Physics 141Mechanics Lecture 20 Fluid Dynamics Yongli Gao A fluid is a substance that can flow. In contrast to a solid, a fluid has no shape, and it takes the form of its container. Gases and liquids are fluids. A gas is compressible. A liquid is almost incompressible. From a molecular point of view, there's a lot empty space between the molecules in a gas, and almost none in a liquid. The molecules in a fluid can move around everywhere at random, whereas the molecules in a solid can only vibrate slightly around the equilibrium positions.

Density and Pressure The density is the mass of unit volume, or, for a non-uniform object, The unit of density in SI is kg/m 3. The density of solids and liquids are almost constant, but the density of gases depend on the pressure and temperature. Pressure is the normal force on unit area,, or, for a non-uniform situation, The unit of pressure in SI is N/m 2, or pascal (Pa).

Hydrostatic Pressure What should be the pressure a diver has to bare ? Why is it difficult to build deep ocean submarines? The pressure increases as we go deeper into the water, described by the equation for hydrostatic pressure The equation applies to all incompressible fluids. For gasses, we'd have to include the compressibility and the resulting changes in . If you consider the free body diagram of a column of water, you'd see how it works.

Pascal's Principle Pascal's principle: A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid. This is the principle behind all the hydraulic machines since You may have a huge mechanical advantage by enlarge the ratio of the areas. You don't gain in term of work since the volume is constant,

Achimedes Principle Due to the difference of pressures at the top and the bottom, any body submerged in a fluid experiences an upward force termed as buoyancy. Achimedes principle: The force of buoyancy acting on a body is the weight of the fluid it displaces. Consider if the body is made of the same fluid, it'd be in equilibrium => the total force by the pressure from all directions equals to the weight of the fluid displaced. This is the principle to all the ships and balloons.

Acting Point of Buoyancy As we've seen in statics for rigid bodies, the action of point of a force is one of the three elements of a force. The acting point of buoyancy is the CM of the replaced fluid. You can understand it by realizing that the replaced fluid was at equilibrium before being replaced by the submerged object. The weight of the replaced fluid is at its CM, so the total force it experiences by the surrounding fluid must also at the CM. If the submerged object is uniform, it may float or sink, but it'll not tilt since its CM is the same as the displaced fluid. Otherwise, it'll tilt such that the two CM's are along the same vertical line.

Continuity of Flow Let's consider an incompressible fluid flowing in a tube of varying cross section. In the same time interval, the quantity of fluid flowing through any cross section must be the same along the tube, This is the equation of continuity, which links the speed and cross section of the flow of an incompressible fluid inside a tube, without any source or sink in between.

Bernoulli's Equation If an incompressible fluid is not in equilibrium, the pressure and flow velocity may change along the tube. The relation is the conservation of energy: the work done by the pressure is converted to the kinetic energy of the fluid.