STM-D-1016Paper1999Designed, not yet built

An extended Navier-Stokes algorithm and the challenges of relativistic fluid dynamics

Paul Murad

Summary and citation · read the original at the source · none found

In one page

Fluid dynamics has a dividing line every engineer knows. Below the speed of sound the governing equations behave one way — smooth, diffusive, everything connected to everything. Above it they behave another way, and shocks appear. Paul Murad, presenting to the AIAA’s 37th Aerospace Sciences Meeting at Reno in January 1999, set out an algorithm that carries the Navier-Stokes conservation equations across that line, and then asked what becomes of the whole picture when the flow itself is relativistic. The method he built — he later called it the Method of Potential Surfaces — rewrites each steady-state conservation equation as a Poisson equation when the flow is subsonic, and as an inhomogeneous wave equation when it is supersonic, so that one piece of machinery covers both regimes. For Murad this was never only a computing convenience. In the papers he wrote next, the same mathematics is the bridge he throws between fluid flow and gravitation: Newtonian gravity plays the subsonic part, wave-equation gravity the supersonic part, and the shock front between them is where he goes looking for propulsion.

Why it matters hereChapter 5 reads the vacuum as a fluid, and a formulation that handles the smooth regime and the shock-carrying regime with one set of potentials is a working tool for exactly that reading. Chapter 8 gets what Murad was after: if a gravitational field can carry a shock the way a gas can, then a craft could push against the field instead of against propellant.

What it claims

01The paper’s declared subject is an extended algorithm for the Navier-Stokes conservation equations, together with the difficulties that relativistic flow presents to that formulation. It was presented as AIAA 99-0562 at the 37th Aerospace Sciences Meeting and Exhibit at Reno, Nevada, in January 1999, with Paul Murad as sole author.Title and bibliographic record, AIAA 99-0562, DOI 10.2514/6.1999-562
Designed, not yet built
02The algorithm is what Murad names the Method of Potential Surfaces: it converts each steady-state conservation equation into a set of Poisson equations for subsonic flow, or into inhomogeneous wave equations for supersonic flow. Murad points back to this 1999 work as the earlier effort in which he defined it.Murad’s own description of the earlier work, in Closed-Form Solutions to the Transient/Steady-State Navier-Stokes Fluid Dynamic Equations, AIP Conference Proceedings 813, pages 1264 to 1271, 2006, opening sentences of the abstract
Published and peer-reviewed
03The change of equation type at the sonic point is ordinary continuum mechanics, and it is the fact the method is built on: steady flow below the speed of sound is governed by elliptic equations, above it by hyperbolic ones, which is why shocks exist on one side of the line and not on the other. What Murad adds is a single potential formulation that spans the two.Standard continuum result, restated as the premise of the method in AIP Conference Proceedings 813, 1264 (2006)
Settled physics
04The intended reach of the equations is far wider than a wind tunnel. Murad lists their range as internal flows inside a propulsion system or a reactor, external flows around spacecraft, plasmas and galactic gas dynamics, and says the same methodology can be carried on to chemical reactions, turbulence, and coupling with Maxwell’s equations for magnetohydrodynamic propulsion.AIP Conference Proceedings 813, 1264 (2006), abstract — the continuation of the 1999 algorithm
Designed, not yet built
05The relativistic half of the title is the harder half, and Murad stayed with it. In the companion paper he published the following year he reports that the pseudo-fluid-dynamic processes approach near steady-state conditions at light speed, argues that Einstein’s assumptions treat only sub-light conditions and can be extended, and proposes an electromagnetic propulser built on Jefimenko’s gravity and cogravity wave equations.Hyper-light dynamics and the effects of relativity, gravity, electricity and magnetism, Acta Astronautica 47, pages 575 to 587, July 2000, abstract
Designed, not yet built
06What to watch is where the fluid analogy is meant to pay off. Murad’s later statement of the programme is that Newtonian gravitation behaves mathematically like subsonic flow, while the wave-equation gravity laws from Jefimenko to Einstein behave like supersonic flow — so a region where distinct gravitational fields of different strength merge could carry a gravitational shock, and a propulsion system exploiting that shock would produce thrust. The observation that would open the question is an experiment that creates an inhomogeneous gravitational field sharp enough for the shock to appear.Gravitational Shocks, Shock Waves, and Exotic Space Propulsion, Journal of Physical Mathematics 8, 2017, abstract
What to watch

The way in

https://doi.org/10.2514/6.1999-562SOURCE NOT REACHED. AIAA holds this conference paper closed and it was not reached in any form on 2026-09-08. Unpaywall and OpenAlex both report open access status closed with no repository location; Semantic Scholar returns the record but states in its own disclaimer that the abstract field has been elided by the publisher; OpenAIRE carries the Crossref and Microsoft Academic Graph records with no description; colab.ws carries the bibliographic stub only; the AIAA reading room answered a bot challenge rather than a page; and a Wayback sweep of the publisher URL found only a redirect capture. So there is no abstract for this paper anywhere the pipeline could reach, and NO TEXT OF ANY KIND IS REPRODUCED HERE. WHAT THE SHEET IS WRITTEN FROM. Two things, and each claim below says which. First, the bibliographic record: AIAA 99-0562, 37th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 1999, DOI 10.2514/6.1999-562, single author Paul Murad. Second, Murad’s own later papers, in which he describes this work as his earlier effort and names the method it introduced — Closed-Form Solutions to the Transient/Steady-State Navier-Stokes Fluid Dynamic Equations, AIP Conference Proceedings 813, pages 1264 to 1271 (2006); Hyper-light dynamics and the effects of relativity, gravity, electricity and magnetism, Acta Astronautica 47, pages 575 to 587 (2000); and Gravitational Shocks, Shock Waves, and Exotic Space Propulsion, Journal of Physical Mathematics 8 (2017). Where a claim rests on one of those rather than on the 1999 paper itself, the locator names it. AUTHOR. Paul A. Murad publishes as Paul Murad and as P. A. Murad; no affiliation is asserted here.

How to cite it

Paul Murad (1999) An extended Navier-Stokes algorithm and the challenges of relativistic fluid dynamics. doi:10.2514/6.1999-562

Where it sits in the curriculum

The vacuum as a quantum fluid Inertial mass reduction and transmedium craft