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Consider the general form of the Reynolds transport theorem (RTT) given by
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EBK FLUID MECHANICS: FUNDAMENTALS AND A
- Consider the general form of the Reynolds transport theorem (RTT) given by dBsys / dt = d/dt ∫CV ρb dV +∫CS ρbV-› r·n-› dAwhere V-›r is the velocity of the fluid relative to the control surface. Let Bsys be the mass m of a closed system of fluid particles. We know that for a system, dm/dt = 0 since no mass can enter or leave the system by definition. Use the given equation to derive the equation of conservation of mass for a control volume.arrow_forwardIn plane stagnation flow, an incompressible fluid occupying the space y>0 has one velocity component given by Vx-x. The flow is two-dimensional and steady, such that V₂=0 and nothing depends on z or time t. (a) Use the continuity equation to determine Vy(x,y), given that Vy(x,0) =0. (This condition for Vy corresponds to the plane y=0 being an impenetrable boundary.) (b) is arbitrary, so you may set Y=0 at any convenient location.) Determine the stream function for this flow, (x,y). (The absolute value ofarrow_forwardA fixed control volume has three one-dimensional boundary sections, as shown. The flow within the control volume is steady. The flow properties at each section are tabulated below. Find the rate of change of energy of the system which occupies the control volume at this instant. Section Туре P. kg/m V, m/s A, m? e, J/kg 1 Inlet 800 5.0 2.0 300 Inlet 800 8.0 3.0 100 3 Outlet 800 17.0 2.0 150 CVarrow_forward
- In this question, assume the "additional displacement" is in the positive u direction. A mass weighing 16 lbs stretches a spring 8 inches. The mass is in a medium that exerts a viscous resistance of 1 lbs when the mass has a velocity of 2 ft/sec. Suppose the object is displaced an additional 5 inches and released. Find an equation for the object's displacement, u(t), in feet after t seconds. u(t) =arrow_forward9- V(D1)^2=V1(D2)^2 mass 10 points continuity equation O true O False 10-stream line is a line giving 10 points direction of velocity at any point. O True O Falsearrow_forwardb) A Newtonian fluid flows in an annular space created by a concentric pipe of radius R, and a rod of radius R;, as shown in Figure Q1(b). The rod is moving at a constant velocity V, while the pipe is stationary. The flow is steady, laminar and incompressible and there is no forced pressure gradient driving the flow. Assuming the velocity components in the radial and tangential directions are zero and ignoring the effects of gravity, derive an expression for the velocity field in the annular space. R. R: Figure Q1(b)arrow_forward
- The flow between two horizontal infinite parallel plates is a two-dimension, steady-state, incompressible and fully- developed flow. The distance between the plates is h m. The bottom plate is stationary and the top plate velocity is U. m/s in the x-direction. The flow is driven by the top moving plate and there is, therefore, no pressure gradient in the direction of the flow. Velocity in the y-direction, v = 0. Note: Align the x-axis to the bottom wall. Use the x-momentum equation to show that the velocity profile equation is (a) u(y) = ay + b and find the values of a and b. Use the energy equation to derive the temperature distribution T(y) for the flow if the surface temperature and temperature gradient on the bottom plate are both zero. (b)arrow_forwardThe well-known Bernoulli equation is used to calculate the Bernoulli constant C in a flow, V² P P + +gz 2 where P is the pressure, p is the fluid density, Vis the magnitude of the velocity, g is the gravitational constant, and z is the elevation. In a water flow experiment (p = 1000. kg/m³), the following measurements are made: V = 15.0 +/- 0.15 m/s, and == 3.42 +/- 0.01 m P= 20.5 +/- 0.5 kPa, The gravitational constant is 9.81 m/s². Calculate the RSS uncertainty for the Bernoulli constant C, and write C in standard engineering notation (C = m²/s²). +/-arrow_forward3. A circular cylinder of radius a is fitted with two pressure sensors to measure pressure at 0 = 180° and at 150°. The intent is to use this cylinder as a stream velocimeter, i.e. a device to determine the velocity of a stream by measuring the pressures at the two taps. The fluid is incompressible with a density of p. Figure for Part (a) U Figure for Part (b) 30 a) Using potential flow approximation, derive a formula for calculating U from the measured pressure difference at the two pressure taps. Note that for accurate measurement, the velocimeter must be aligned to have one of the taps exactly facing the stream as shown in the figure. (Ans: 2|Aptaps|/p ) b) Suppose the velocimeter has been misaligned by ổ degrees so that the two pressure taps are now at 180° + 8 and 150° + 8. Derive an expression for the percent error in stream velocity measurement. Then, calculate the error for 8 = 5°,10° and –10°. (Ans: [2/(sin2(150 + 8) – sin²(180 + 8) )– 1] × 100 )arrow_forward
- The velocity of a fluid for (x, y, z) is given by v = (ax + by, cx + dy, 0) a. Find the conditions on the constants a, b, c and d such that there exists: ii. Irrotational flow i. Incompressible flow b. Verify that, in this case v = V(ax² + 2bxy — ay²).arrow_forwardQ2/ Suppose you have crude oil flows through an annulus between two horizontal pipes with the same center, if the velocity distribution v, and the average velocity Vavg are expressed by: ΔΡ Rr²+ R² In (R₂/R₁) (¹) In 4μL ΔΡ Vavg = R₁ + R₂ R - R In (R₂/R₁ 8µL Where AP: is the pressure drop through the annulus, μ: is the fluid viscosity, L: is the pipe length, R₁ and R₂: are the inside radius of inner and outer pipes, respectively. Write a program in a script file that calculates the velocity distribution and the average velocity. When the script file is executed, it requests the user to input AP, μ, L, R₁ and R₂ where r has many values between R₁ and R₂. The program displays the inputted values and the calculated average velocity (using fprintf) followed by a table with the values r in the first column and the corresponding values of the velocity distribution in the second column.arrow_forward1.6 An incompressible Newtonian fluid flows in the z-direction in space between two par- allel plates that are separated by a distance 2B as shown in Figure 1.3(a). The length and the width of each plate are L and W, respectively. The velocity distribution under steady conditions is given by JAP|B² Vz = 2µL B a) For the coordinate system shown in Figure 1.3(b), show that the velocity distribution takes the form JAP|B? v, = 2μL Problems 11 - 2B --– €. (a) 2B (b) Figure 1.3. Flow between parallel plates. b) Calculate the volumetric flow rate by using the velocity distributions given above. What is your conclusion? 2|A P|B³W Answer: b) For both cases Q = 3µLarrow_forward
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