Hydrostatic Equilibrium & Buoyancy

• DECEMBER 2019

Fluid Statics – Lesson 2 Hydrostatic Equilibrium

• Consider a small, differential cubic region submerged in a fluid in a gravity field acting in the negative y 푦 direction as shown. The forces acting on the boundaries are the pressure forces (퐹푝), while the volume 푔Ԧ itself has weight (퐹푤) which is a body force acting in the direction of gravity (푔Ԧ). • The weight of the body force is given by:

Ԧ 퐹푤 = −휌푔 푑푥 푑푦 푑푧 푗Ƹ 푑푦 • The forces acting on the cube walls are fluid pressure forces. Assuming a smooth pressure variation in the differential limit we can write the following equations for the equilibrium force balance in each coordinate direction: 푑푧 푥 휕푝 푑푥 x−direction: 푝 푑푦 푑푧 − 푝 + 푑푥 푑푦 푑푧 = 0 휕푥 푧

휕푝 y−direction: 푝 푑푥 푑푧 − 푝 + 푑푦 푑푥 푑푧 − 휌푔 푑푥 푑푦 푑푧 = 0 휕푦

휕푝 z−direction: 푝 푑푥 푑푦 − 푝 + 푑푧 푑푥 푑푦 = 0 휕푧 휕푝 휕푝 휕푝 = = 0 = −휌푔 • The resulting equations are the basic hydrodynamic equilibrium equations of fluid statics: 휕푥 휕푧 휕푦

2 Integration of the Hydrostatic Equation

• We can apply the hydrostatic equilibrium equations to determine the pressure distribution acting on submerged surfaces due to gravity. For example, if we consider a liquid of constant 푔Ԧ 푦 density 휌 acting on a surface, integrating in the 푦 direction from 푦 = ℎ1 to 푦 = ℎ2 gives:

ℎ2 ℎ2 휕푝 න 푑푦 = − න 휌푔푑푦 휕푦 푝(푦) ℎ1 ℎ1

푝ℎ2 = 푝ℎ1 − 휌푔(ℎ2 − ℎ1)

• Note that this is a linear pressure variation in 푦, and that the pressure in the 푥 and 푧 directions is zero (since the gradient is zero from the previous slide). • The foregoing can be written more generally as: 푦0

푝 푦 = 푝0 − 휌푔 푦 − 푦0 Datum (푝 = 푝0)

• The subscript 0 denotes any (arbitrary) datum or reference point in the fluid.

3 Atmospheric Pressure (Isothermal Air Column)

• What if the density is not constant? An example of this would be the pressure variation in the atmosphere for a column of air at uniform temperature. • Air at standard conditions behaves as an ideal gas, and therefore the density can be calculated as a function of temperature using the ideal gas law: 푝 휌 = 푅푇

where 푅 = gas constant for air (287 J/(kg∙K)) • Inserting this into the hydrostatic equilibrium relation gives the ordinary differential equation: 푑푝 푔 = 푑푦 푝 푅푇

• Integrating both sides (assuming constant temperature) and evaluating the constant as 푝(푦 = 0) = 푝0 yields

𝑔푦 − 푝 푦 = 푝0푒 푅푇

• Thus, pressure falls off exponentially with height in an isothermal atmosphere. In reality, the temperature also generally decreases with height, making the pressure variation more complex.

4 Air Pressure on Mt. Everest (Altitude = 29,035 ft.)

The atmospheric pressure decreases with altitude from sea level to 29,000 ft.

The exponential expression given previously will not apply exactly due to the decrease in air temperature but would be a reasonable approximation at lower altitudes.

5 Hydrostatic Forces on General Submerged Surfaces

• Knowing the pressure distribution in the fluid allows us to determine the net pressure force and moments acting on any submerged surface. • Let 푛ො denote the surface normal pointing into the fluid and 푝 is the local surface pressure. The net pressure force vector can be computed on surface 퐴 by integrating over the surface: 퐹Ԧ푝

Ԧ 퐹푝 = − ඵ 푝푛ො푑퐴 푛ො 퐴 푝(푥, 푦, 푧) 푑퐴

• We need to know mathematically the shape of the surface in order to evaluate the integral, which can be done for simple shapes. • The moment vector can likewise be calculated about a specified point knowing the pressure distribution.

푀푝 = −푟Ԧ × ඵ 푝푛ො푑퐴 퐴

6 Archimedes’ Principle

• Archimedes’ Principle states that the pressure force acting on a body immersed in a static fluid equals the weight of the fluid it displaces.

