Fluids

Properties of Fluid

To study the behavior and forces produced by fluids, we look at properties such as density, temperature, pressure, viscosity, specific volume, and specific gravity.

• Density, \( \rho\), is the amount of mass present in a substance per unit volume. Its corresponding formula is \( \rho=\frac{m}{V} \), where \( m\) is mass and \( V \) is volume.

• Pressure, \( P \), is the force exerted on a fluid per unit area.

  • Pressure, due to gravity, exerted on a fluid at any point is called Hydrostatic pressure. The formula corresponding to hydrostatic pressure is \( P= {\rho}gh \), where \( P\) is fluid pressure, \( \rho\) is density, \( g\) is the acceleration due to gravity, and \( h\) is the depth of the fluid.

  • Gauge Pressure is the difference between total pressure and atmospheric pressure. Its corresponding formula is \( P_G=P_T - P_A \), where \( P_G\) represents gauge pressure, \( P_T \) represents total pressure, and \( P_A\) represents atmospheric pressure.

  • Absolute pressure is the sum of gauge pressure and atmospheric pressure.

• Temperature is a thermodynamic property indicating how hot or cold a fluid is. Temperature refers to the average kinetic energy within a substance.

• Viscosity is a fluid's resistance to motion. If a fluid has a negligible viscosity or no viscosity, it is said to be an inviscid fluid.

• Specific Volume is the ratio of a substance's volume to mass. It is the reciprocal of density. Its corresponding formula is \( v=\frac{V}{m}=\frac{1}{\rho} \).

• Specific Weight is the weight per unit volume. Its corresponding formula is \( \gamma =\rho{g} \). Note that this property varies with temperature.

Fluid Categories

Fluids are broken down into multiple categories depending on different properties, such as flow or viscosity. Flow refers to the action of moving.

Key Concepts of Fluids

Besides being aware of fluid properties and categories, one should also be aware of two key concepts associated with fluids.

1. Buoyancy and Archimedes' principle.

2. Conservation of energy and the Bernoulli equation.

Buoyancy and Archimedes Principle

Buoyancy is the tendency for objects to float within a fluid.

On a microscopic level, fluids consist of atoms that are bound together. Due to a fluid's ability to move, when an object is placed inside a fluid, the atoms are pushed aside and bend around the object. Since the fluid wants to be back in its original state, the atoms exert interatomic electric forces to push against the object, which causes an upward buoyant force. To fully understand buoyancy, we must discuss Archimedes' principle, which is the physical law of buoyancy.

To better understand this principle, let's look at the following example of a cube submerged in water, where we have simplified the forces due to the pressure of the water into a single downward force and a single upward force.

Force is equal to pressure, \( P \) times the area, \( A \), where the pressure is being exerted.

$$F=PA.$$

Now we know that pressure is equal to the density of the fluid times gravity times the height of the fluid.

$$P=\rho_\mathrm{f}gh.$$

Therefore, we can express the force acting on the top and bottom of the cube as follows:

$$F_1=\rho_\mathrm{f}gh_1A,$$

$$F_2=\rho_\mathrm{f}gh_2A.$$

To find the buoyant force, we must find the difference between the force acting on the top and the force acting on the bottom.

$$F_2-F_1=\rho_\mathrm{f}g(h_2-h_1)A.$$

where \( h_2-h_1 \) is the height of the cube. When we multiply this by the area of the face of the cube, \( A \), we get the volume of the cube, \(V_\mathrm{f}\), or in this case, the amount of water displaced by the cube. Consequently, we can derive the following equation for the buoyant force,

$$F_\mathrm{b}=\rho_\mathrm{f}V_\mathrm{f}g,$$

and as we know that mass is equal to density times volume, we can substitute mass into the equation. This yields the equation

$$F_\mathrm{b}=m_\mathrm{f}g,$$ and since weight is equal to mass times gravity, the above result indicates that the buoyant force is equal to the weight of the displaced fluid, as stated by Archimedes' principle.

Conservation of Energy and the Bernoulli Equation

An important equation associated with fluids is Bernoulli's equation which describes the conservation of energy in fluid flow. This equation emphasizes the relationship between velocity and pressure. It states that when discussing moving fluids, a point where velocity is low means that the pressure is high, and a point where velocity is high, pressure is low.

$$\text{Bernoulli's Equation} \rightarrow P_1 + \frac{1}{2}\rho{v_1}^2 + {\rho}gh_1=P_2 + \frac{1}{2}\rho{v_2}^2 + {\rho}gh_2.$$

To help our understanding, let's look at the following example and derive the Bernoulli equation.

To derive this equation, three assumptions are made.

1. A streamlined flow indicates that all particles in the fluid follow the same path.

2. Constant density which indicates an incompressible fluid.

3. No viscosity while moving.

In order to proceed, we must calculate the work required to move a volume of fluid from one position to another. Note that the fluid at point one travels a distance of \( \Delta{l}_1 \) while the fluid at point two travels a distance of \( \Delta{l}_2 \).

