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Note for Fluid Mechanics - FM By Himanshu Chauhan

  • Fluid Mechanics - FM
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Himanshu Chauhan
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DIMENSIONAL ANALYSIS 1.The pressure drop ẟp in a pipe of diameter D and length l depends on mass density ρ and viscosity Β΅ of the flowing fluid, mean velocity of flow v and average height k of roughness projections on the pipe surface. Obtain a dimensionless expression for 𝑓𝑙𝑉 2 βˆ†π‘ βˆ†p. Hence show that β„Žπ‘“ = where β„Žπ‘“ is the head loss due to friction (= ), w is 2𝑔𝐷 𝑀 specific weight of the fluid and f is coefficient of friction. 2. Show by method of dimensional analysis that the resistance R to the motion of a sphere of diameter D moving with uniform velocity v through a fluid having density Β΅ ρ and viscosity Β΅ may be expressed as R=(𝐷2 𝑉 2 ρ) ΓΈ( ) also show that the above ρ𝑉𝐷 expression reduces to R=kΒ΅VD when the motion is through viscous fluid at low velocity where k is a dimensionless constant.

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Fluid Mechanics 3. Find the viscosity in poise of a liquid through with a steel ball of diameter 1mm falls, with a uniform velocity of 20 mm/s .The specific gravity of the liquid is 0.91 and that of steel is 7.8. Given that k=3ΒΆ 𝑁 𝑃 4. The equation for specific speed for a turbine is given by 𝑁𝑠 = 5ΰ΅— by ΒΆ-theorem 𝐻 4 and using variables such as power p, speed N, head H, diameter D of the turbine, density ρ of the fluid and acceleration due to gravity g, deduce the above expression for 𝑁𝑠 .

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Fluid Mechanics 5. Water at 15Β°C flows at 4 m/s in a 150 mm pipe. At what velocity must oil at 30Β°C flow in a 75 mm pipe for the two flows to be dynamically similar? Take kinematic viscosity for water at 15Β°C as 1.145x10βˆ’6 π‘š2 /s and that for oil at 30Β°C as 3.0x10βˆ’6 π‘š2 /s. Answer: 20.96 π‘š2 /s 6.A model with length scale ratio, model to prototype, equal to x, of a Mach 2 supersonic aircraft is tested in a wind tunnel, wherein air is maintained at atmospheric temperature and a pressure of y times atmospheric pressure. Determine the speed of the model in the tunnel. Given that the velocity of sound in atmospheric air = z. Answer π‘‰π‘š = 2𝑧 π‘₯𝑦

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Fluid Mechanics 7. A solid sphere of diameter 100 mm moves in water at 5m/s . It experiences a drag of magnitude 19.62 N .What would be the velocity of 5m diameter sphere moving in air in order to ensure similarity? What will be the drag experienced by it ? State which law governs the similarity. Take ρ𝑀 =1000kg/π‘š3 ;Οπ‘Žπ‘–π‘Ÿ =1.2kg/π‘š3 ;kinematic viscosity of air = 13kinematic viscosity of water. Answer: 𝐹𝑃 = 872 N 8. A 1:10 scale model of a submarine moving far below the surface of water is tested in a water tunnel. If the speed of the prototype is 8 m/s, determine the corresponding velocity of water in tunnel. Also determine the ratio of the drag for the model and prototype.ρ𝑀 =1000kg/π‘š3 ; Οπ‘ π‘’π‘Ž π‘€π‘Žπ‘‘π‘’π‘Ÿ =1027kg/π‘š3 ¡𝑀 = 0.001 ; Β΅π‘ π‘’π‘Ž π‘€π‘Žπ‘‘π‘’π‘Ÿ =1.151267x10βˆ’3 Answer: 71.365 m/s,0.775

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