In fluid dynamics, Green's functions play a crucial role in solving steady flow problems governed by the Euler and Navier-Stokes equations. These functions, denoted by G(x, y) for a point source at (x, y), represent the perturbation in the velocity and pressure fields induced by a unit point force acting at that location.

Steady Euler Equation

For the steady Euler equation, which describes inviscid incompressible fluid flow, the Green's function satisfies the following equation:

∇² G(x, y) + δ(x, y) = 0

where δ(x, y) is the Dirac delta function representing the point source. The solution to this equation is given by:

G(x, y) = -1/(2π) ln(|x - y|)

This Green's function represents the velocity potential due to a unit point source in the steady Euler flow.

Steady Navier-Stokes Equation

For the steady Navier-Stokes equation, which incorporates the effects of viscosity, the Green's function satisfies a modified equation:

∇² G(x, y) + δ(x, y) - Re∇² G(x, y) = 0

where Re is the Reynolds number, a dimensionless parameter characterizing the flow regime. The solution to this equation is given by:

G(x, y) = K_0(Re |x - y|)

where K_0 is the modified Bessel function of the second kind of order zero. This Green's function represents the velocity perturbation due to a unit point force in the steady Navier-Stokes flow.

Applications

Green's functions are widely used in various fluid dynamics applications, including:

• Solving steady flow problems: By superposing the contributions from multiple point sources, one can construct the velocity and pressure fields for complex flow geometries.
• Analyzing flow singularities: Green's functions provide insights into the behavior of the flow near singularities, such as corners and cusps.
• Investigating flow stability: Green's functions can be used to study the stability of steady flow solutions and identify potential sources of instability.

In summary, Green's functions serve as powerful tools for understanding and analyzing steady fluid flows governed by the Euler and Navier-Stokes equations. Their applications span a wide range of fluid dynamics problems, from fundamental theoretical studies to practical engineering applications.

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