Fluid mechanics is the field of physics that deals with the physical mechanics of fluids (plasmas, gases, and liquids) and forces acting on them. It has a wide variety of applications in fields like engineering, oceanography, astrophysics, geophysics, biology, and meteorology. Fluid mechanics can be categorised into fluid statics and fluid mechanics. Fluid statics is the study of fluids at the state of rest. Fluid dynamics is the study of the impacts of forces on fluids in motion. It is a section of continuum mechanics, a field that deals with the matter without concerning the information that comes out of the inherent properties of atoms. It only models matter from a macroscopic perspective rather than from an atomic or molecular viewpoint.
Fluid dynamics is a prolific research field which is generally mathematically complex. Numerous problems are wholly or partly unsolved and are efficiently addressed by numerical techniques, usually using computers. A cutting-edge discipline known as computational fluid dynamics is dedicated to this approach. Particle image velocimetry is an experimental technique for analysing and visualising the flow of fluids. It also takes into account the visual nature of the fluid flow.
Now let’s explore interesting equations that deal with the flow and forces of viscous fluids.
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Table of Contents |
Navier Stokes Equations – Definition
In fluid mechanics, the Navier-Stokes equations are partial differential equations that express the flow of viscous fluids. These equations are generalisations of the equations developed by Leonhard Euler (18th century) to explain the flow of frictionless and incompressible fluids. In 1821, Claude-Louis Navier put forward the component of viscosity (friction) for a more realistic and difficult problem of viscous fluids. During the entire middle period of the 19th century, George Gabriel Stokes refined this work even though entire solutions were found only in the case of basic two-dimensional flows. The complicated turbulence or vortices, or chaos that happens in three-dimensional fluid flows as velocities rise, has become intractable to any but numerical analysis techniques. The Navier–Stokes equations numerically describe the conservation of mass and the conservation of momentum for Newtonian fluids.
The Navier Stokes momentum equation
The Navier–Stokes momentum equation can be mathematically deduced as a distinct type of the Cauchy momentum equation. The general convective structure is
by making the Cauchy stress tensor σ be the sum of a viscosity term τ (the deviatoric stress) and a pressure quantity -pI (volumetric stress), we arrive at,
Cauchy momentum equation (convective structure):
Where
- \(\begin{array}{l}\frac{D}{Dt} \textup{ is the material derivative, stated as} \end{array} \)
- \(\begin{array}{l}\frac{\partial }{\partial t} + u\cdot \bigtriangledown , \end{array} \)
- ρ = density,
- u = flow velocity,
- ▽ = divergence,
- p = pressure,
- t = time,
- τ = deviatoric stress tensor (order 2),
- “g” denotes material accelerations acting on the continuum (like electrostatic accelerations, inertial acceleration, gravity, etc.)
Continuity Equation
The additional equation that represents the behaviour of fluid is the continuity equation. The equation to the conservation of mass implies the mass of the fluid is neither created nor destroyed in motion. The concept of conservation is an essential principle used throughout classical physics.
Continuity equation for flow density,
Cauchy momentum equation (conservation structure)
Every non-relativistic balance equation, such as the Navier–Stokes equations, can be constructed by starting with the Cauchy equations and citing the stress tensor with a constitutive relation. By describing the deviatoric stress tensor with fluid velocity gradient and viscosity, and taking fixed viscosity, the Cauchy equations will result in the Navier–Stokes equations.
The video about the equation of continuity
Applications of Navier Stokes Equations
The Navier–Stokes equations can be very useful in applied physics. Primarily, they help to describe the mechanics of various engineering and scientific phenomena. They could be applied to model ocean currents, weather, air flow around wings, and the flow of water in pipes. These equations, in their simplified and full forms, help out with the modelling of vehicles and aircraft. They are also applied in the analysis of dense liquids, the examination of pollution, the design of power, and other processes related to fluids. Along with Maxwell’s equations, these equations can be applied to study and model magnetohydrodynamics.
