Identify the expression that denotes the modified Reynolds analogy.
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- Henri Darcy, a French engineer, proposed that the pressuredrop ∆ p for fl ow at velocity V through a tube of length Lcould be correlated in the form ∆p ___________ = α LV2 ρ If Darcy’s formulation is consistent, what are the dimensionsof the coefficient α ?1. Any equation that comes out of the differential approach is valid at every point in the flow field. On the other hand, the integral approach gives us a point by point value and not an average value. True or False 2. During the analysis of fluid flow, we are more often interested in obtaining the average velocity, and one of the common ways of obtaining the average velocity is to take an average across the flow cross-section. This is called area-average velocity, average along the direction parallel to the principal direction of flow. True or False 3. A crude oil with a density of 875 kilogram per cubic meter is pumped and flowing at 1.5 meter per second through a pipeline 1 meter in diameter. The maximum viscosity of the oil to exhibit a laminar flow behavior is 625 centipoise. True or FalseOrange JO A O X 91|4 2:26 ch1_introductio.. Shear stress (t) is the resistance per unit area of the upper plate t = R/A=T/A Water responds to shear stress by continuously yielding in angular deformation in the direction of the shear. IThe rate of angular deformation in the fluid, d(8)/dt ,is proportional to the shear Istress, as shown in Figure 1.1. do dt dx ,and v = dy dx Angular deformation (Shear strain), 0 = dt do Rate of shear strain = dt dx dv (Velocity gradient) dy dy dt dv Therefore, to dv T = constant dy dv T = - dy The proportionally constant, u, is called the absolute viscosity of the flyid Example A flat plate of 50 cm² is being pulled over a fixed flat surface at a constant velocity of 45 cm/sec (Figure 1.1). An oil film of unknown viscosity separates the plate and the fixed surface by a distance of 0.1 cm. The force (T) required to pull the plate is measured to be 31.7 N, and the viscosity E of the fluid is constant. Determine the viscosity (absolute). 22 Example A flat…
- (b) One form of fluid movement is rotation and deform angularly. Figure Q1(b) shows the rotation and angular deformation caused by velocity variation about z-axis. Based on Table 1 and setting given to you, derive an equation of rotation. ди Sy St ây > B' ĉu B B ôy dy A' ↑ Sa v+. ôx A ôx Figure Q1(b) : Rotation and Angular Deformation Table 1: Axis of Rotation Setting Axis of Rotation 2 у-ахisThe wind flutter on the wing of a newly proposed jet fighter is given by the following 1st order differential equation: dy/dx = 2yx With the Boundary Condition: y(0) = 1 (remember this means that y = 1 when x = 0) Determine the vertical motion (y) in terms of the span (x) of the wing. The frequency of fluctuations of the wing at mach 2 is given by the non-homogenous 2nd order differential equation: y'' + 3y' - 10y = 100x With the boundary conditions: y(0) = 1 and y(1) = 0 (i.e., y = 1 when x = 0 and y = 0 when x = 1) By solving the homogenous form of this equation, complete the analysis and determine the amplitude (y) of vibration of the wing tip at mach 2. Critically evaluate wing flutter and fluctuation frequency amplitude determined by solving the two differential equations above.1. Answer the following questions: (a) What is the physical meaning of the following: D a +V.v at Dt where V is the velocity vector of the flow field. (b) Let the viscous stress tensor be denoted by 7. How is the surface (vector) force f, acting by the fluid on a surface element ds (with unit normal în ) computed? Give your answer in vector notation and also in index notation. What is the physical meaning of Ty ? (c) Write down the work done on a material volume of fluid by the viscous surface force in vector notation and also in index notation. (d) Write down the amount of conduction heat flux 'q' (a scalar) on a surface element ds (with unit normal în ) in vector notation and also in index notation.
- 4. The velocity vectors of three flow fileds are given as V, = axĩ + bx(1+1)}+ tk , V, = axyi + bx(1+t)j , and V3 = axyi – bzy(1+t)k where coefficients a and b have constant values. Is it correct to say that flow field 1 is one-, flow filed 2 is two-, and flow filed 3 is three-dimensional? Are these flow fields steady or unsteady?Q1:: Explain all the terms of the Continuity Equation and their physical meanings with the help of examples.1. Which of the following is a linear equation? (Select all that apply) (а) 12 + у — 19 (c) sin x + 9y = -4 (e) x² – y? = 1 -3x + 4y 2x (d) (b) х 3 —у + 2 Зу - (f) VT = y 10 5 2. Give an example of the following of a linear combination in t, u, v, w.
- For the flow of a viscous fluid, with the velocity V = f(x)g(y)h(z)i (where f, g, h are arbitrary functions), the following conditions are given: . The flow is adiabatic. • The quantities v = 2 and 3 = $ are constants. • The velocity circulation is conserved for the flow, irrespective of the values of vand 3. What is the general solution for the functions f, g, h?Tp = Fq +°P/Q• (1) Here ip/Q is the "position of point P relative to point Q." Similarly the velocities of the two points are related by õp = bq + Up/Q- (2) The quantity õp/Q is the velocity of point P relative to point Q. I want you to use these ideas to solve the following problems. 1. The figure below shows a view from above of a large boat in the middle of the ocean. So that the crew on the ship can get exercise on long journeys, there is a circular walking/running track on the back deck. CA B- -D Suppose that the radius of the track is R = 6 m, and a person is running on the track at a constant speed of v = 3m/s as measured with a stopwatch by a crew-mate on board the ship. Suppose the runner is running counter-clockwise around the track when viewed from above. Write the velocity vector of the runner in terms of basis (ê1, ê2) as perceived by a crew-mate on the ship. (a) What is the velocity vector when the runner is at point A? (b) What is the velocity vector when the runner is…The wind flutter on the wing of a newly proposed jet fighter is given by the following 1st order differential equation: With the Boundary Condition: y(0) = 1 (remember this means that y = 1 when x = 0) Determine the vertical motion (y) in terms of the span (x) of the wing. The frequency of fluctuations of the wing at mach 2 is given by the non-homogenous 2nd order differential equation: With the boundary conditions: y(0) = 1 and y(1) = 0 (i.e., y = 1 when x = 0 and y = 0 when x = 1) By solving the homogenous form of this equation, complete the analysis and determine the amplitude (y) of vibration of the wing tip at mach 2. Critically evaluate wing flutter and fluctuation frequency amplitude determined by solving the two differential equations above.