Fluid Mechanics Instructor: Xiao, Yong ( 肖湧 ) ， Wang Kai( 王凯 ) TA: Li, Yueyan （李跃岩） Recitation TA: Zhai, Chenyu （翟宸宇） General Physics

Phase of Matter The three common phases, or states, of matter are solid, liquid, and gas. A solid maintains a fixed shape and a fixed size, even if a large force is applied to a solid, it does not readily change in shape or volume, like rigid body which we have discussed in previous classes. In fact, there is fourth state called plasma, which occurs only at very high temperatures and consists of ionized atoms, such as the sun.

Fluid I Unlike solid matters, last three states do not have fixed shapes, thus they can be regarded as fluids. In this section, we are mainly concerned with liquid and gas.

Fluid II Density The major difference between gas and liquid is density. For a gas, it can be easily compressed. However, density of a liquid maintains constant, which means the liquid is not compressible. Therefore it is also called incompressible liquid:

Density, Flux and continuity equation If density is a function of displacement x and time t, we have:

Density, Flux and continuity equation Define Flux J as, so the equation above becomes: Which is called continuity equation.

Fluid II Pressure and Pascal’s principle A tube is full of liquid water. If we press the fluid at area A in to flow a distance d in, then the fluid at area A out will move up a distance d out. Since the water is incompressible, so The work done by F in should be the same as the work done by F out.

Fluid II---cont Pressure We have We introduce a new quantity pressure P. It is defined as force per unit area:

Fluid II—cont Pascal’s principle If an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount.

Fluid II--cont Atmospheric pressure There is a large mount of air around the Earth. Because air is not an incompressible fluid, its density is not constant. But we can use average density to calculate the atmospheric pressure on our bodies:

Fluid II Example We have a common sense that the water level goes up when we drink juice through a straw. Similarly if the air is removed from a glass tube, the liquid moves up. Given the pressure at the top of the tube becomes zero, the level remains unchanged, since atmospheric pressure can support a column of mercury only about 76cm high. So the atmosphere pressure is:

Fluid III Buoyant Force and Archimedes’ Principle The buoyant force occurs because the pressure in a fluid increases with depth. In other words, the buoyant results from the difference between the pressure on the bottom surface of a submerged object and the pressure on its top surface. The pressure on the top is: The pressure on the bottom is:

Fluid III---cont Net force is This net force, called the buoyant force, acts upward. V is the volume of the object, the product is the mass of the fluid displaced. Archimedes’ principle The buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by that object.

Archimedes’ Principle Example The density of ice is only 90% of water. When ice floats at rest on the water, what is the percentage of the volume is in the water? The buoyant force is The buoyant force has magnitude equal to the weight:

Archimedes’ Principle II Example A man on the boat throws a stone into the lake, please determine that the level of the lake goes up, comes down or is unchanged. Before throw: After throw: so the lake comes down.

Fluid IV Bernoulli’s principle: Where the velocity of a fluid is high, the pressure is low, and where the velocity is low, the pressure is high. Use work-energy principle, the work done on a system is equal to its change in mechanical energy. The work:

Bernoulli’s equation Mechanical energy: For an incompressible fluid: We rearrange to get Bernoulli’s equation:

Bernoulli’s equation Airplanes experience a lift force on their wings, keeping them up in the air. The area for air flow between any two streamlines is reduced as the streamlines get closer together, so from the equation of continuity.The air speed is greater above the wing than below it. From Bernoulli's equation, the pressure above the wing is less than the pressure below the wing. Hence there is a net upward force on the wing called dynamic lift

Homework #5 Density, Flux and continuity equation P363/4 Pressure and Pascal’s principle P363/10 17 Buoyant Force and Archimedes’ Principle P364/28 34 Bernoulli’s equation P365/49 55