Static Dark Fluid Thin Shells in Schwarzschild-de Sitter Spacetimes: Stability and Black Hole Shadows

We study the existence and radial stability of static, spherically symmetric thin shells separating two Schwarzschild–de Sitter spacetimes with parameters $(m_{\pm },{\mathrm{\Lambda }}_{\pm })$. Using the Israel junction formalism and a linear barotropic equation of state $p=\lambda (\sigma -{\sigma }_1)c^2$, we decouple the sound speed $c_s^2=\lambda c^2$ from the equilibrium equation-of-state parameter $w_0\equiv p_0\mathrm{/}({\sigma }_0{c}^2)$ and derive the effective potential governing radial dynamics. For observationally motivated parameters, stable configurations with ${\sigma }_0>0$ and $0<\lambda \le 1$ exist only when $m_{+}\mathrm{/}{m}_{-}>1$. Three distinct stability windows emerge, when $\lambda =1$: $-3\mathrm{/}7\lesssim w_0\lesssim 1\mathrm{/}2$ for ${\mathrm{\Lambda }}_{+}={\mathrm{\Lambda }}_{-}$, $-2\mathrm{/}3\lesssim w_0\lesssim 1\mathrm{/}2$ for ${\mathrm{\Lambda }}_{+}>{\mathrm{\Lambda }}_{-}$ and $0\lesssim w_0\lesssim 1$ for ${\mathrm{\Lambda }}_{+}<{\mathrm{\Lambda }}_{-}$. Positive-pressure shells ($w_0>0$) reside near the photon sphere, whereas negative-pressure shells ($w_0<0$) extend outward, reaching either the cosmological horizon or the static radius. Stability relies on the variation of $w(\sigma )$ with the surface energy density. Negative pressure (tension) stabilizes the system because the tension increases during expansion. Conversely, positive pressure stabilizes the system because the pressure increases during contraction. Finally, a static, stable, dark, fluid thin shell acts as a gravitational refractive layer that enlarges the black hole’s shadow for a distant, static observer outside the shell. The effect depends on the shell radius $R_0$, the background parameters $(m_{\pm },{\mathrm{\Lambda }}_{\pm })$, and the equation-of-state. Dark fluid shells can be considered as theoretical toy models that illustrate qualitative effects. Future high-resolution black hole shadow observations could, in principle, use such models to explore how different equations of state might influence observable signatures.