The Newton's problem assuming non-constant density of the fluid

TL;DR

This paper studies the minimal resistance problem for a body in a fluid with exponentially decreasing density, proving existence, regularity, and finite domain of radial solutions.

Contribution

It extends Newton's minimal resistance problem to variable fluid density, establishing existence and properties of solutions under these conditions.

Findings

Radial solutions exist locally with specified initial conditions.

Solutions have a finite maximal domain ending at a critical slope.

The problem is analyzed using fixed-point theorem techniques.

Abstract

This paper investigates the Newton's problem of minimal resistance for a body moving through a fluid whose density decreases exponentially with altitude. We prove the local existence and regularity of radial solutions $u(r)$ satisfying the initial conditions $u(0)=u'(0)=0$ using a fixed-point theorem. We show that the maximal domain of the solution is finite, $[0, r_M)$, terminating at a critical slope $u'(r_M) = \frac{1}{\sqrt{3}}$.