The study of fluid dynamics involves the movement of liquids and gases. No matter what fluid is flowing over a car, moving through a pipe, or passing through our bodies, fluid dynamics provides the framework for understanding and predicting how fluids behave. Several essential equations describe the key properties of fluid motion in this field.
Link: Wang FZ, Animasaun IL, Muhammad T, Okoya SS. Recent advancements in fluid dynamics: drag reduction, lift generation, computational fluid dynamics, turbulence modelling, and multiphase flow. Arab J Sci Eng 2024:1–13.
We will explore the most important fluid dynamics equations, explaining the basic concepts and covering more advanced topics for a well-rounded understanding.
Governing Equations of Fluid Flow and Heat Transfer Dynamics
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The Continuity Equation
Control volume discretization to examine continuity equation
In fluid dynamics, the continuity equation is one of the most fundamental concepts. In other words, it ensures that for any fluid passing through a volume, the amount entering equals the amount leaving, assuming there is no accumulation.
For a steady flow of incompressible fluid, the continuity equation is expressed mathematically as follows:
![\[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0 \]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-a0e96cd7e73dbf00d9c875487eddfa86_l3.png "Rendered by QuickLaTeX.com")
Where:
- is the fluid density,
- \vec{v} is the velocity field,
- \nabla \cdot (\rho \vec{v}) represents the divergence of the flow field.
For incompressible flows, where the density remains constant, this equation simplifies to:
![\[ \nabla \cdot \vec{v} = 0 \]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-4aa2537e1c79d1f150a8fbc4df4bd8bc_l3.png "Rendered by QuickLaTeX.com")
![\[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-fb794fed3bc60b0318e601f260865b3b_l3.png "Rendered by QuickLaTeX.com")
In simpler terms, the continuity equation ensures that mass is conserved as the fluid flows. If you imagine water flowing through a pipe, the same amount of water must flow into one end as flows out the other, unless there is some storage or loss of fluid. The equation captures this idea by relating the fluid’s velocity and the cross-sectional area through which it flows. More information about conservation of mass in Fluid Mechanics.
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Navier-Stokes Equations
In fluid dynamics, the Navier-Stokes equations provide a detailed mathematical description of the motion of viscous fluids. As a result of these equations, momentum is conserved, and a complete framework for modeling complex fluid flows is provided.
The general form of the Navier-Stokes equation for an incompressible fluid is:
![\[ \rho \left( \frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla) \vec{v} \right) = -\nabla P + \mu \nabla^2 \vec{v} + \vec{f} \]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-f5ab999835e199fca8f72b846b6d3459_l3.png "Rendered by QuickLaTeX.com")
Where:
- r is the fluid density,
- \vec{v} is the velocity field,
- P is the pressure,
- μ is the dynamic viscosity,
- \vec{f} represents any external body forces (like gravity),
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![\[\nabla^2\]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-19e1ca6c5bc299730552a3f66b4da79e_l3.png "Rendered by QuickLaTeX.com")
is the Laplacian operator, accounting for the diffusion of momentum.
The Navier-Stokes equations are often regarded as the fluid equivalent of Newton’s second law. They explain how a fluid’s velocity changes in response to internal forces (such as viscosity and pressure gradients) and external forces (such as gravity). It is possible to use these equations to model everything from weather patterns to the flow of air over aircraft wings. Nevertheless, they are notoriously difficult to solve and often require numerical simulations to be performed.
The Navier-Stokes equations are the cornerstone of fluid dynamics, governing the motion of viscous fluid substances. These equations describe how the velocity of a fluid evolves over time under the influence of forces like pressure, viscous stress, and external forces such as gravity. In Computational Fluid Dynamics (CFD) simulations, the Navier-Stokes equations form the foundation for predicting fluid behavior in complex systems, from airflow over aircraft wings to the flow of water in pipes. By solving these equations numerically, CFD tools like ANSYS Fluent can simulate real-world fluid dynamics scenarios, enabling engineers to analyze factors such as velocity distribution, pressure fields, and turbulence in a variety of industries including aerospace, automotive, and environmental engineering. Although challenging to solve due to their nonlinear nature, the Navier-Stokes equations are essential for accurately modeling fluid flow in both laminar and turbulent regimes.
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The Energy Equation
Energy equation enables this way in Ansys Fluent software
Energy conservation is as important as mass conservation and momentum conservation in fluid dynamics. A fluid system’s energy balance is represented by the energy equation, which takes into account internal, kinetic, and potential energy. In the case of compressible flows or heat transfer, this equation is fundamental.
