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The Navier-Stokes equations control the motion of fluids and can be seen as Newton's second law of motion for fluids. The continuity equation reflects the conservation of mass, whereas the Navier-Stokes equations indicate the conservation of momentum. The study of the physical mechanics of fluids (plasmas, gasses, and liquids), as well as the forces acting on them, is known as fluid mechanics in the area of physics. There are two different branches of fluid mechanics: fluid statics and fluid mechanics. Studies of fluids in a rest state are called fluid statics. Studying the effects of forces on fluids in motion is known as fluid dynamics.
Read more: Angular momentum
KeyTerms: Navier Stokes Equations, Fluid viscosity, Fluid flow, Velocity, Navier Stokes momentum, Continuity equation, Density, Pressure.
Navier Stokes Equations - Definition
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The Navier-Stokes equations, which are partial differential equations, are used in fluid mechanics to describe the flow of viscous fluids.
- These equations are generalizations of the ones Leonhard Euler created in the eighteenth century to describe the flow of incompressible and frictionless fluids.
- Claude-Louis Navier proposed the viscosity (friction) component for a more challenging and realistic viscous fluids problem in 1821.
- Although complete solutions were only discovered in the case of simple two-dimensional flows, George Gabriel Stokes continued to develop this work during the middle decades of the 19th century.
- With the exception of numerical analysis approaches, the complex turbulence, vortices, or chaos that occurs in three-dimensional fluid flows as velocities increase is unsolvable.
- The mass and energy conservation laws are mathematically described by the Navier-Stokes equations.
Navier Stokes momentum equation
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Mathematically, the Navier-Stokes momentum equation is a special case of the Cauchy momentum equation. The general convective structure is
Where,
- \(\frac{D}{Dt}\)is the material derivative, stated as
- ρ = density,
- u = flow velocity,
- ⛛ = divergence
- p = pressure
- t = time
- ፖ = deviatoric stress tensor (order 2)
- “G” denotes the action of the material acceleration on the continuum (like electrostatic accelerations, inertial acceleration, gravity, etc.)
Navier’s stoke equation
Continuity Equation
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The continuity equation is an additional equation that depicts fluid behavior. According to the equation for the conservation of mass, a fluid's mass is neither created nor destroyed while it is in motion. The idea of conservation is a fundamental idea utilized in classical physics.
Cauchy momentum equation (conservation structure)
The Cauchy equations provide the foundation for all non-relativistic balance equations, including the Navier-Stokes equations, and the stress tensor with a constitutive relation is used to illustrate this. The Navier-Stokes equations are obtained by converting the Cauchy equations into a fixed viscosity and defining the deviatoric stress tensor with fluid velocity gradient and viscosity.
Caunchy’s Stress Tensor
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Applications of Navier Stokes Equations
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In applied physics, the Navier-Stokes equations play a significant role. Some of the applications are as follows:
- Navier stokes describes the mechanics of various engineering and scientific phenomena.
- They could be used to simulate weather, air flow around wings, water flow in pipelines, and ocean currents.
- These equations, in both their streamlined and comprehensive versions, are useful for modeling automobiles and aeroplanes.
- Additionally, they are used in the study of pollution, the design of power, and other fluid-related activities, as well as the investigation of dense liquids.
- These equations can be used to analyze and model magnetohydrodynamics along with Maxwell's equations.
In pure mathematics, the Navier-Stokes equations are quite significant. The equations are indefinitely differentiable at every point in the domain, despite the fact that they have a wide variety of applications. The Navier-Stokes smoothness and existence problem is the name given to it. This has been referred to as one of the most important unresolved mathematical puzzles.
Flow Velocity
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A normal fluid's flow velocity is a vector field that gives all places in the fluid a vector with magnitude and direction equal to the fluid's velocity at that particular point in time and space. Usually, it is looked at in three dimensions. Even though fluid’s flow velocity is typically studied in three spatial dimensions, two-dimensional and steady-state scenarios are often used as models and their higher-dimensional analogues are addressed in both applied and pure mathematics.
