Fluid FlowFluid Flow

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

Daniel Bernoulli

Equation of Continuity

A11 = A22

The product of the cross-sectional area of a pipe and the fluid speed is a constant.

Equation of Continuity

A11 = A22

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: m3/s

Equation of Continuity

A11 = A22

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

Equation of Continuity

A11 = A22

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.

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)

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.

2 21 1 1 2 2 2

1 1

2 2 constant

gy gy

P P

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: A11 = A22

Example. Suppose, for the diameters: d1=2d2. Then A1 = 4A2 and

Application: Venturi Tube

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

Fluid Flow QuestionsFluid Flow Questions

Fluid FlowFluid Flow

At any point in an ideal fluid(1) The volume flow rate is constant:

(2) Bernoulli’s equation applies:

1 1 2 2A A

21 constant2

gy P

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,(A11/A22), is

A. 1/1 B. 2/1 C. 4/1 D. 1/2 E. 1/4

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,(A11/A22), 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: d1=3d2.

The fluid velocity ratio 1/2 is

A. 1/1 B. 1/3 C. 1/9 D. 3/1 E. 9/1

Water flows through two connected sections of pipe. The diameters are related by: d1=3d2.

The fluid velocity ratio 1/2 is

A. 1/1 B. 1/3 C. 1/9 D. 3/1 E. 9/1

2 22 1 2 1A A d d

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

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 A11=A22

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 P1 at the lower end is

21 1 o

3

2gy P P

Water flows out of a hole at the bottom of a tank. The depth of the water is h, the tank diameter is d1

and the hole diameter is d2.

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. d1>>d2 B. d1<<d2 C. h is very large D. h is very small

2gh

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. d1>>d2 B. d1<<d2 C. h is very large D. h is very small

2gh