What makes the Navier Stokes equation so difficult?

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Thread starter jedishrfu
Start date Jan 16, 2018
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Navier stokes Navier stokes equation Stokes
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In summary, the article discusses the math behind the Navier Stokes equations, how they are difficult to solve and whether they truly represent fluid flow. There are many assumptions that have to be met in order to obtain solutions to the equations, and they are still a challenge to solve.

Jan 28, 2018

boneh3ad

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I assure you that myself and everyone I know who works with or for NASA, the military, Boeing, Lockheed, Northrop, Raytheon, Ford, GM, Ferrari, Mercedes, and any other organization dealing very heavily with fluids treats the continuum assumption as being "perfect" except in rare circumstances. We will all continue not worrying about individual molecules and using that continuum assumption to design your planes, rockets, missiles, and cars like we always have. Your can worry about individual molecules if you wish.

<h2>1. What is the Navier Stokes equation?</h2><p>The Navier Stokes equation is a set of partial differential equations that describe the motion of fluids, such as liquids and gases. It takes into account factors such as pressure, velocity, and viscosity to predict how a fluid will behave in a given situation.</p><h2>2. Why is the Navier Stokes equation important?</h2><p>The Navier Stokes equation is important because it is the fundamental equation that governs the behavior of fluids. It is used in many fields of science and engineering, including aerodynamics, weather forecasting, and oceanography. Solving the Navier Stokes equation allows us to understand and predict the behavior of fluids in various scenarios.</p><h2>3. What makes the Navier Stokes equation difficult to solve?</h2><p>The Navier Stokes equation is difficult to solve because it is a set of nonlinear partial differential equations, meaning that the equations involve multiple variables and their derivatives. Additionally, the equations are highly sensitive to initial conditions and can exhibit chaotic behavior, making it challenging to find exact solutions.</p><h2>4. Are there any simplifications or assumptions made in the Navier Stokes equation?</h2><p>Yes, there are several simplifications and assumptions made in the Navier Stokes equation. One common simplification is assuming that the fluid is incompressible, meaning that its density remains constant. Other assumptions may include neglecting certain forces, such as surface tension, or assuming that the fluid is in a steady state.</p><h2>5. How do scientists and engineers solve the Navier Stokes equation?</h2><p>There are various methods for solving the Navier Stokes equation, including analytical, numerical, and computational techniques. Analytical solutions involve finding exact mathematical expressions for the variables in the equation, while numerical solutions use approximations and algorithms to solve the equations. Computational methods use computer simulations to solve the equations and can provide more accurate results for complex scenarios.</p>

1. What is the Navier Stokes equation?

The Navier Stokes equation is a set of partial differential equations that describe the motion of fluids, such as liquids and gases. It takes into account factors such as pressure, velocity, and viscosity to predict how a fluid will behave in a given situation.

2. Why is the Navier Stokes equation important?

The Navier Stokes equation is important because it is the fundamental equation that governs the behavior of fluids. It is used in many fields of science and engineering, including aerodynamics, weather forecasting, and oceanography. Solving the Navier Stokes equation allows us to understand and predict the behavior of fluids in various scenarios.

3. What makes the Navier Stokes equation difficult to solve?

The Navier Stokes equation is difficult to solve because it is a set of nonlinear partial differential equations, meaning that the equations involve multiple variables and their derivatives. Additionally, the equations are highly sensitive to initial conditions and can exhibit chaotic behavior, making it challenging to find exact solutions.

4. Are there any simplifications or assumptions made in the Navier Stokes equation?

Yes, there are several simplifications and assumptions made in the Navier Stokes equation. One common simplification is assuming that the fluid is incompressible, meaning that its density remains constant. Other assumptions may include neglecting certain forces, such as surface tension, or assuming that the fluid is in a steady state.

5. How do scientists and engineers solve the Navier Stokes equation?

There are various methods for solving the Navier Stokes equation, including analytical, numerical, and computational techniques. Analytical solutions involve finding exact mathematical expressions for the variables in the equation, while numerical solutions use approximations and algorithms to solve the equations. Computational methods use computer simulations to solve the equations and can provide more accurate results for complex scenarios.