Fluid Flow Daniel Bernoulli 1700 1782 Swiss physicist

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

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

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

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

Equation of Continuity A 1 u 1 = A 2 u 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.

Au = constant (Equation of Continuity) 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.

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.

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.

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.

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

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

Fluid Flow Questions

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

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 u 1/A 2 u 2), is A. 1/1 B. 2/1 C. 4/1 D. 1/2 E. 1/4

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

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 uupper/ulower will be A. 2/1 B. >2/1 C. <2/1

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 uupper/ulower will be A. 2/1 B. >2/1 C. <2/1 …because A 1 u 1=A 2 u 2

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 uupper/ulower is 2/1. Now use Bernoulli’s equation to show that the pressure P 1 at the lower end is

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.

The velocity u 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