The Navier–Stokes equations also have great importance in pure mathematics. Despite their extensive range of applications, there is no proof for the consistent existence of smooth solutions in three dimensions; the equations are infinitely differentiable at every point in the domain. It is known as the Navier–Stokes smoothness and existence problem. This has been called one of the most significant unsolved problems in mathematics. A university has offered prize money of 1 million US dollars for whoever finds a solution for it.
Flow Velocity
Flow velocity is a vector field; to all points in a normal fluid, at any instance in a time period, it provides a vector whose magnitude and direction are of the fluid’s velocity at that instance in time and at that point in space. It is generally examined in three dimensions. Even though two-dimensional and steady-state scenarios are usually employed as models, and greater dimensional analogues are analysed both in applied and pure mathematics, it is generally examined in three spatial dimensions. When the measurement of velocity field is done, other quantities such as temperature or pressure could be found utilising dynamical relations and equations. This is much different from what is commonly seen in early mechanics, where derived solutions are generally trajectories of a particle’s position or deflection.
Solution of Navier-Stokes Equations
In the most general form, there are no analytical solutions to the Navier-Stokes equations. In other words, it is only possible to get some form of analytical solutions in particular approximate scenarios. The outcomes may not ever be realised in a real system. More geometrically sophisticated systems will need a numerical technique to find some form of a solution which is achieved with CFD simulations.
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Frequently Asked Questions – FAQs
What is fluid mechanics?
Fluid mechanics is the branch of physics that deals with the behaviour of fluids and the various forces that they generate. Fluids include plasma, gases, and liquids. Fluid mechanics dwells on the behaviour of fluids that are in motion and at the state of rest. It is applied in many disciplines, including geophysics, aerospace, mechanical engineering, biomedical engineering, chemical engineering, aeronautics, etc.
What are the basic properties of fluids?
The basic properties of fluids are density, temperature, pressure, viscosity, specific volume, specific weight, specific gravity, and surface tension.
What are Euler’s equations of motion?
Euler’s equations are named after the mathematician Leonhard Euler. They are a group of equations that govern the inviscid and adiabatic flow of fluids. Euler’s equations give a relationship between the density, velocity, and pressure of a moving fluid. They represent the conservation of energy, mass, and momentum in fluids. They are derived from Newton’s second law of motion.
What are compressibility and bulk modulus?
Compressibility is the measurement of the variation in a fluid’s volume when it is subjected to a change in pressure. For a particular mass of fluid, a rise in pressure will induce a reduction in the volume of the fluid. Compressibility can also be defined as the ratio of variation in volume to the variation in pressure. Bulk modulus is stated as the ratio of variation in pressure to the variation in volume. It is the inverse of compressibility.
What are the two methods of describing fluid motion?
There are two fundamental methods for describing fluid motion – Eulerian and Lagrangian. In the Lagrangian method, one follows every fluid particle and describes the changes around every fluid particle along its trajectory. In the Eulerian method, the changes are described at every fixed station as a time function.
Explain Navier-Stokes equations.
In fluid mechanics, the Navier-Stokes equations are partial differential equations that express the flow of viscous fluids. These equations are generalisations of the equations developed by Leonhard Euler (18th century) to explain the flow of frictionless and incompressible fluids. The Navier–Stokes equations analytically describe the conservation of mass and the conservation of momentum for Newtonian fluids.
What is the Navier Stokes momentum equation?
The Navier–Stokes momentum equation can be mathematically deduced as a distinct type of the Cauchy momentum equation. The general convective structure is,
What is the main application of Navier Stokes equations?
It can be applied to model ocean currents, weather, air flow around wings, and the flow of water in pipes. They are also applied in the examination of liquid flow, the study of pollution, the design of power, and other processes related to fluids. Along with Maxwell’s equations, these equations can be applied to study and model magnetohydrodynamics.
Is there any consistent solution to the Navier-Stokes equations?
In the most general form, there are no analytical solutions to the Navier-Stokes equations. In other words, it is only possible to get some form of an analytical solution in particular approximate scenarios. The outcomes may not ever be realised in a real system.
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