Lecture notes Fluid Mechanics

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Regulation 2013 Mechanics of fluids . Mechanics of fluids . Regulation 2013 Mechanics of fluids Objectives : To understand the basic properties of the  Fluid  Fluid kinematics  Fluid dynamics To analyze & appreciate the complexities involved in solving the fluid flow problems. UNIT – I Fluid properties & Fluid statics. INTRODUCTION Fluid Mechanics is the branch of science which deals with the study of behavior of the fluids at rest as well as in motion .it has two branches .,  Fluid statics (Study of Fluid at rest )  Fluid dynamics (Study of fluid at motion ) CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 1 Mechanics of fluids . Regulation 2013 PROPERTIES OF FLUIDS Mass Density : 𝑠𝑎𝑠𝑀 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 Mass Density (𝜌 ) = 𝑙𝑢𝑚𝑒𝑉𝑜 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 Weight Density : 𝑤𝑒𝑖𝑔 ℎ𝑡 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 Weight Density (w) = 𝑚𝑒𝑣𝑜𝑙𝑢 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑠𝑎𝑠𝑀 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 ×𝑔 = 𝑚𝑒𝑣𝑜𝑙𝑢 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 =𝜌 × 𝑔 3 Weight Density of Water = 1000 × 9.81 N/𝑚 Specific Volume : 𝑚𝑒𝑣𝑜𝑙𝑢 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 Specific Volume = 𝑠𝑎𝑠𝑀 𝑜𝑓 𝑓 𝑙𝑢𝑖𝑑 1 = 𝜌 3 It is expressed in 𝑚 /kg Specific Gravity : 𝑊𝑒𝑖𝑔 ℎ𝑡 𝑖𝑡𝑦𝑠𝑒𝑛𝑑 𝑜𝑓 𝑖𝑑𝑢𝑙𝑖𝑞 Specific Gravity (for Liquids) = 𝑔𝑤𝑒𝑖 ℎ𝑡 𝑖𝑡𝑦𝑠𝑒𝑛𝑑 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑊𝑒𝑖𝑔 ℎ𝑡 𝑖𝑡𝑦𝑠𝑒𝑛𝑑 𝑜𝑓 𝑒𝑠𝑎𝑠𝑔 Specific Gravity (for Gases) = 𝑤𝑒𝑖𝑔 ℎ𝑡 𝑖𝑡𝑦𝑠𝑒𝑛𝑑 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 2 Mechanics of fluids . Regulation 2013 VISCOSITY Property of a fluid which offers resistance to the movement of one later of the fluid over the other adjacent layer of the fluid . 𝜏 𝛼 𝜏 = 𝜇 𝜏 → Sheer Force. du/dy→ rate of change of velocity with respect y. µ→constant of proportionality Unit of viscosity 𝑓𝑘𝑔 −𝑒𝑐𝑠 MKS unit of viscosity = 2 𝑚 𝑦𝑛𝑒𝑑 −𝑒𝑐𝑠 CGS unit of viscosity = 2 𝑁𝑠 SI unit of viscosity = 2 𝑚 CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 3 𝑐𝑚 𝑑𝑦 𝑑𝑢 𝑑𝑦 𝑑𝑢Mechanics of fluids . Regulation 2013 1 𝑁𝑠 One poise = 2 10 𝑚 Kinematic viscosity 𝑖𝑜𝑡𝑦𝑠𝑐𝑣𝑖𝑠 µ 𝜐 = = 𝑖𝑡𝑦𝑠𝑒𝑛𝑑 𝜌 Newton’s law of viscosity It states that the sheer strees on the fluid element layer is directely proportional to the rate of sheer strees . 𝑑 𝑢 𝜏 = 𝜇 Variation of viscosity with temperature 1 (i) For liquids , 𝜇 = 𝜇 ∙ 2 1+ +𝛽 𝑡 𝜇 →viscosity of liquid at t℃ , in poise 𝜇 ∙→viscosity of liquid at 0℃ , in poise 𝛼 , 𝛽 →constants For water ., −3 𝜇 ∙→1.79× 10 poise 𝛼 →0.03368 𝛽 → 0.000221 2 (ii) For gases , 𝜇 = 𝜇 ∙ +𝛼𝑡 − 𝛽 𝑡 CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 4 𝛼𝑡 𝑑𝑦Mechanics of fluids . Regulation 2013 For air ., 𝜇 ∙→0.000017 𝛼 →0.000000056 −9 𝛽 → 0.1189× 10 Types of Fluids 1. Ideal Fluid - incompressible & having no viscosity . 2. Real Fluid – possesses viscosity. 3. Newtonian Fluid – a real fluid in which sheer strees 𝛼 rate of sheer strain . 4. Non_Newtonian Fluid - a real fluid in which sheer strees is not proportional to the rate of sheer strain . CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 5 Mechanics of fluids . Regulation 2013 5. Ideal plastic fluid –sheer strain is more than the yield value & Sheer strees 𝛼 rate of sheer strain . COMPRESSIBILITY Compressibility is the reciprocal of the bulk modulus of elasticity, K which is defined as the ratio of compressive stress to the volumetric strain. 