaving, as shown in Fig. given below. If <em>v<\/em>1 and <em>v<\/em>2 are the speeds of the fluid at the ends A and B respectively, and \u03c1 is the density of the fluid, then the liquid entering the tube at A covers a distance <em>v<\/em>1 in one second. So, the volume of the liquid entering per second= <em>A<\/em>1 \u00d7 <em>v<\/em>1. Therefore<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"180\" src=\"https:\/\/app.kapdec.com\/questions-images\/dKqKPti52AN41728557458.png?time=1728557458\" width=\"413\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>Mass of the liquid entering per second at point A = A1v1 \u03c1 Similarly, the mass of the liquid leaving per second at point B = A2v2 \u03c1 Since there is no accumulation of fluid inside the tube, the mass of the liquid crossing any section of the tube must be same. Therefore, we get<\/p>\n<p>A1v1 \u03c1 = A2v2 \u03c1<\/p>\n<p>or A1v1 = A2v2<\/p>\n<p>This expression is called <strong>an equation of continuity<\/strong>.<\/p>\n<p>\u00a0<\/p>\n<p><strong>Critical Velocity and Reynolds\u2019s Number<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>The value of the critical velocity of any liquid depends on the<\/p>\n<ul>\n<li>nature of the liquid, i.e. coefficient of viscosity ( \u03b7 ) of the liquid;<\/li>\n<li>diameter of the tube (d) through which the liquid flows; and<\/li>\n<li>the density of the liquid (\u03c1).<\/li>\n<\/ul>\n<p>Hence, we can write<\/p>\n<p>vc = R.\u03b7\/\u03c1d<\/p>\n<p>\u00a0<\/p>\n<p>Where R is the constant of proportionality and is called Reynolds&#8217;s Number. It has no dimensions. Experiments show that if R is below 1000, the flow is laminar. The flow becomes unsteady when R is between 1000 and 2000 and the flow becomes turbulent for R greater than 2000.<\/p>\n<p><strong>Bernoulli\u2019s principle:<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>Bernoulli\u2019s Principle states that where the velocity of a fluid is high, the pressure is low and where the velocity of the fluid is low, pressure is high.<\/p>\n<p>\u00a0<\/p>\n<ol>\n<li><strong>The energy of a Flowing Fluid<\/strong><\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<p>Flowing fluids possess three types of energy. We are familiar with the kinetic and potential energies. The third type of energy possessed by the fluid is pressure energy. It is due to the pressure of the fluid. The pressure energy can be taken as the product of pressure difference and its volume. If an element of liquid of mass m, and density d is moving under a pressure difference p, then Pressure energy = p \u00d7 (m\/d) joule<\/p>\n<p>Pressure energy per unit mass = (p\/d) J kg\u20131<\/p>\n<p>\u00a0<\/p>\n<p><strong>2. Bernoulli\u2019s Equation<\/strong><\/p>\n<p>Bernoulli developed an equation that expresses this principle quantitatively. Three important assumptions were made to develop this equation:<\/p>\n<p>1. The fluid is incompressible, i.e. its density does not change when it passes from a wide bore tube to a narrow bore tube.<\/p>\n<p>2. The fluid is non-viscous or the effect of viscosity is not to be taken into<\/p>\n<p>account.<\/p>\n<p>3. The motion of the fluid is streamlined.<\/p>\n<p>\u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"297\" src=\"https:\/\/app.kapdec.com\/questions-images\/A70kfLUeAAfm1728557459.png?time=1728557460\" width=\"395\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>We consider a tube of varying cross-sections shown in Fig. given above. Suppose at point A the pressure is P<sub>1<\/sub>, the area of cross-section A<sub>1<\/sub>, the velocity of flow v<sub>1<\/sub>, height above the ground h<sub>1<\/sub> and at B, the pressure is P<sub>2<\/sub>, area of cross-section A<sub>2<\/sub> velocity of flow = v<sub>2<\/sub>, and height above the ground h<sub>2<\/sub>.<\/p>\n<p>\u00a0<\/p>\n<p>Since points A and B can be any two points along a tube of flow, we write<\/p>\n<p>Bernoulli\u2019s equation<\/p>\n<p>\u00a0<\/p>\n<p>P + 1\/2 dv<sup>2<\/sup> + h dg = Constant.<\/p>\n<p>That is, the sum of pressure energy, kinetic energy and potential energy of a fluid<\/p>\n<p>remains constant in streamlined motion.<\/p>\n<p>\u00a0<\/p>\n<p><strong>Applications of Bernoulli\u2019s Theorem<\/strong><\/p>\n<p>Bernoulli\u2019s theorem finds many applications in our lives. Some commonly observed phenomena can also be explained on the basis of Bernoulli\u2019s theorem.