help a. Nondimensionalize the continuity and Navier-Stokes equations (for two-dimensional flow, x-momentum only). Review section 7.10 in the text. Start from Eqs. (7.28) and (7.29). Considerthe reference scales for velocity, length, time, and pressure to be: U, L, L/U, pU², respectively.Show all work.b. Comment on the significance of the result in terms of the concept of Similarity. Why is theReynolds number so prevalent in characterizing the behavior of flows?c. Review sections 7.1-7.3, which considers as an example, pressure drop (per unit length) for flowthrough a smooth pipe.d. Consider SAE 10W oil at 77°F flowing through a 1-in. diameter horizontal pipe, at an averagespeed of 3 ft/s, produces a pressure drop of 7 psi over a 500-ft length. Water at 60°F flowsthrough the same pipe under dynamically similar conditions. Using the text results reviewed inc., determine the average speed of the water flow [ft/s] and the corresponding pressure drop [psi]. distorted model flows and the real thing.7.10Similitude Based on Governing Differential EquationsSimilarity lawscan be directlydeveloped fromthe equationsgoverning thephenomenon ofIn the preceding sections of this chapter, dimensional analysis has been used to obtain similaritylaws. This is a simple, straightforward approach to modeling, which is widely used. The use ofdimensional analysis requires only a knowledge of the variables that influence the phenomenon ofinterest. Although the simplicity of this approach is attractive, it must be recognized that omissionof one or more important variables may lead to serious errors in the model design. An alternativeapproach is available if the equations (usually differential equations) governing the phenomenon areknown. In this situation similarity laws can be developed from the governing equations, even thoughit may not be possible to obtain analytic solutions to the equations.To illustrate the procedure, consider the flow of an incompressible Newtonian fluid. Forsimplicity we will restrict our attention to two-dimensional flow, although the results are applicableto the general three-dimensional case. From Chapter 6 we know that the governing equations arethe continuity equationinterest.диdo+= 0ду(7.28)dxand the Navier-Stokes equationsдидидрди+ udt(7.29)--dxdxdoдрPg + uдуdvdv+ u+ v+(7.30)ддуdx² ™ dy²,дхwhere the y axis is vertical, so that the gravitational body force, pg, only appears in the “y equation."To continue the mathematical description of the problem, boundary conditions are required. Forexample, velocities on all boundaries may be specified; that is, u = ug and v = vg at all boundarypoints x = xg and y = yg. In some types of problems it may be necessary to specify the pressureover some part of the boundary. For time-dependent problems, initial conditions would also have tobe provided, which means that the values of all dependent variables would be given at some time(usually taken at t = 0).

Non dimensional the continuity (for 2D) and x- momentum nevier stoke…

iðhal flow, Nondimensionalize the continuity and Navier-Stokes equations (for tw momentum only). Review section 7.10 in the text. Start from Eqs. (7.28) and (7.29). Consider the reference scales for velocity, length, time, and pressure to be: U, L, L/U, pƯ², respectively. Show all work. . Comment on the significance of the result in terms of the concept of Similarity. Why is the Reynolds number so prevalent in characterizing the behavior of flows? c. Review sections 7.1-7.3, which considers as an example, pressure drop (per unit length) for flow through a smooth pipe. 1. Consider SAE 10W oil at 77°F flowing through a l-in. diameter horizontal pipe, at an average speed of 3 ft/s, produces a pressure drop of 7 psi over a 500-ft length. Water at 60°F flows through the same pipe under dynamically similar conditions. Using the text results reviewed in