In the scenario depicted below, water overflows a dam. It is assumed that the flow rate per unit length q along the dam varies with the height H, width b, gravitational acceleration g, fluid density p, and fluid viscosity μ. Using b, g, and p as repeating variables, determine the appropriate set of dimensionless parameters for this problem. H
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- 1. The thrust of a marine propeller Fr depends on water density p, propeller diameter D, speed of advance through the water V, acceleration due to gravity g, the angular speed of the propeller w, the water pressure p, and the water viscosity μ. You want to find a set of dimensionless variables on which the thrust coefficient depends. In other words CT = FT · = ƒen(#1, #2, ...) pV2D2 (a) What is k? Explain. (b) Find the 's on the right-hand-side of equation 1 if one of them HAS to be a Froude number gD/V², (1)Please don't providehandwritten solution ......The differential equation for small-amplitude vibrations y(r, f) of a simple beam is given by a*y + E = 0 ax pA where p = beam material density A = cross-sectional area I= area moment of inertia E = Young's modulus Use only the quantities p, E, and A to nondimensionalize y, x, and t, and rewrite the differential equation in dimensionless form. Do any parameters remain? Could they be removed by further manipulation of the variables?
- Please solve this problem, Thank you very much! Figure is attached 1. liquids in rotating cylinders rotates as a rigid body and considered at rest. The elevation difference h between the center of the liquid surface and the rim of the liquid surface is a function of angular velocity ?, fluid density ?, gravitational acceleration ?, and radius ?. Use the method of repeating variables to find a dimensionless relationship between the parameters. Show all the steps.3- Consider laminar flow over a flat plate. The boundary layer thickness & grows with distance x down the plate and is also a function of free-stream velocity U, fluid viscosity u, and fluid density p. Find the dimensionless parameters for this problem, being sure to rearrange if necessary to agree with the standard dimensionless groups in fluid mechanics.Buckingham Pi. A mechanical stirrer is used to mix chemicals in a large tank. The required shaft power P is a function of liquid density p, viscosity μ, stirrer blade diameter D, and angular speed w of the spinning blades. (a) use repeating variables p, D, u to find a relation between dimensionless power (1) and w (m2); (b,c) a small 1/3 scale model is used in water to predict the actual required power in a viscous liquid with SG =2 and μ = 12μwater. Find (b) the ratio of speeds, wwater/ wactual, necessary for dynamic similarity and then (c) the predicted ratio of powers Pwater/ Pactual. expecting unit : (a) π1 ~ D ; π^2 ~ D^2; (b) wwater/ wactual: 10^0; (c) Pwater/ Pactual 10^-3 SI constant Patm = 10^5 Pa; pwater - 1000 kg/m^3; pair ~ 1.2kg/m^3; µwater ~ 10^-3 N•s/m^2; pair - 2 x 10^-5 N•s/m^2 ; g = 9.8 m/s^2 =
- The velocity V of propagation of ripples on the surface of a shallow liquid depends on the gravitational acceleration g and the liquid depth h. If Buckingham's Theorem is used to identify the salient dimensionless group(s), how many dimensionless group(s) will be obtained? Number of dimensionless group(s) = 1. {1} (Enter your answer as a number.)During World War II, Sir Geoffrey Taylor, a British fluid dynamicist, used dimensional analysis to estimate theenergy released by an atomic bomb explosion. He assumed that the energy released E, was a function of blastwave radius R, air density ρ, and time t. Arrange these variables into single dimensionless group, which we mayterm the blast wave number.When a fluid flows slowly past a vertical plate of height h and width b, pressure develops on the face of the plate. Assume that the pressure, p, at the midpoint of the plate is a function of plate height and width, the approach velocity V, and the fluid viscosity u and fluid density p. Make use of dimensional analysis to determine how the pressure, p will be formed with a dimensionless group. (Take b, V. p as repeating variables). Select one: O a. n1 = p /V² p O b. n1 = p/Ve O c.n1 = p/Vp? O d. 11 = p/V² p²
- Q1: If an air stream flowing at velocity (U) pasta body of length (L) causes a drag force (F) on the body which depends only upon U, L, and fluid viscosity μ. Formulate the suitable dimensionless parameter of the air drag force.Use image below Solve carefully QUESTION DOWN BELOWThe time t d to drain a liquid from a hole in the bottom of atank is a function of the hole diameter d , the initial fluidvolume y 0 , the initial liquid depth h 0 , and the density ρ andviscosity μ of the fluid. Rewrite this relation as a dimensionlessfunction, using Ipsen’s method.