Pressure Poisson Equation is just another form of satisfying incompressibility constraint.

The Pressure Poisson Equation is derived from the Navier-Stokes equation, which describes the motion of fluid. The incompressibility constraint states that the divergence of the velocity field must be zero, which means that the fluid cannot be compressed or expanded. Mathematically, this can be expressed as:

$$ \nabla . u = 0 $$

where ∇ is the divergence operator and u is the velocity field.

To satisfy this constraint, we introduce a scalar function p called the pressure. The Pressure Poisson Equation can then be derived as follows:

1. We take the divergence of the Navier-Stokes equation:

$$ \nabla . (\rho u) = \nabla . (-\nabla p + \rho f) $$

where ρ is the fluid density and f is the body force per unit mass.

1. Using the product rule for the divergence, we can simplify this to:

$$ \rho (\nabla . u) = -\nabla^2 p + \nabla.(\rho f) $$

1. We substitute the incompressibility constraint (∇ · u = 0) into the equation:

$$ \nabla^2 p = \nabla.(\rho f) $$

1. Finally, we solve for p by taking the divergence of both sides..,

$$ \nabla^2 p = \nabla.(\rho f) $$

This is the Pressure Poisson Equation, which relates the pressure field to the body force per unit mass. It is commonly used in numerical simulations of fluid flow to ensure that the incompressibility constraint is satisfied.

For more details on math behind PPE, look at http://www.thevisualroom.com/poisson_for_pressure.html

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Yes, this is helpful! The detailed explanation and proofs provided for the interpretation of pressure as a Lagrange multiplier in both the Stokes and Euler equations give a clear mathematical foundation for understanding this concept. Here’s a summary and further clarification of the key points:

Stokes Equation