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Fluid Flow 1700 – 1782 Swiss physicist and mathematician. Wrote Hydrodynamica. Also did work that was the beginning of the kinetic theory of gases. Daniel Bernoulli

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Equation of Continuity A 1 1 = A 2 2 The product of the cross-sectional area of a pipe and the fluid speed is a constant.

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Equation of Continuity A 1 1 = A 2 2 The product of the cross-sectional area of a pipe and the fluid speed is a constant. A is called the volume flow rate. Unit: m 3 /s

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Equation of Continuity A 1 1 = A 2 2 The product of the cross-sectional area of a pipe and the fluid speed is a constant. A is the mass flow rate. Unit: kg/s

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Equation of Continuity A 1 1 = A 2 2 The product of the cross-sectional area of a pipe and the fluid speed is a constant. Speed is high where the pipe is narrow and speed is low where the pipe has a large diameter.

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This is a consequence of conservation of mass and a steady flow. This is equivalent to the fact that the volume of fluid that enters one end of the tube in a given time interval equals the volume of fluid leaving the tube in the same interval. Assumes the fluid is incompressible and there are no leaks. A = constant (Equation of Continuity)

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Bernoulli’s Equation Relates pressure to fluid speed and elevation. Bernoulli’s equation is a consequence of Conservation of Energy applied to an ideal fluid. Assumes the fluid is incompressible and nonviscous, and flows in a nonturbulent, steady-state manner.

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Bernoulli’s Equation States that the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume has the same value at all points along a streamline.

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Application: Measuring Speed A fluid flows through a horizontal constricted pipe. Speed changes as diameter changes. Used to measure the speed of the fluid flow. Swiftly moving fluids exert less pressure than do slowly moving fluids.

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Application: Measuring Speed Swiftly moving fluids exert less pressure than do slowly moving fluids. Continuity: A 1 1 = A 2 2 Example. Suppose, for the diameters: d 1 =2d 2. Then A 1 = 4A 2 and

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Application: Venturi Tube Fluid column height is higher in the constricted area of the tube. This indicates that the pressure is lower.

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Fluid Flow Questions

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Fluid Flow At any point in an ideal fluid (1) The volume flow rate is constant: (2) Bernoulli’s equation applies:

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Two rings sit in a stream of water, oriented perpendicular to the current. Loop 2 has twice the area of loop 1. The ratio of volume flow rates, (A 1 1 /A 2 2 ), is A. 1/1 B. 2/1 C. 4/1 D. 1/2 E. 1/4

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Two rings sit in a stream of water, oriented perpendicular to the current. Loop 2 has twice the area of loop 1. The ratio of volume flow rates, (A 1 1 /A 2 2 ), is A. 1/1 B. 2/1 C. 4/1 D. 1/2 E. 1/4

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Water flows through two connected sections of pipe. The diameters are related by: d 1 =3d 2. The fluid velocity ratio 1 / 2 is A. 1/1 B. 1/3 C. 1/9 D. 3/1 E. 9/1

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Water flows through two connected sections of pipe. The diameters are related by: d 1 =3d 2. The fluid velocity ratio 1 / 2 is A. 1/1 B. 1/3 C. 1/9 D. 3/1 E. 9/1

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Two sections of pipe are joined as shown here. The area ratio is 2/1 (lower to upper section). The upper end is at atmospheric pressure. The velocity ratio upper / lower will be A. 2/1 B. >2/1 C. <2/1

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Two sections of pipe are joined as shown here. The area ratio is 2/1 (lower to upper section). The upper end is at atmospheric pressure. The velocity ratio upper / lower will be A. 2/1 B. >2/1 C. <2/1 …because A 1 1 =A 2 2

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Two sections of pipe are joined as shown here. The area ratio is 2/1 (lower to upper section). The upper end is at atmospheric pressure. The velocity ratio upper / lower is 2/1. Now use Bernoulli’s equation to show that the pressure P 1 at the lower end is

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Water flows out of a hole at the bottom of a tank. The depth of the water is h, the tank diameter is d 1 and the hole diameter is d 2.

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The velocity 2 of the water flowing out of the hole can be expressed by the approximation known as Torricelli’s equation, if (choose one or more) A. d 1 >>d 2 B. d 1 <<d 2 C. h is very large D. h is very small

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The velocity 2 of the water flowing out of the hole can be expressed by the approximation known as Torricelli’s equation, if (choose one or more) A. d 1 >>d 2 B. d 1 <<d 2 C. h is very large D. h is very small

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