<\/p>\n<p>\u00a0<\/p>\n<p>It is a device used to measure the rate of flow of liquids through pipes. The device is inserted in the flow pipe, as shown in the Fig. below.<\/p>\n<p>\u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"147\" src=\"https:\/\/app.kapdec.com\/questions-images\/XHIzmNWS8JTn1728557459.png?time=1728557460\" width=\"456\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>It consists of a manometer, whose two limbs are connected to a tube having two different cross-sectional areas say A1 and A2 at A and B, respectively. Suppose the main pipe is horizontal at a height <em>h <\/em>above the ground. Then applying Bernoulli\u2019s theorem for the steady flow of liquid through the venturi meter at A and B, we can write<\/p>\n<p>Total Energy at A = Total Energy At B<\/p>\n<p>\u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"73\" src=\"https:\/\/app.kapdec.com\/questions-images\/TDEHiXGaariF1728557459.png?time=1728557460\" width=\"417\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>On rearranging terms, we can write,<\/p>\n<p>\u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"67\" src=\"https:\/\/app.kapdec.com\/questions-images\/21K49fs2iHWu1728557459.png?time=1728557460\" width=\"379\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>It shows that points of higher velocities are the points of lower pressure (because of the sum of pressure energy and K.E. remain constant). This is called Venturi\u2019s<\/p>\n<p>Principle.<\/p>\n<p>For steady flow through the venturi meter, the volume of liquid entering per second at A = liquid volume leaving per second at B. Therefore<\/p>\n<p>A1v1 = A2v2<\/p>\n<p>\u00a0<\/p>\n<p>\u00a0<\/p>\n<p>(The liquid is assumed incompressible i.e., velocity is more at narrow ends and<\/p>\n<p>vice versa.<\/p>\n<p>Using this result in Eqn.<\/p>\n<p>\u00a0<\/p>\n<p>\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"67\" src=\"https:\/\/app.kapdec.com\/questions-images\/aMlbuK2A8sNb1728557460.png?time=1728557460\" width=\"379\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>we conclude that pressure is lesser at the narrow ends;<\/p>\n<p>\u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"281\" src=\"https:\/\/app.kapdec.com\/questions-images\/WB2tRl1AVQJr1728557460.png?time=1728557461\" width=\"293\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>\u00a0<\/p>\n<p>If <em>h <\/em>denotes the level difference between the two limbs of the venturi meter, then<\/p>\n<p>\u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"76\" src=\"https:\/\/app.kapdec.com\/questions-images\/N6LMVlI91Egz1728557460.png?time=1728557461\" width=\"265\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>From this, we note that <em>v<\/em>1 <em>h<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"20\" src=\"https:\/\/app.kapdec.com\/questions-images\/spJYobzQKlzS1728557460.png?time=1728557461\" width=\"17\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><em>\u00a0 <\/em>since all other parameters are constant for a<\/p>\n<p>given venturi meter. Thus<\/p>\n<p>\u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"37\" src=\"https:\/\/app.kapdec.com\/questions-images\/iwvyyaOaqYdl1728557460.png?time=1728557461\" width=\"120\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>where K is constant.<\/p>\n<p>The volume of liquid flowing per second is given by<\/p>\n<p>\u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"65\" src=\"https:\/\/app.kapdec.com\/questions-images\/ClQO1mSD51gE1728557460.png?time=1728557461\" width=\"238\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>where K<sup>\u2019 <\/sup>\u00a0= K A1 is another constant.<\/p>\n<p>\u00a0<\/p>\n<p><strong>Example 1: <\/strong>Water flows out of a small hole in the wall of a large tank near its bottom (Fig. given below). What is the speed of efflux of water when the height of the water level in the tank is 2.5m?<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"137\" src=\"https:\/\/app.kapdec.com\/questions-images\/Sg0r6HCoJkFJ1728557461.png?time=1728557462\" width=\"190\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p><strong>Solution: <\/strong>Let B be the hole near the bottom. Imagine a tube of flow A to B for the water to flow from the surface point A to hole B. We can apply Bernoulli&#8217;s theorem to points A and B for the streamlined flow of small mass m. Total energy at B = Total energy at A<\/p>\n<p>\u00a0<\/p>\n<p>At A, v<sub>A<\/sub>= 0, p<sub>A<\/sub>= p = atmospheric pressure, h = height above the ground.