Step 1: Calculate the work done on the fluid at point one at its cross-sectional area, \( A_1 \), by the fluid to the left of it. The fluid to the left of point one forces the fluid to move toward point two.

$$W_1= F_1\Delta{l}_1= P_1A_1\Delta{l}_1.$$

Step 2: Calculate the work done at point two on its cross-sectional area, \( A_2\). Work here will be negative because the force acting on the fluid is opposite to the motion of the fluid.

$$W_2= -F_2\Delta{l}_2= -P_2A_2\Delta{l}_2.$$

Step 3: Calculate the work done by the fluid against gravity.

$$W_3= -mg(y_2-y_1)=-mg\Delta{y}.$$

Step 4: Calculate the total work done on the system.

\begin{align}W_T&= W_1 + W_2 + W_3\\W_T&= P_1{A_1}\Delta{l}_1 -P_2{A_2}\Delta{l}_2 -mg\Delta{y}\\\end{align}

Now recall the work-energy theorem, which states that the total kinetic energy, where kinetic energy is \( K=\frac{1}{2}m{v^2}\), of a system, is equal to the total work done on that system. Hence, this yields the equation

$$\frac{1}{2}m{v_2}^2-\frac{1}{2}m{v_1}^2=P_1A_1\Delta{l}_1 -P_2A_2\Delta{l}_2 -mg\Delta{y},$$

where we can write mass in terms of density and volume using the density equation, \( \rho=\frac{m}{V}\implies m=\rho{V} \), and replace the term \( A{\Delta{l}} \) by \( V\) because the product of those two terms gives us volume.

$$\frac{1}{2}\rho{V}{v_2}^2 - \frac{1}{2}\rho{V}{v_1}^2 = P_1{V}-P_2{V} - \rho{V}g\Delta{y}.$$

The \( V's \) will cancel out, thus yielding the equation,

$$P_1 + \frac{1}{2}\rho{v_1}^2 + {\rho}gh_1=P_2 + \frac{1}{2}\rho{v_2}^2 + {\rho}gh_2,$$

which is Bernoulli's equation. However, two key points to recognize and understand are that this equation only holds true if our assumptions hold true and that since we used the work-energy theorem, we can logically deduce that this equation is essentially derived as a result of the conservation of energy.

Shearing Stress in Fluids

When parallel objects slide past one another, this action is known as shearing. This phenomenon occurs in fluids and results in shearing stress.

Shearing stress is one of two types of stress fluids undergo. In physics, stress refers to a force per unit area acting on an infinitesimal surface. Stresses are vector quantities and are divided into normal stresses and tangential stresses. Normal stresses include pressures that act inward and perpendicular to the surface. Tangential stresses include shear stresses. The main reason shear stresses are present in fluids is friction due to viscosity. Fluids cannot resist shear stress. This means that when shear stress is applied to a fluid at rest, the fluid moves as it is unable to remain at rest.

Experiments in Fluids

Reading scientific jargon to fully understand scientific concepts, like fluids, is sometimes hard to wrap your mind around. Hence, let's discuss two easy experiments that can help you better understand concepts like Archimedes' principle and the principle behind Bernoulli's equation.

Archimedes' Principle

To visualize Archimedes' principle, take a glass of warm water and put a grape in the water. The grape sinks because it is denser than the water. However, if we add salt to the water, the grape begins to float. Recall that Archimedes' principle states that if an object's weight is more than its own volume when placed in a fluid, the object sinks. Adding salt to the water increases the water's mass per unit volume until it is equal to or greater than the density of the grape, thus allowing the grape to float.

Bernoulli's Equation

To demonstrate the principles behind Bernoulli's equation, we can perform some balloon magic. This idea requires us to blow up two balloons of the same size and attach strings of the same length to each balloon. Using tape, we attach the balloons to the underside of the top part of our door frame. Make sure the balloons are separated by some distance. Finally, take a hairdryer and blow a steady stream of air in between the balloons using both the low and high settings. What do you see? On the low setting, the balloons most likely did not move too much. However, on the high setting, did you notice that the balloons move toward one another without being touched? This effect is Bernoulli's principle, which demonstrates that the air surrounding the balloons is exerting the same amount of pressure on all sides of the balloon. By blowing more forcible air between the balloons, an area of low pressure is created. Faster-moving air creates less pressure. Therefore, the pressure between the balloons decreased compared to when balloons were at rest with no airflow separating them. As a result, the balloons move toward each other as high pressure pushes low pressure.

Examples of Fluid

As we discussed fluids and some key corresponding concepts, let's complete some quick examples to drive home what we've learned.

Let's try a slightly more complicated example.