Solutions of Navier Stokes Equations
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The Navier-Stokes equations don't have any analytical solutions in their most basic form. In other words, only certain approximations allow for the possibility of some degree of analytical solutions. The results might never materialize in a practical system. More complex geometrical systems will require a numerical approach to arrive at some sort of solution, which is accomplished through CFD simulations.
Things to remember
- Fluid mechanics is the field of physics that deals with the mechanics of fluids and forces acting on them.
- Fluid mechanics are further categorized into fluid statics and fluid dynamics.
- 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.
- Navier Stokes equations are partial differential equations expressing the flow of viscous fluids.
- Cauchy momentum equation is
- Continuity equation represents the behavior of fluids.
- The equations help in modelling vehicles and aerospace.
- Flow velocity provides the vector whose magnitude and direction are of the fluid’s velocity at a particular instance in time and that point in space.
Sample Questions
Ques. What is fluid mechanics? (2 marks)
Ans. The area of physics known as fluid mechanics deals with the behavior of fluids and the many forces that they produce. Plasma, gasses, and liquids all qualify as fluids. The behavior of fluids in motion and at rest is the focus of fluid mechanics. Numerous fields use it, including geophysics, aerospace, mechanical engineering, biomedical engineering, chemical engineering, and aeronautics.
Ques. What are the two methods of describing fluid motion? (2 marks)
Ans. The Eulerian and Lagrangian approaches are the two primary ways to describe fluid motion.
- The Lagrangian approach involves tracking each fluid particle along its course and describing the changes that occur in and around it.
- In the Eulerian technique, a time function is used to describe the changes at each fixed station.
Ques. What are the basic properties of fluids? (1 mark)
Ans. Density, temperature, pressure, viscosity, specific volume, specific weight, specific gravity, and surface tension are the fundamental characteristics of fluids.
Ques. Explain Navier- Stokes equations. (3 marks)
Ans. The Navier-Stokes equations, which are partial differential equations, are used in fluid mechanics to describe the flow of viscous fluids. These equations are generalizations of the ones Leonhard Euler created in the eighteenth century to describe the flow of incompressible and frictionless fluids. For Newtonian fluids, the mass and momentum conservation laws are analytically explained by the Navier-Stokes equations.
Ques. What are Euler’s equations of motion? (3 marks)
Ans. Euler’s equations are named after the mathematician Leonhard Euler.
- They are a collection of mathematical formulae that control the adiabatic and inviscid flow of fluids.
- The density, velocity, and pressure of a moving fluid are related by Euler's equations.
- In fluids, they stand for the conservation of energy, mass, and momentum.
- Newton's second law of motion serves as the foundation for them.
Ques. What are compressibility and bulk modulus? (2 marks)
Ans. The ability of a fluid to change its volume in response to a change in pressure is known as compressibility. A rise in pressure will cause the volume of a fluid with a certain mass to decrease. The ratio of volume variation to pressure variation is another way to describe compressibility. The ratio of pressure variation to volume variation is the definition of the bulk modulus. It is the opposite of compressibility.
Ques. Is there any consistent solution to the Navier Stokes equations? (2 marks)
Ans. The Navier-Stokes equations don't have any analytical solutions in their most basic form. In other words, only specific approximations of the problem can lead to some sort of analytical solution. The results might never materialize in a practical system.
Ques. What is the Navier Stokes momentum equation? (2 marks)
Ans. The Navier–Stokes momentum equation can be mathematically deduced as a distinct type of the Cauchy momentum equation. The general convective structure is,
Ques. What is the main application of the Navier Stokes equation? (3 marks)
Ans. The applications of Navier Stokes equations are as follows:
- It can be used to simulate weather, air flow around wings, ocean currents, and water movement through pipes.
- They are also used in the analysis of liquid flow, the investigation of pollution, the development of power, and other fluid-related operations.
- These equations can be used to analyze and model magnetohydrodynamics along with Maxwell's equations.
Ques. Is the Navier Stokes problem easy to solve? Explain. (1 mark)
Ans. No, the Navier stokes equation is not easy to solve because it is nonlinear. The complicated equations can be solved only when they have linear equations by adding up many simple equations.
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