n physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases..pptx

n physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases.

1. Fluid Dynamics Fluid Dynamics is the branch of fluid mechanics which deals with the study of velocity and acceleration of the particles of fluids in motion and their distribution in space with considering forces causing the motion.

2. Equation of motion According to newton's law of motion, the net force 𝐹 𝑥 acting on a fluid element in the direction of x is equal to mass m of the fluid element multiplied by the acceleration 𝑎 𝑥 in x direction. Thus mathematically, 𝑭 𝒙 = 𝒎. 𝒂 𝒙 ------- (I) In the fluid flow, the following forces are present: 𝐹 𝑔 , Gravity force: Due to weight of the fluid 𝐹 𝑝 , the pressure force: Due to pressure gradient between two points in the flow. 𝐹 𝑣 , force due to viscosity : Due to viscous property of flowing fluid. 𝐹 𝑡 , force due to turbulence : Due to turbulence of the flow. 𝐹 𝑐 , force due to compressibility : Due to compressibility property of fluid. Thus, in equation (I) the net force 𝐹𝑥 = (𝐹𝑔) 𝑥 + (𝐹𝑝) 𝑥 + (𝐹𝑣 )𝑥 + (𝐹𝑡 )𝑥 + (𝐹𝑐 ) 𝑥

3. Euler’s equation of motion 𝐹𝑥 = (𝐹𝑔) 𝑥 + (𝐹𝑝) 𝑥 + (𝐹𝑣 ) 𝑥 + (𝐹𝑡 ) 𝑥 + (𝐹𝑐 ) 𝑥 1)If 𝐹𝑐 is negligible , 𝐹𝑥 = (𝐹𝑔) 𝑥 + (𝐹𝑝) 𝑥 + (𝐹𝑣 )𝑥 + (𝐹𝑡 )𝑥……..Reynolds equations of motion 2) If 𝐹𝑡 is negligible, 𝐹𝑥 = (𝐹𝑔) 𝑥 + (𝐹𝑝) 𝑥 + (𝐹𝑣 )𝑥 …………………Navier-Stokes Equation 3) If 𝐹𝑣 is zero, 𝐹𝑥 = (𝐹𝑔) 𝑥 + (𝐹𝑝) 𝑥 ……………………………… Euler’s equations of motion Euler's equation of motion: This is equation of motion in which the forces due to gravity and pressure are taken into consideration. This is derived by considering the motion of a fluid element along the stream line…..

4. Consider a stream of cross section in which flow is taking place in S direction as shown in figure Assume the cylindrical element of cross-section dA and length dS . The forces acting on the cylindrical element are: Pressure force pdA in the direction of flow. Pressure force(𝒑 + 𝒅𝒑)𝒅𝑨 opposite to the direction of flow. Weight of element ρgdAds.

5. Let θ is the angle between the direction of flow and the line of action of the weight of element. The resultant force on the fluid element in the direction of S must be equal to the mass of fluid element x acceleration in the X direction S. Net Pressure force in the direction of flow is 𝑝𝑑𝐴 − (𝑝 + 𝑑𝑝)𝑑𝐴 = −𝑑𝑝. 𝑑𝐴 ………………..…………………….(I) 2) Component of the weight of the fluid element in the direction of flow is = 𝜌𝑔𝑑𝐴𝑑𝑠𝑐𝑜𝑠𝜃 = 𝜌𝑔𝑑𝐴𝑑𝑠 (𝑐𝑜𝑠𝜃 = ) = 𝜌𝑔𝑑𝐴𝑑z ……………………………………..(II) 3) Mass of the fluid element = 𝜌 𝑑 𝐴 𝑑 ………………………………………(III) 4) The acceleration of the fluid element 𝑎= = × = 𝑉. …………………..(Iv)  

6. Now according to Newton's second law of motion, Force = mass x acceleration ∴ −𝑑𝑝. 𝑑𝐴 − 𝜌𝑔𝑑𝐴𝑑𝑧 = 𝜌𝑑𝐴𝑑s × 𝑉. Dividing both sides by 𝜌. 𝑑𝐴,we get - − 𝑔. 𝑑z = 𝑉. 𝑑𝑉 + 𝑽. 𝒅𝑽 + 𝒈. 𝒅𝐳 = 0…… Euler's equation for motion Bernoulli's Equation: After integrating Euler's equation for motion as energy equation. ∫ 𝑑𝑝 + ∫ 𝑉. 𝑑𝑉 + ∫ 𝑔. 𝑑𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 + + 𝑔𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Dividing by g, + + 𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 In other words,  

7. Assumptions of Bernoulli’s equation: 1) The flow of liquid is steady and continuous. 2) The fluid is ideal (non-viscous) and Incompressible. 3) The flow is along the streamline, i.e. in one dimensional. 4) The velocity is uniform over the section and is equal to the mean velocity. 5) The only forces acting on the fluid are the gravity forces and the pressure forces. Limitations of Bernoulli’s equation: 1) In actual practice, fluid is not ideal fluid therefore, due to slip condition, velocity at the fixed boundary. Thus, velocity is not uniform across the section as assumed. 2) In actual practice, some forces like viscous forces are involved in addition to gravity and pressure forces. 3) Some energy addition or subtraction may take place when fluid passes from one section to the other.

8. The Bernoulli equation is an approximate relation between pressure, velocity, and elevation and is valid in regions of steady, incompressible flow where net frictional forces are negligible. Equation is useful in flow regions outside of boundary layers and wakes. Practical applications of Bernoulli's Theorem: 1) Venturimeter 2) Orifice meter 3) Pitot Tube

9. Venturimeter: A Venturimeter is one of the most important practical applications of Bernoulli’s theorem. It is an instrument used to measure the rate of flow or discharge in a pipe line ant is often fixed permanently at different section of the pipe line to know the discharge there. A Venturimeter has been named after the 18th century Italian Engineer Venturi. Principle of Venturimeter: The basic principle on which a Venturimeter works is that by reducing the cross-sectional area of the flow passage, a pressure difference enables the determination of the discharge through the pipe.

10. A Venturimeter fitted in horizontal pipe through which a fluid is flowing 𝑑 1 = 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑎𝑡 𝑖𝑛𝑙𝑒𝑡 𝑜𝑟 𝑎𝑡 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 1 𝑎 1 = 𝑎𝑟𝑒𝑎 𝑎𝑡 𝑖𝑛𝑙𝑒𝑡 ( 𝑑 1 2 ) 𝑝 1 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑖𝑛𝑙𝑒𝑡 𝑜𝑟 𝑎𝑡 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 1 𝑉 1 = 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑎𝑡 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 1 𝑑 2 , 𝑎 2 , 𝑝 2 , 𝑉 2 are corresponding values at section 2  

11. Orificemeter The Orificemeter is a device which is used for measuring flow rate of liquid through pipes. Its basic principle is that by reducing the cross-sectional area of flow passage, a pressure difference is created and its measuring enables the determination of discharge through the pipes. It consists of a flat plate which has a circular sharp edged hole called orifice, which is concentric with the pipe. The orifice diameter is kept 0.5 times the diameter of pipe, though it may vary from 0.4 to 0.8 times pipe diameter. Q=