𝑟𝑒𝑎𝑠𝑒𝑐𝑖𝑛 𝑖𝑛 𝑟𝑒𝑠𝑠𝑝 𝑟𝑒𝑢 K = 𝑚𝑒𝑡𝑟𝑖𝑣𝑜𝑙𝑢𝑐 𝑡𝑟𝑎𝑖𝑛𝑠 K= −𝑑 ∀/∀ − K= ∀ 𝑑 ∀ 1 Compressibility = 𝐾 SURFACE TENSION It is defined as the tensile force acting on the surface a liquid in contact with a gas or on the surface between two immiscible liquids such that the contact surface behaves a membrane under tension . i) tensile force due to the surface tension acting around the circumference of cut portion is equal to = 𝜎 × 𝜋𝑑 CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 6 𝑑𝑝 𝑑𝑝Mechanics of fluids . Regulation 2013 2 2 𝜋 𝑑 𝜋 𝑑 ii) Presence of force on area = p × 4 4 2 𝜋 𝑑 p × = 𝜎 × 𝜋𝑑 4 𝜎 × 4𝜎 p = = 2 𝜋 𝑑 𝑑 4 iii) surface tension on a hollow bubble . 2 𝜋 𝑑 p × = 2( 𝜎 × 𝜋𝑑 ) 4 2( 𝜎 × ) 8𝜎 p = = 2 𝜋 𝑑 𝑑 4 iv) surface tension on a liquid jet . Force due to pressure = p × area = p × L × d Force due to surface tension = 𝜎 × 2L CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 7 𝜋𝑑 𝜋𝑑Mechanics of fluids . Regulation 2013 Equating the two forces ., p × L × d = 𝜎 × 2L 2𝜎 p = 𝑑 CAPILLARITY Capillarity is defined as the phenomenon of rise or fall of a liquid surface in a small tube relative to the adjacent general level of liquid when the tube is held vertically in the liquid . The rise of the liquid surface is known as capillary rise while the fall in the liquid surface is called capillary depression. Rise of water is given by., 4𝜎 h = 𝜌𝑔𝑑 CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 8 Mechanics of fluids . Regulation 2013 Height of depression in the liquid., 4𝜃𝑜𝑠𝑐𝜎 h = 𝜌𝑔𝑑 The value of 𝜃 for mercury & glass tube is 128°. VAPOUR PRESSURE When vapourization takes place, the molecules escapes from the free surface of the liquid . these vapour molecules get accumulated in the space between free liquid surface and top of the vessel . these accumulated vapours exert a pressure on the liquid surface . this pressure is known as vapour pressure. CAVITATION The cavitation is the phenomenon of formation of vapour bubbles of a flowing liquid in a region where the pressure of the liquid falls below the vapour pressure and sudden collapsing of these vapour bubbles in a region of higher pressure . FLUID PRESSURE P = 𝐹 P = 𝐴 Force = pressure × area CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 9 𝑑𝐴 𝑑𝐹Mechanics of fluids . Regulation 2013 2 Units: MKS units – kgf/𝑚 2 SI units – N/𝑚 ( pascal ) PASCAL’S LAW It states that the pressure or intensity of pressure at a point in a fluid at rest is equal in all directions. 𝑝 = 𝑝 = 𝑝 𝑥 𝑦 𝑧 MEASUREMENT OF PRESSURE The pressure of a fluid is measured by the following devices., 1.Manometers . 2.Mechanical gauges . Manometers : i) Simple Manometers ii) Differential manometers a) U_tube differential Manometers b) Inverted U_tube differential Manometers SIMPLE MANOMETER i) Piezometer ii) U-tube Manometer iii) Single column Manometer CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 10 Mechanics of fluids . Regulation 2013 U-tube Manometer : a) For gauge pressure : P = ( 𝜌 𝑔 ℎ − 𝜌 𝑔 ℎ ) 2 2 1 1 b) For vacuum pressure : P = - ( 𝜌 𝑔 ℎ + 𝜌 𝑔 ℎ ) 2 2 1 1 Single column Manometer : a) Vertical single column manometer. 𝑎 ×ℎ 2 𝑝 = 𝜌 𝑔 − 𝜌 𝑔 + ℎ 𝜌 𝑔 − ℎ 𝜌 𝑔 𝐴 2 2 2 2 1 1 𝐴 b) Inclined single column manometer. 𝑝 = 𝑖𝑛𝜃𝑠 × 𝜌 𝑔 − ℎ 𝜌 𝑔 𝐴 2 1 1 DIFFERTIAL MANOMETER a) U-tube differential manometer. 𝑝 − 𝑝 = 𝑔 ( 𝜌 − 𝜌 ) 𝐴 𝐵 2 1 b) Inverted U-tube differential manometer. 𝑝 − 𝑝 = 𝜌 𝑔 ℎ − 𝜌 𝑔 ℎ − 𝜌 𝑔 ℎ 𝐴 𝐵 1 1 2 2 𝑠 CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 11 Mechanics of fluids . Regulation 2013 CENTER OF PRESSURE Total pressure (F) : It is defined as the force exerted by a static fluid on a surface either plane or a curved when the fluid comes in contact with the surfaces. This force acts always normal to the surface. Center of pressure (h) : It is defined as the point of application of the total pressure on the surface . There are four cases of submerged surfaces on which the total pressure force and center of pressure is to be determined . The submerged surfaces may be ., 1. Vertical plane surface. 2. Horizontal plane surface. 3. Inclined plane surface. 