Fluids

A fluid is a substance that cannot maintain its own shape but takes the shape of its container. Fluid laws assume idealized fluids that cannot be compressed.

Density and pressure

The density (ρ) of a substance of uniform composition is its mass per unit volume: ρ = m/ V. In the SI system, density is measured in units of kilograms per cubic meter.

Imagine an upright cylindrical beaker filled with a fluid. The fluid exerts a force on the bottom of the container due to its weight. Pressure is defined as the force per unit area: P = F/ A, or in terms of magnitude, P = mg/A, where mg is the weight of the fluid. The SI unit of pressure is N/m 2, called a pascal. The pressure at the bottom of a fluid can be expressed in terms of the density (ρ) and height (h) of the fluid:

or P = ρ hg. The pressure at any point in a fluid acts equally in all directions. This concept is sometimes called the basic law of fluid pressure.

Pascal's principle

Pascal's principle may be stated thus: The pressure applied at one point in an enclosed fluid under equilibrium conditions is transmitted equally to all parts of the fluid. This rule is utilized in hydraulic systems. In Figure 1, a push on a cylindrical piston at point a lifts an object at point b.



Figure 1

Pascal's principle is used to easily lift a car.

Let the subscripts a and b denote the quantities at each piston. The pressures are equal; therefore, P a = P b . Substitute the expression for pressure in terms of force and area to obtain f a / A a = ( F b / A b ). Substitute π r 2 for the area of a circle, simplify, and solve for F b : F b =( F a )( r b 2/ r a 2). Because the force exerted at point a is multiplied by the square of the ratio of the radii and r b > r a , a modest force on the small piston a can lift a relatively larger weight on piston b.

Archimedes' principle

Water commonly provides partial support for any object placed in it. The upward force on an object placed in a fluid is called the buoyant force. According to Archimedes' principle, the magnitude of a buoyant force on a completely or partially submerged object always equals the weight of the fluid displaced by the object.

Archimedes' principle can be verified by a nonmathematical argument. Consider the cubic volume of water in the container of water shown in Figure 2. This volume is in equilibrium with the forces acting on it, which are the weight and the buoyant force; therefore, the downward force of the weight ( W) must be balanced by the upward buoyant force ( B), which is provided by the rest of the water in the container.



Figure 2

Weight is balanced by buoyant force within a volume of water.



If a solid floats partially submerged in a liquid, the volume of liquid displaced is less than the volume of the solid. Comparing the density of the solid and the liquid in which it floats leads to an interesting result. The formulas for density are D s = m s / V s and D l = m l / V l , where D is the density, V is the volume, m is the mass, and the subscripts s and l refer to quantities associated with the solid and the liquid respectively. Solving for the masses leads to m s = D s V s and m l = D l V l . According to Archimedes' principle, the weights of the solid and the displaced liquid are equal. Because the weights are simply mass times a constant (g), the masses must be equal also; therefore, D s V s = D l V l or D s / D l = V l V l . Now, V = Ah, where A is the cross‐sectional area and h is the height. For a solid floating in liquid, A l = A s and h l is the height of the solid that is submerged, h sub. With these substitutions, the above relationship becomes D s / D l = h sub/ h s; therefore, the fractional part of the solid that is submerged is equal to the ratio of the density of the solid to the density of the surrounding liquid in which it floats. For example, about 90 percent of an iceberg is beneath the surface of sea water because the density of ice is about nine‐tenths that of sea water.

Bernoulli's equation

Imagine a fluid flowing through a section of pipe with one end having a smaller cross‐sectional area than the pipe at the other end. The flow of liquids is very complex; therefore, this discussion will assume conditions of the smooth flow of an incompressible fluid through walls with no drag. The velocity of the fluid in the constricted end must be greater than the velocity at the larger end if steady flow is maintained; that is, the volume passing per time is the same at all points. Swiftly moving fluids exert less pressure than slowly moving fluids. Bernoulli's equation applies conservation of energy to formalize this observation: P + (1/2) ρ v 2 + ρ gh = a constant. The equation states that the sum of the pressure (P), the kinetic energy per unit volume, and the potential energy per unit volume have the same value throughout the pipe.