The general form of the energy equation is:
![\[ \frac{\partial}{\partial t} (\rho e) + \nabla \cdot (\rho \vec{v} e) = -\nabla \cdot \vec{q} + \tau∶ \nabla \vec{v} + \rho \dot{q} \]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-99294eea02537d06d6afd38a9649b94d_l3.png "Rendered by QuickLaTeX.com")
Where:
- e is the specific total energy (including internal and kinetic energy),
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![\[\vec{q}\]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-90e736e13b1931b313d145dc43a269b1_l3.png "Rendered by QuickLaTeX.com")
is the heat flux vector,
- τ is the viscous stress tensor,
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![\[\dot{q}\]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-76744dc833eddaebc8031bc0fe78d874_l3.png "Rendered by QuickLaTeX.com")
is the rate of heat addition.
The energy equation ensures that all forms of energy within the system—kinetic, thermal, and potential—are accounted for. It becomes especially critical when dealing with situations involving heat transfer or temperature changes, such as combustion engines, heating systems, or cooling flows. More information about energy conservation Equation.
Basic Fluid Dynamics Equations
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Fluid Dynamics Bernoulli Equation
One of the most famous equations in fluid dynamics is the Bernoulli equation, named after Swiss mathematician Daniel Bernoulli. This equation demonstrates that mechanical energy can be conserved in a fluid system by relating pressure, velocity, and height (or elevation) in a steady, incompressible flow.
The Bernoulli equation can be written as:
![\[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-52947ef6527bca2857a56ed67e9260e8_l3.png "Rendered by QuickLaTeX.com")
Where:
- P is the pressure,
- r is the fluid density,
- n is the velocity of the fluid,
- g is the acceleration due to gravity,
- h is the height above a reference point.
According to Bernoulli’s equation, pressure energy, kinetic energy (related to speed), and potential energy (related to height) remain constant along a streamline. The increase in one of these variables must be accompanied by a decrease in the other. For example, if the velocity of a fluid increases, its pressure decreases. There are numerous applications of this principle, including the mechanism by which airplanes generate lift or the mechanism by which shower curtains are sucked inward by flowing water.
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Euler’s Equations
As a simplified version of the Navier-Stokes equations, Euler’s equations apply when viscosity is negligible, i.e., when the fluid is considered to be inviscid. In specific aerospace applications, these equations are ideal for understanding high-speed flows with minimal viscosity effects.
The Euler equations for an incompressible fluid are:
![\[ \rho \left( \frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla) \vec{v} \right) = -\nabla P + \vec{f} \]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-df750cfb6eab5edb8551aefef0a13a36_l3.png "Rendered by QuickLaTeX.com")
Notice the absence of the viscosity term (
![\[\mu \nabla^2 \vec{v}\]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-199fb46f09ab1cf8883dc5fed5a15510_l3.png "Rendered by QuickLaTeX.com")
that appears in the Navier-Stokes equations.
Euler’s equations describe how fluid velocity evolves when the flow is frictionless. These equations are less accurate for most practical applications, where viscosity plays a role, but they provide a simpler model for specific idealized scenarios.
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Darcy-Weisbach Equation
The Darcy-Weisbach equation becomes an essential tool when studying fluid flow through pipes. It relates the pressure drop due to friction to the fluid’s velocity, the pipe’s length, and the pipe’s diameter.
The equation is:
![\[ \Delta P = f \cdot \frac{L}{D} \cdot \frac{\rho v^2}{2} \]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-b72ed5322654c284791d605d264fd8eb_l3.png "Rendered by QuickLaTeX.com")
Where:
- ΔP is the pressure loss,
- f is the Darcy friction factor,
- L is the length of the pipe,
- D is the pipe’s diameter,
- r is the fluid density,
- n is the fluid velocity.
This equation helps engineers design piping systems, ensuring that the pressure drop due to friction is considered. It’s widely used in industries ranging from water distribution to oil pipelines.
Conclusion
The equations discussed in this blog form the basis of fluid dynamics, providing insight into how fluids move, how forces affect them, and how energy is transferred within them. These fundamental equations provide the tools you need to understand and predict fluid behavior, regardless of whether you are calculating how air flows over an airplane wing or how water moves through a pipe. For those who are just beginning their study of fluid dynamics, mastering these equations will provide a solid foundation. If you are already familiar with the basics, diving deeper into the nuances of these equations will help you solve complex, real-world problems.
Besides the fundamental equations of fluid dynamics, there are also more advanced and specialized equations for complex fluid behaviors encountered in real-world applications. As an example, turbulence is modeled by Reynolds-Averaged Navier-Stokes (RANS) equations or Large Eddy Simulation (LES), which introduce additional equations to account for chaotic and fluctuating flows. For multiphase flows, where fluids interact with different phases (like gas-liquid or liquid-solid interactions), models such as the Volume of Fluid (VOF) and Eulerian-Eulerian approaches are commonly used. In species transport, equations governing the diffusion and reaction of chemical species become crucial, especially in combustion or reactive flows.
Moreover, in Lagrangian frameworks, fluid particles are tracked individually, which is essential for understanding particle-laden flows or dispersed phases. These more advanced equations and models play a key role in fields like aerospace, chemical engineering, and environmental science. We will explore these specialized equations in more detail in upcoming discussions, delving deeper into each specific area.
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