<\/p>\n<p>At B, v<sub>B <\/sub>= v = ?, p<sub>B<\/sub> = p, h<sub>B<\/sub> = height of the hole above the ground.<\/p>\n<p>\u00a0<\/p>\n<p>Let h<sub>A<\/sub> \u2013 h<sub>B<\/sub> = H = height of the water level in the vessel = 2.5m and d = density of the water. Applying Bernoulli&#8217;s Principle and substituting the values we get,<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"214\" src=\"https:\/\/app.kapdec.com\/questions-images\/9on5YKhNgWTD1728557461.png?time=1728557462\" width=\"257\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p><strong>Key points:<\/strong><\/p>\n<p><strong>Bernoulli&#8217;s Principle:<\/strong><\/p>\n<p>Bernoulli&#8217;s principle states that as the speed of a fluid (such as air or water) increases, its pressure decreases, and vice versa.<\/p>\n<p>This principle is based on the idea that energy in a fluid is conserved, and is expressed mathematically as P + (1\/2)\u03c1v<sup>2<\/sup> + \u03c1gh = constant, where P is the pressure, \u03c1 is the density, v is the velocity, h is the height above a reference point, and g is the acceleration due to gravity.<\/p>\n<p>Bernoulli&#8217;s principle can be used to explain a wide range of phenomena, from how aeroplanes fly to how perfume sprayers work.<\/p>\n<p><strong>Equation of Continuity:<\/strong><\/p>\n<p>The Equation of Continuity is a mathematical expression of the principle of conservation of mass for fluids.<\/p>\n<p>This principle states that the mass flow rate of a fluid is constant through any cross-section of a pipe or duct, regardless of changes in the fluid&#8217;s velocity or density.<\/p>\n<p>The Equation of Continuity is expressed mathematically as A<sub>1<\/sub>v<sub>1<\/sub> = A<sub>2<\/sub>v<sub>2<\/sub>, where A is the cross-sectional area of the pipe or duct, and v is the fluid velocity.<\/p>\n<p>The Equation of Continuity is used in many practical applications, including designing piping systems, measuring fluid flow rates, and understanding the behaviour of blood flow in the human body.<\/p>\n<p>\u00a0<\/p>\n<p><!--kapdec-footer-start--><\/p>\n<style>.kapdec-article-footer{font-family:Arial,Helvetica,Calibri,sans-serif;color:#444;}.kapdec-footer-grid{display:flex;align-items:stretch;border:1px solid #e5e7eb;border-radius:6px;overflow:hidden;}.kapdec-footer-left,.kapdec-qr-block{flex:1 1 50%;width:50%;box-sizing:border-box;min-width:0;}.kapdec-footer-left{padding:22px 28px;border-right:1px solid #e5e7eb;}.kapdec-citation-block{line-height:1.6;font-size:9pt;color:#333;margin:0;}.kapdec-citation-block p{margin:0 0 10px 0;}.kapdec-citation-block a{color:#0066cc;text-decoration:underline;}.kapdec-copyright-block{margin-top:18px;padding-top:14px;border-top:1px solid #e5e7eb;font-size:7.5pt;color:#777;line-height:1.55;text-align:left;}.kapdec-copyright-block p{margin:0 0 5px 0;}.kapdec-qr-block{padding:22px 28px;display:flex;flex-direction:column;align-items:center;justify-content:center;text-align:center;}.kapdec-qr-label{margin:0 0 8px 0;font-size:8.5pt;font-weight:600;color:#444;line-height:1.35;letter-spacing:.02em;}.kapdec-qr-url{margin:0 0 14px 0;font-size:7.5pt;line-height:1.4;color:#777;word-break:break-word;max-width:100%;}.kapdec-qr-url a{color:#777;text-decoration:underline;}@media (max-width:640px){.kapdec-footer-grid{flex-direction:column;}.kapdec-footer-left,.kapdec-qr-block{width:100%;flex-basis:100%;border-right:none;}.kapdec-footer-left{border-bottom:1px solid #e5e7eb;}}<\/style>\n<div class=\"kapdec-article-footer\" style=\"margin-top: 28px; padding-top: 4px;\">\n<div class=\"kapdec-footer-grid\">\n<div class=\"kapdec-footer-left\">\n<div class=\"kapdec-citation-block\">\n<p>A Kapdec&reg; learning guide &#8211; Crafted by elite STEM mentors for ambitious learners.<\/p>\n<p><a href=\"https:\/\/kapdec.com\" target=\"_blank\" rel=\"noopener noreferrer\">Learn more at https:\/\/kapdec.com<\/a><\/p>\n<\/div>\n<div class=\"kapdec-copyright-block\">\n<p>Author: Kapdec | Publisher: Kapdec | Copyright: &copy; Kapdec. 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