A toy model of viscous relativistic geometrically thick disk in Schwarzschild geometry

In this earliest study of thick accretion disks with viscosity effects, we construct stationary solutions of a relativistic geometrically thick accretion disk in the Schwarzschild spacetime under the influence of shear viscosity and the curvature of the black hole by solving the general relativistic causal Navier-Stokes equation. Motivated by the causal prescription of relativistic hydrodynamics initially introduced in M\"uller-Israel-Stewart theories, our approach adopts a simplistic path and takes into account of only shear viscosity, discarding influences of bulk viscosity and heat flow. This work investigates possible impacts of both the shear viscosity tensor and black hole curvature on the shape of a thick disk characterized by constant specific angular momentum distribution. The existence of the integrability condition of the Navier-Stokes equation has been examined in our study which further supports the existence of stationary solutions in the given set-up.