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Navier-Stokes

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Navier-Stokes

Project

Navier-Stokes

Year

2007

Type of project

Music + Visuals

Location

Navier-Stokes is a project born from a fascination for fluid dynamics, entropy and especially the transition from order to chaos, prevalent in fluid dynamics and the onset of turbulence in fluid motion. Combining abstract sounds and visuals, I've tried to convey both the sense of wonder and the feeling of alienation I get from contemplating the juxtaposition of order and chaos that defines our perception of reality.

The Navier-Stokes equations are a set of partial differential equations that describe the motion of viscous fluid substances. They were developed by Claude-Louis Navier and George Gabriel Stokes in the 19th century. The Navier-Stokes equations are are derived from the conservation laws of mass and momentum, and they form the foundation for the mathematical modeling of fluid flow.

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Continuity Equation: ∇⋅v=0

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This equation expresses the conservation of mass. It states that the divergence of the fluid velocity vector (∇v) must be zero, indicating that the rate of change of mass within any given region of the fluid must be balanced by the flow of mass in and out of that region.

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Momentum Equation (Navier-Stokes Equation): ρ ( (∂v/∂t) + (v⋅∇v) ) = −∇p + μ∇²v + ρg

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‍This equation represents the conservation of momentum and is often written in component form for each dimension (x, y, z).

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Breaking it down:

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ρ is the fluid density.

v is the velocity vector.

t is time.

p is pressure.

μ is dynamic viscosity.

∇ represents the gradient operator.

∇² is the Laplacian operator.

g is the gravitational acceleration vector.

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The terms on the right side of the equation represent the forces acting on the fluid element: pressure gradient force, viscous forces, and gravitational force.

These equations are applicable to incompressible Newtonian fluids, where the fluid density is constant and the viscosity is linearly proportional to the rate of strain. Solving these equations numerically is a common approach for understanding and predicting fluid flow in a wide range of engineering and scientific applications, from aerodynamics to weather modeling. It's worth noting that obtaining analytical solutions to these equations for most practical cases is a challenging problem, and various computational methods are often employed for simulation and analysis.

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The open problem associated with the Navier-Stokes equations is known as the "Navier-Stokes existence and smoothness problem." This problem is one of the seven "Millennium Prize Problems" designated by the Clay Mathematics Institute, each of which comes with a one-million-dollar reward for a correct solution.

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Navier-Stokes

Continuity Equation: ∇⋅v=0

This equation expresses the conservation of mass. It states that the divergence of the fluid velocity vector (∇v) must be zero, indicating that the rate of change of mass within any given region of the fluid must be balanced by the flow of mass in and out of that region.

‍

Momentum Equation (Navier-Stokes Equation): ρ ( (∂v/∂t) + (v⋅∇v) ) = −∇p + μ∇2v + ρg

‍This equation represents the conservation of momentum and is often written in component form for each dimension (x, y, z).

‍

Breaking it down:

ρ is the fluid density.

v is the velocity vector.

t is time.

p is pressure.

μ is dynamic viscosity.

∇ represents the gradient operator.

∇2 is the Laplacian operator.

g is the gravitational acceleration vector.

‍

The terms on the right side of the equation represent the forces acting on the fluid element: pressure gradient force, viscous forces, and gravitational force.

Graffiti

Momentum Equation (Navier-Stokes Equation): ρ ( (∂v/∂t) + (v⋅∇v) ) = −∇p + μ∇²v + ρg

∇² is the Laplacian operator.