Stationary Couette-type flows in relativistic fluids

TL;DR

The paper addresses stationary, planar-symmetric Couette-type flows in relativistic fluids and shows that heat flux cannot be neglected because the inertia of heat contributes to momentum density. By analyzing a generalized Couette setup with finite chemical potential in the Eckart frame and transforming to the Landau frame, it derives an exact analytic symmetric solution with explicit temperature and heat-flux profiles, revealing a universal velocity profile that is independent of the heat conductivity when η is uniform. In the Landau frame, it derives coupled equations for the velocity components and provides both analytic small-R approximations and a method to solve with boundary conditions, highlighting how energy transport causes boundary-crossing of fluid parcels. The work emphasizes the interplay of heat inertia, viscous heating, and boundary conditions in relativistic hydrodynamics, with potential relevance to high-energy or astrophysical relativistic flows, while acknowledging that turbulence may arise at large velocities and the results are idealized stationary solutions. $u(x) = anigg(rac{2x}{L} an^{-1}igg(rac{v}{ oot 0pt ext{...}}{}igg)igg)$ provides a concrete symmetric profile, and the Landau-frame expressions show explicit coupling between velocity components through the heat flux.

Abstract

We investigate a class of stationary, planar-symmetric solutions of relativistic hydrodynamics, in which a dissipative fluid is confined between two parallel plates that move relative to each other and/or are maintained at different temperatures. We find that neglecting the heat flux leads to qualitatively incorrect flow profiles, even in systems with temperature-independent viscosity. This arises from the fact that, in special relativity, the heat flux itself contributes to the momentum density (the so-called "inertia of heat"). This effect is most evident in the Landau frame, where the fluid removes the excess energy generated by viscous heating by streaming across the boundaries. The analysis is further extended to the limit of vanishing chemical potential.

Figures (6)

• Figure 1: Left panel: Classical Couette flow. Two parallel plates (brown) are in relative motion, forcing a fluid confined between them (blue) to undergo a shear flow. Right panel: Generalization that accounts for heat exchanges. Now the plates have some temperatures $T_\pm$ (often equal), and they can absorb the excess heat (red arrows) generated by viscous dissipation.
• Figure 2: Velocity profile of a relativistic fluid with uniform viscosity undergoing a generalized Couette flow (see Fig. \ref{['fig:sketch']}), for equal boundary temperatures ($T_+=T_-$) and with plate velocities $v=0.9$ (left panel) and $v=0.9999$ (right panel). The solid blue line shows the exact solution \ref{['exactone']}, the red line Rogava's model \ref{['rogG']} (which neglects the heat flux Rogava1996), and the dashed line the Newtonian result, where the speed increases linearly with $x$.
• Figure 3: Temperature (left panel) and heat flux (right panel) profiles of a relativistic fluid with uniform $\eta$ and $\chi$ undergoing a generalized Couette flow (see Fig.\ref{['fig:sketch']}), for equal boundary temperatures ($T_+=T_-$), and with plate velocities $v=0$ (dashed), $0.9$ (blue), $0.999$ (magenta), and $0.99999$ (red). The analytical expressions are provided in Eq. \ref{['temperatueandheattone']}.
• Figure 4: Comparison between Eckart's flow velocity $u^j/u^t$ (left panel) and Landau's flow velocity $u_L^j/u_L^t$ (right panel) of a fluid with uniform viscosity undergoing a generalized Couette flow (see Fig. \ref{['fig:sketch']}), assuming equal boundary temperatures ($T_+=T_-$) and luminal plate velocities ($v\rightarrow 1$). The analytical expressions are given by Eq.s \ref{['exactone']} and \ref{['landizia']}, where the ratio \ref{['theSloope']} has been taken to be $0.2$, to highlight the differences (this ratio should be much smaller than $1$, for hydrodynamics to be applicable). Intuitively, the left panel describes the flow of particles, while the right panel describes the flow of energy.
• Figure 5: Velocity (left column) and temperature (right column) of a relativistic fluid with uniform $\eta$ and $\chi$ undergoing a generalized Couette flow (see Fig.\ref{['fig:sketch']}), for unequal boundary temperatures, and with plate velocities $v=0.6$ (upper row), and $0.999$ (lower row). The analytical expressions are provided in Eq. \ref{['uasymm']}, where we took $h=0$ (blue), $-1$ (magenta), $-100$ (red).