Fix sign of the radial-velocity term in the documented mixture pressure

[sbryngelson]

Description

docs/documentation/equations.md gave the Euler-Euler modified mixture pressure as

p = (1 - alpha) p_l + alpha ( <R^3 p_bw>/<R^3> - rho <R^3 Rdot^2>/<R^3> )

with a minus on the radial-velocity term. That sign is wrong. Ando (Caltech thesis, 2010) writes the mixture momentum flux as p_l - ptilde in Eq. (2.2), with

ptilde = alpha ( p_l - <R^3 p_bw>/<R^3> - rho <R^3 Rdot^2>/<R^3> )   Eq. (2.5)

Both terms are negative inside the parenthesis, so

p_l - ptilde = (1 - alpha) p_l + alpha ( <R^3 p_bw>/<R^3> + rho <R^3 Rdot^2>/<R^3> )

and the radial-velocity contribution is positive. Eq. (A.71) of the same thesis carries the two contributions in a single bracket with the same sign, which is an independent check.

This matches what m_riemann_solver_hllc.fpp computes, where the correction is folded into pres_L and pres_R in place. The documentation was wrong, not the solver.

Type of change

• Documentation

Scope

• This PR comprises a set of related changes with a common goal

Testing

Documentation only. No solver code is touched and no golden files move. ./mfc.sh precheck passes.

Notes

Reported by R. Jason Scharff in #1739, which this resolves.

[Copilot]

Pull request overview

Updates MFC’s governing-equations documentation for Euler–Euler bubbles to match the solver’s implemented modified mixture pressure formulation by correcting the sign of the radial-velocity-squared contribution.

Changes:

• Corrected the documented modified mixture pressure equation by flipping the sign of the radial-velocity term contribution from negative to positive.

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