Overview
This paper presents a function of two variables derived through the application of the Lambert W Function. This function is demonstrated to satisfy Euler's Equation of Inviscid Motion under specific boundary conditions where pressure exhibits independence from spatial variables, operating within a defined domain. The work aims to address a simplified instance of the Navier-Stokes equations.
Research Context
The Navier-Stokes equations describe the motion of viscous fluid substances, and their general solution remains an open problem in mathematics and physics. Euler's Equation of Inviscid Motion represents a simplification of the Navier-Stokes equations, specifically for fluid flows where viscosity is neglected. The challenge often lies in finding analytical solutions, even for these simplified forms, particularly in complex geometries or under varying physical conditions.
Approach
The core of the methodology involves conceiving and constructing a novel two-variable function. The explicit form of this function incorporates the Lambert W Function. The researchers then generalize this function to demonstrate its property of satisfying Euler's Equation of Inviscid Motion. This generalization is subject to the conditions that the motion occurs over a specific domain and that the pressure within this system remains independent of the space variables involved.
Findings
The primary finding is the identification of a two-variable function, expressible using the Lambert W Function, which fulfills the conditions of Euler's Equation of Inviscid Motion. This satisfaction occurs under the explicit constraint that the pressure within the system is not dependent on the spatial coordinates. The domain over which this function applies is specified in the research, indicating a targeted applicability rather than a universal solution. This function thus offers a direct means to solve a specific, simplified case of the Navier-Stokes equations where inviscid flow and spatially independent pressure are assumed.