QUESTIONS ON DIMENSIONAL ANALYSIS BY SANDRA O.O

Answer question 1-3 using Indices method and 4-6 using Buckingham's method. 1) When fluid in a pipe is accelerated linearly from rest, it begins as laminar flow and then undergoes transition to turbulence at a time ttr which depends upon the pipe diameter D, fluid acceleration a, density ρ, and viscosity µ. Arrange this into a dimensionless relation.

2) The drag force 𝐹_𝑑 exerted on submerged sphere as it moves through a viscous fluid is a function of the following variables: the sphere diameter, 𝑑; the velocity of the sphere, 𝑢; the density of the fluid, 𝜌; and viscosity of the fluid, 𝜇. Obtain the dimensionless group for these variables. 3) A drop of liquid spreads over a horizontal surface. Given that the rate at which the liquid spreads, 𝑢_𝑅 (𝑚∕𝑠), will be influenced by viscosity of the fluid, 𝜇; volume of the drop, V; density of the fluid, 𝜌; acceleration due to gravity, g; and surface tension of the liquid, 𝜎 (𝑁𝑚^(−1)) Obtain the dimensionless groups of the variables which will influence the rate at which the liquid spreads,𝑢_𝑅.

4) A steady stream of liquid in turbulent flow is heated by passing it through a long, straight, heated pipe. The temperature of the pipe is assumed to be greater by a constant amount than the average temperature of the liquid. Using dimensional analysis, it is desired to find a relationship that can be used to predict the rate of heat transfer from the wall of the liquid. Thetheoretical equation for this problem may be written in the general form as: 𝑞 𝐴 =𝑓 (𝐷, 𝑉, 𝜌, 𝜇, 𝐶 𝑝 , 𝐾, ∆𝑇) Symbol Quantity Dimensions 𝑞 𝐴 = 𝐻𝑒𝑎𝑡 𝑓𝑙𝑜𝑤 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑎𝑟𝑒𝑎 HL-2T-1 D = Diameter of pipe (inside) - V = Average velocity of liquid - 𝜌= 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 - 𝜇 = Viscosity of liquid - 𝐶 𝑝 = 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 ℎ𝑒𝑎𝑡, 𝑎𝑡 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒, 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 HM-1θ-1 K = Thermal conductivity of liquid HL-1T-1 θ-1 ∆𝑇= Temperature difference between wall and fluid θ

5) An engineering class is organizing a get-together for an upcoming event and decided that a self produced drink would be served. An observer on the drink extraction process concluded that the power, ∅ (J/s) required in mixing the content for proper extraction is dependent on the diameter, d (m) of the stirrer, the rotation, n (s-1), of the stirrer, the density, ρ (kg/m3), and the viscosity, μ (Pa.s) of the zobo mixture. Use dimensional analysis to show an expression representing this observation is: ∅ 𝜌 𝑛 3 𝑑 5 =𝑓 𝜌𝑛 𝑑 2 𝜇 𝛼 𝑛 2 𝑑 𝑔 𝛽

6) It is found experimentally that the terminal settling velocity uo of a spherical particle in a fluid is a function of the following quantities: particle diameter, d; buoyant weight of particle (weight of particle - weight of displaced fluid), W; fluid density, ρ, and fluid viscosity, µ. Obtain a relationship for uo using dimensional analysis. Stokes established, from theoretical considerations, that for small particles which settle at very low velocities, the settling velocity is independent of the density of the fluid except in so far as this affects the buoyancy. Show that the settling velocity must then be inversely proportional to the viscosity of the fluid.