4. Curved surface. CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 12 Mechanics of fluids . Regulation 2013 BUOYANCY When a body is immersed in a fluid , an upward force is exerted by the fluid on the body. This upward force is equal to the weight of the fluid displaced by the body and is called the force of buoyancy. Center of buoyancy : It is defined as the point, through which the force of buoyancy is supposed to act. As the force buoyancy is a vertical force and is equal to the weight of the fluid displaced by the body, the center of buoyancy will be the center of gravity of fluid displaced. CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 13 Mechanics of fluids . Regulation 2013 CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 14 Mechanics of fluids . Regulation 2013 CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 15 Mechanics of fluids . Regulation 2013 CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 16 Mechanics of fluids . Regulation 2013 UNIT – II Fluid kinematics & dynamics. (A) KINEMATICS OF FLOW INTRODUCTION Kinematics is the branch of science which deals with motion of particles without considering the forces causing the motion. DESCRIBING FLUID MOTION The fluid motion is described by two methods., (i) Lagrangian Method. - a single fluid particle is followed during its motion and its velocity, acceleration, density, etc., are described. (ii) Eulerian Method. - the velocity, acceleration, density, etc., are described at a point in flow field. TYPES OF FLUID FLOW : i) Steady flow. - in which the fluid characteristics like velocity, pressure, density, etc., at a point do not change with time. Unsteady flow. - in which the fluid characteristics like velocity, pressure, density, etc., at a point changes with respect to time. CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 17 Mechanics of fluids . Regulation 2013 ii) Uniform flow. - in which the velocity at any given time does not change with respect to space. Non-uniform flow. - in which the velocity at any given time changes with respect to space. iii) Laminar flow. - in which the fluid particles moves along well-defined paths or stream line and all the stream lines are straight and parallel. Turbulent flow. - in which the fluid particles moves in a zig-zag way. iv) Compressible flow. - in which the density of fluid changes from point to point . Incompressible flow. - in which the density of fluid is constant for fluid flow. v) Rotational flow. - in which the fluid particles while flowing along stream-lines, also rotate about their own axis. Irrotational flow. - in which the fluid particles while flowing along stream-lines, do not rotate about their own axis. CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 18 Mechanics of fluids . Regulation 2013 vi) One dimensional flow. - in which the flow parameter such as velocity is a function of time and one space co_ordinate only, say x. u = f(x), v = 0, w = 0 . Two dimensional flow. - in which the velocity is the function of time and two rectangular space co_ordinates say x and y. u = 𝑓 (x,y), v = 𝑓 (x,y) ,w =0. 1 2 Three dimensional flow: - in which the velocity is the function of time and three mutually perpendicular directions. u = 𝑓 (x,y,z), v =𝑓 (x,y,z) , w = 𝑓 (x,y,z) 1 2 3 RATE OF FLOW OR DISCHARGE It is defined as the quantity of fluid flowing per second through a section of a pipe or a channel. For an incompressible fluid, the rate of flow is expressed as the volume of fluid flowing across the section per second. For compressible fluids, the rate of flow is usually expressed as the weight of fluid across the section. 3 i) For liquids the units of Q are 𝑚 /s or litre/sec. ii) For gases the units of Q is kgf/s or Newton/sec. Q = A× V A – cross sectional area of pipe. V – average velocity of fluid across the section. CIVIL Engg.dept _KPRIET aravint7530gmail.com Page 19

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