The Physics of Fluid Flows

The Navier-Stokes equations are a set of partial differential equations (PDEs)  that describe the motion of fluid substances. Named after Claude-Louis Navier and George Gabriel Stokes, who contributed significantly to fluid dynamics in the 19th century, these equations provide a fundamental framework for understanding and analyzing fluid flow phenomena. They are equations used by engineers to model the conservation of mass and momentum in Newtonian liquids.

In this section, we will delve into the Navier-Stokes equations, their derivation, significance, and applications in various fields.

The Navier-Stokes equations are derived from the principles of conservation of mass, momentum, and energy, combined with Newton’s second law of motion.

They can be expressed in vector form as follows:

∂v/∂t+ (v⋅∇)v  = −(1/ρ)*​∇p+ν∇²v+f

$\nabla \cdot \mathbf{v}=0$

Where:

• $\mathbf{v}$ is the velocity vector field of the fluid,
• $t$ is time,
• $p$ is pressure,
• $\rho$ is fluid density,
• $\nu$ is the kinematic viscosity of the fluid,
• $\nabla$ is the gradient operator,
• $\mathbf{f}$ represents external body forces per unit mass (such as gravitational or electromagnetic forces).

The first equation is the momentum equation, while the second equation is the continuity equation, ensuring the conservation of mass. Together, these equations govern the behavior of fluid motion in both spatial and temporal domains.

The momentum equation describes the acceleration of fluid particles as a result of various forces acting upon them. The first term on the left-hand side represents the local rate of change of velocity with respect to time, known as convective acceleration. The second term represents the advection term, which accounts for the transport of momentum by the fluid flow itself. The third term on the right-hand side represents the pressure gradient force, which accelerates fluid particles from regions of high pressure to low pressure. The fourth term represents viscous forces, which arise due to the internal friction within the fluid. The final term, $\mathbf{f}$, accounts for any external forces acting on the fluid, such as gravitational or electromagnetic forces.

The continuity equation ensures that mass is conserved within the fluid domain by stating that the divergence of the velocity field ($\nabla \cdot \mathbf{v}$) must be zero. In other words, it asserts that the rate of change of mass within any volume of fluid must equal the net rate of mass flux across its boundaries.

The Navier-Stokes equations are highly nonlinear and coupled, making them notoriously difficult to solve analytically for most practical flow problems. These PDEs are non-linear because of convective acceleration which is relates to the change of velocity with position. They commonly exhibit time-dependent chaotic behaviour which we described as turbulence.

Despite their complexity, the Navier-Stokes equations have profound implications and applications across various fields. In fluid mechanics and engineering, they serve as the foundation for understanding and predicting the behavior of fluid flows in applications ranging from aerospace and automotive engineering to environmental and biomedical sciences.

One of the key phenomena described by the Navier-Stokes equations is turbulence, which arises due to the nonlinear interactions between fluid particles at high Reynolds numbers. Turbulence is ubiquitous in natural and engineered fluid flows and plays a crucial role in phenomena such as mixing, heat transfer, and drag reduction. Understanding and predicting turbulence is essential for optimizing the design and performance of engineering systems, such as aircraft wings, propellers, and gas turbines.

As a result of their complexity, numerical methods, such as finite difference (Godunov & Bohachevsky, 1959), finite volume (Eymard et al., 2000), finite element (Zienkiewicz et al., 1977), spectral methods (Canuto et al., 2012) and computational fluid dynamics (CFD) techniques, are often employed to approximate solutions to these equations. These numerical methods discretize (using these methods of different orders), the fluid domain into a grid and solve the equations iteratively to obtain an approximate solution. Turbulence can be simulated to different levels of ‘computational cost and fidelity’.

These numerical methods can be expensive from a computational perspective. They are almost impossible to solve at high Reynolds numbers because of the wide range of scales on space and time necessary to resolve these flows. 

The Navier-Stokes equations also play a vital role in weather prediction and climate modeling. By simulating the atmospheric circulation patterns and interactions between air masses, scientists can forecast weather phenomena such as hurricanes, tornadoes, and rainfall patterns. Additionally, climate models utilize the Navier-Stokes equations to simulate the global circulation of the atmosphere and oceans, helping scientists understand the complex interactions driving climate variability and change.

In the field of biomedical engineering, the Navier-Stokes equations are used to model blood flow in the cardiovascular system. By simulating blood flow patterns in arteries and veins, researchers can gain insights into the hemodynamic factors influencing cardiovascular diseases such as atherosclerosis and aneurysms. These simulations aid in the design of medical devices such as stents and artificial heart valves, as well as in surgical planning for interventions such as bypass surgeries and angioplasty.

The Navier-Stokes equations are fundamental equations that govern the motion of fluid substances. Despite their complexity and nonlinear nature, these equations provide a powerful framework for understanding and analyzing fluid flow phenomena in various fields, including engineering, environmental science, and biomedical research. By utilizing numerical methods and computational techniques, researchers and engineers can leverage the Navier-Stokes equations to solve practical flow problems and optimize the design and performance of complex systems.