DIMENSIONAL ANALYSIS 1.The pressure drop ẟp in a pipe of diameter D and length l depends on mass density ρ and viscosity µ 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 ∆𝑝 ∆p. Hence show that ℎ𝑓 = where ℎ𝑓 is the head loss due to friction (= ), w is 2𝑔𝐷 𝑤 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 µ ρ and viscosity µ may be expressed as R=(𝐷2 𝑉 2 ρ) ø( ) also show that the above ρ𝑉𝐷 expression reduces to R=kµVD when the motion is through viscous fluid at low velocity where k is a dimensionless constant.
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=3¶ 𝑁 𝑃 4. The equation for specific speed for a turbine is given by 𝑁𝑠 = 5ൗ by ¶-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 .
Fluid Mechanics 5. Water at 15°C flows at 4 m/s in a 150 mm pipe. At what velocity must oil at 30°C flow in a 75 mm pipe for the two flows to be dynamically similar? Take kinematic viscosity for water at 15°C as 1.145x10−6 𝑚2 /s and that for oil at 30°C as 3.0x10−6 𝑚2 /s. Answer: 20.96 𝑚2 /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 𝑉𝑚 = 2𝑧 𝑥𝑦
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 ρ𝑤 =1000kg/𝑚3 ;ρ𝑎𝑖𝑟 =1.2kg/𝑚3 ;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.ρ𝑤 =1000kg/𝑚3 ; ρ𝑠𝑒𝑎 𝑤𝑎𝑡𝑒𝑟 =1027kg/𝑚3 µ𝑤 = 0.001 ; µ𝑠𝑒𝑎 𝑤𝑎𝑡𝑒𝑟 =1.151267x10−3 Answer: 71.365 m/s,0.775