Fluid Flow

Fluid Flow Rate Formula

To calculate a fluid's rate of flow, the corresponding equation depends on the parameters of a given situation.

In accordance with this definition, the corresponding mathematical equation is

$$Q=\frac{V}{t},$$

where \( V \) is volume and \( t \) is time. Let's try a quick example.

Flow rate is also calculated in terms of velocity and area

\[ Q= vA, \]

where \( v \) is velocity and \( A \) is the cross-sectional area. Flow rate has corresponding SI units of \( \mathrm{\frac{m^3}{s}} \); however, other units are used depending on different scenarios. For instance, when referring to blood flowing through our bodies, the flow rate is measured in units of liters per minute \( \left(\mathrm{\frac{L}{min}} \right)\). Other common units include cubic meters per second, gallons per minute, and cubic feet per second.

Viscous Fluid Flow

The word viscous refers to viscosity.

In the case of viscous liquids, viscosity is the result of electrical forces between adjacent molecules that arise due to the attraction of electrons and protons between these adjacent molecules. Viscosity in gases, however, is slightly different as it comes from the collision between molecules. This behavior of fluids can be quantified using the viscosity coefficient.

Viscosity Coefficient

The viscosity of a fluid is denoted by the Greek symbol \( \eta \). This coefficient has a corresponding formula of

\[ \eta=\frac{F\,l}{A\,v}, \]

where \( F \) is force, \(A\) is the cross-sectional area, \( v\) is velocity, and \( l \) is distance. Higher viscosity coefficients indicate a fluid has a higher viscosity due to the presence of more internal friction. However, to fully understand this term and its corresponding formula, let us look at the following example.

Characteristics of Fluid Flow

When looking at the behavior of fluids, we consider characteristics such as density, temperature, pressure, viscosity, specific volume, and specific gravity. However, the behavior of fluid flow is largely determined by pressure, density, and viscosity.

• Pressure is the force exerted on a fluid.

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

• Viscosity is resistance to motion or the amount of internal friction present in a fluid.

Fluid Flow Types

Due to fluid behavior, the flow of fluids is divided into four categories.

Bernoulli's Equation

An important equation related to fluids is Bernoulli's equation, which describes the conservation of energy in fluid flow. Mathematically, it can be expressed as

\[P_1+ \rho{g}h_1+\frac{1}{2}\rho{v_1^2}=P_2+ \rho{g}h_2+\frac{1}{2}\rho{v_2^2}.\]

Here, \(P\) is the static pressure measured in pascals (\(\mathrm{Pa}\)), \(\rho\) is the fluid density \(\left(\frac{\mathrm{kg}}{\mathrm{m^3}}\right)\), \(g\) is the acceleration due to gravity \(\left(\frac{\mathrm{m}}{\mathrm{s}^2}\right)\), \(y\) is the elevation of the fluid measured in meters (\(\mathrm{m}\)), and \(v\) is the velocity of the fluid \(\left(\frac{\mathrm{m}}{\mathrm{s}}\right)\).

Fig. 2 - A diagram of a fluid flowing between two points to demonstrate Bernoulli's equation.

In other words, the sum of the pressure energy, potential energy (the second term corresponds with \(U=mgh\)), and kinetic energy (the third term corresponds with \(K=\frac{1}{2}mv^2\)) stays constant along a streamline.

Continuity Equation

Another important relation to consider in addition to the Bernoulli equation, and fluid mechanics in general, is the continuity equation, derived using the principle of conservation of mass. Mathematically, it can be expressed as

\[ A_1v_1=A_2v_2,\]

where \(A_1\) and \(A_2\) are the cross-sectional areas of the pipe in two different locations, corresponding to the velocity \(v\) of the fluid at each point. In other words, for an incompressible fluid in streamline flow, the mass of the fluid passing through the different cross-sections every second remains equal. Thus, this equation basically says that fluids don't spontaneously appear or disappear in the pipe. The continuity equation is visualized in Figure 3 below.

Importance of Fluid Flows

Without fluid flows, life on Earth would be significantly different. To demonstrate just how different our lives could be, let us look at ocean currents.

Currents play a major role in the distribution of heat, which significantly impacts climate, water, and oxygen across our planet. Currents also are important for sea life as they carry nutrients and food to sea life permanently bound to specific regions as well as transport ocean life to different places Without currents, temperatures would change drastically. Areas closer to the equator would have extremely hot temperatures while areas near the poles would have frigid temperatures. As a result, the amount of habitable land on Earth would significantly decrease, thus affecting life as we know it.