Did you know that we rely on the flow of fluids within our own bodies to survive? Fluid flow is all around us, however, what matters most to humans is the ability of our blood to flow. Blood flow is an essential bodily function that enables our vital organs to work properly by supplying them with oxygen and nutrients. Nutrients, transported to cells by our blood vessels, are broken down in our bodies by complex chemical reactions, which in turn help build up our energy supply. Keeping this example in mind, let’s dive deeper into fluid flows.
Fluid Flow Definition
Fluid and flow, are two terms each with their own definitions. Fluids, i.e. liquids and gases, are substances with no distinct shape and change easily relative to the presence of external pressure. Flow, on the other hand, refers to movement. When combined, the term fluid flow refers to the movement of liquids and gases.
Fluid Flow describes the movement of fluids produced by a pressure gradient, where pressure gradients are differences in pressure across a surface.
Fluid flow falls under the study of fluid dynamics, the study of fluid motion, which is a subcategory of fluid mechanics. Fluid mechanics is the study of forces that arise due to the behavior of fluids.
Fluid Flow Rate Formula
To calculate a fluid's rate of flow, the corresponding equation depends on the parameters of a given situation.
Flow rate refers to the volume of a fluid passing through a given cross-sectional area during a specific time interval.
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.
Calculate the volume of a fluid during a \( 60\,\mathrm{min} \) time interval, assuming the flow rate is \( 500\,\mathrm{\frac{m^3}{s}} \).
Solution
Before solving this problem, we must first convert minutes to seconds:
\begin{align}60\,\mathrm{min}\left(\frac{60\,\mathrm{s}}{1\,\mathrm{min}}\right)=3600\,\mathrm{s}.\\\end{align}
Now ,rearranging our flow rate equation in terms of volume, our calculations are as follows:
\begin{align}Q&=\frac{V}{t},\\V&=Qt,\\V&=\left(500\,\mathrm{\frac{m^3}{s}}\right)(3600\,\mathrm{s}),\\V&=1.8\,\times 10^6 \,\mathrm{m^3}.\\\end{align}
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.
Viscosity is the internal resistance to motion due to internal friction.
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.
Consider two solid plates with a liquid in between. One plate remains stationary while the other moves.
Fig. 1 - A diagram of two solid plates to aid in the understanding of the viscosity coefficient.
The liquid in direct contact with the moving plate is held together by adhesive forces between molecules within both the liquid and solid. Consequently, the liquid closest to the moving plate moves with a velocity equal to that of the moving plate. Similarly, the liquid closest to the stationary plate is also stationary, meaning its velocity is zero. This means that the velocity of the liquid between the two plates falls between \( v=0 \) and \( v\). The ratio of \( v\) is defined by the distance between the two plates which yields \( \frac{v}{l} \), the velocity gradient.
To move the moving plate along the surface of the liquid, a force is required. The corresponding equation to this force is
$$F={\eta}{A}{\frac{v}{l}},$$
which we can rearrange to yield the equation of the viscosity coefficient mentioned earlier:
\[\eta=\frac{F\,l}{A\,v}. \]
The larger the viscosity, the greater the force needed to move the plate. The larger the distance between the plates, the lower the force.
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.
| Flow Type | Description |
| Steady or Unsteady | Steady flows indicate that conditions such as velocity and pressure vary but do not change with respect to time. In contrast, unsteady suggests that at some point these conditions change with respect to time. |
| Uniform or Non-uniform | Uniform flow indicates that the velocity of a fluid has the same magnitude and direction at all points within the fluid. Non-uniform flow indicates that velocity does not have the same magnitude or direction at all points within the fluid. |
| Compressible or Incompressible | Compressible flows indicate fluids whose volume and density change due to pressure. Incompressible flows indicate fluids whose volume and density do not change due to pressure. |
| Laminar or Turbulent | Laminar flows indicate fluids that flow in a streamlined path, meaning particles follow a straight path and are in parallel with one another. Turbulent flows indicate random and chaotic behavior. |
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.
A streamline is a traceable path taken by a single particle within a fluid.
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 are moving bodies of water in a specific direction.
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.
Fluid Flow - Key takeaways
- Fluid Flow describes the movement of fluids produced by a pressure gradient, where pressure gradients are differences in pressure across a surface.
- Flow rate refers to the volume of a fluid passing through a given cross-sectional area during a specific time interval.
- Flow rate has a corresponding equation of \( Q=\frac{V}{t} \).
- Viscosity is the internal resistance to motion due to internal friction.
- The behavior of fluid flows is largely determined by pressure, density, and viscosity.
- Fluid flow is divided into four types: steady or unsteady, uniform or non-uniform, compressible or incompressible, and laminar or turbulent.
- Bernoulli's equation describes the conservation of energy in fluid flow.
References
- Fig. 1 - Moving plates, StudySmarter Originals.
- Fig. 2 - The Bernoulli equation applied to a pipe system, StudySmarter Originals.
- Fig. 3 - The continuity equation applied to a pipe system, StudySmarter Originals.
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