• This can be derived mathematically by integrating the pressure force vector over an entire surface of a 퐹Ԧ푝 푔Ԧ body (surface area 퐴 and volume 훺): 푛ො 푑퐴 퐹Ԧ푝 = − ඾ 푝푛ො푑퐴 퐴 Ω • Let 푓 be a scalar function of (푥, 푦, 푧) and 퐴 is a closed surface surrounding the body. Using the gradient theorem from vector calculus we have:

඾ 푓푛ො푑퐴 = ම ∇푓푑Ω 푝 푦 퐴 Ω • Recalling that the pressure gradient in fluid statics is:

휕푝 휕푝 휕푝 = = 0 = −휌푔 휕푥 휕푧 휕푦

• Then we have 퐹Ԧ = − ඾ 푝푛ො푑퐴 = − ම ∇푝푑Ω = ම 휌푔푗Ƹ푑Ω 푝 퐹Ԧ푝 = +휌푔Ω푗Ƹ 퐴 Ω Ω

7 Archimedes’ Principle (History)

• Archimedes’ Principle was originally developed by one of the leading scientists of antiquity, Archimedes of Syracuse (287 BC – 212 BC), and was published in an ancient text entitled “On Floating Bodies.” • Archimedes is credited with numerous inventions and discoveries, including the Archimedes screw pump and the development of levers and pulleys for doing useful work. • As the tale goes, Archimedes discovered his principle after being requested by King Hieron II of Syracuse to determine if his crown was pure gold. Archimedes hit upon the solution after contemplating why a certain volume of water had spilled out of a tub in which he was bathing. He deduced that the volume of water that spilled out must be related to his weight and volume. Thus, if the crown were made of pure gold, then if it were dropped into a vessel filled to the top with water, the volume of water displaced (spilled) would equal the volume displaced by a bar of gold of equal weight. If it were less dense, the crown would displace more water. Archimedes was so thrilled with this discovery that he hopped out of his bath and ran through the streets naked shouting “Eureka! Eureka!” (“I have found it! I have found it!”) As for the king's crown, it was found to contain silver as a filler instead of being pure gold.

8 Buoyancy

• Buoyancy is the net difference between the upward and downward forces acting on a body immersed in a fluid. • The upward forces are given by Archimedes' Principle, whereas the downward force is simply the weight of the body. 푔Ԧ • A body is said to have positive buoyancy if weight of the body is less than the upward force. The body will thus rise upward until it reaches the surface of the fluid (i. e., the object floats). • In contrast, negative buoyancy means that the weight of the Positive Buoyancy object is larger than the upward force, and thus the object sinks in the fluid. 퐹푝 • If the forces are equal, the object is said to be neutrally buoyant. Neutral • In a constant force field, e. g., gravity, positive, neutral or Buoyancy negative buoyancy is defined by the ratio of the body’s density to the density of the fluid: a less dense body will float, a more 푊 dense body will sink and an equally dense body will float Negative Buoyancy neutrally. • The concept of specific gravity compares different substances with respect to their buoyancy properties.

9 Specific Gravity

• One way of comparing the densities of different substances is to employ the specific gravity (SG), which is defined as the fluid density divided by reference density. For liquids, the reference density is normally the density of water at 4 C (1000 푘푔/푚3).

휌 푆퐺 = 휌퐻2푂

• We can determine whether or not an object will float in a fluid by comparing their specific gravities. • For example, an object will float in water if its specific gravity is less than one, or it will sink if its specific gravity is greater than one.

10 Hot Air Balloons

• Buoyancy is the main principle behind hot air balloon flight. • A hot air balloon rises if the balloon is filled with a hot gas whose density is less than that of the surrounding air. • Balloon pilots control the hot gas density by using a propane heater to keep the density low. To reduce lift or cause the balloon to descend, vents are used to let hot gas escape. • For level flight, the balloon pilot controls the gas temperature to obtain a neutrally buoyant condition for the balloon and basket (the weight of which can vary).

11 Summary

• The equations for hydrostatic equilibrium were derived in this lesson. • Knowing the pressure variation in space permits pressure forces and moments to be calculated on any surface submerged in a given fluid. • Knowing these forces and moments enables an engineer to understand the loads acting on these surfaces and therefore to design a structure that is strong enough to withstand these forces. • Prediction of an object’s buoyancy is important in the design of marine surface and submergible vessels, lighter-than-air high-altitude atmospheric probes, hot air balloons and airships.