We explore the possibility of formation of steady internal shocks in jets around black holes. We consider a fluid described by a relativistic equation of state, flowing about the axis of symmetry (θ = 0) in a Schwarzschild metric. We use two models for the jet geometry, (i) a conical geometry and (ii) a geometry with non-conical cross-section. Jet with conical geometry is smooth flow. While the jet with non-conical cross section undergoes multiple sonic point and even standing shock. The jet shock becomes stronger, as the shock location is situated further from the central black hole. Jets with very high energy and very low energy do not harbour shocks, but jets with intermediate energies do harbour shocks. One advantage of these shocks, as opposed to shocks mediated by external medium is that, these shocks have no effect on the jet terminal speed, but may act as possible sites for particle acceleration. Typically , a jet with energy 1.8 c 2 , will achieve a terminal speed of v ∞ = 0.813c for jet with any geometry. But for a jet of non-conical cross-section for which the length scale of the inner torus of the accretion disc is 40r g , then in addition, a steady shock will form at r sh ∼ 7.5r g and compression ratio of R ∼ 2.7. Moreover , electron-proton jet seems to harbour the strongest shock. We discuss possible consequences of such a scenario.
General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. The existence of exact coherent structures in stably-stratified plane Couette flow (gravity perpendicular to the plates) is investigated over Reynolds-Richardson number (ReRi b ) space for a fluid of unit Prandtl number (P r = 1) using a combination of numerical and asymptotic techniques. Two states are repeatedly discovered using edge-tracking -EQ7 and EQ7-1 in the nomenclature of Gibson & Brand (2014) -and found to connect with 2-dimensional convective roll solutions when tracked to negative Ri b (the RayleighBenard problem with shear). Both these states and Nagata's (1990) original exact solution feel the presence of stable stratification when Ri b = O(Re −2 ) or equivalently when the Rayleigh number Ra := −Ri b Re 2 P r = O(1). This is confirmed via a stratified extension of the Vortex-Wave-Interaction (VWI) theory of Hall & Sherwin (2010). If the stratification is increased further, EQ7 is found to progressively spanwise and crossstream localise until a second regime is entered at Ri b = O(Re −2/3 ). This corresponds to a stratified version of the Boundary Reduced Equation (BRE) regime of Deguchi, Hall & Walton (2013). Increasing the stratification further, appears to lead to a third, ultimate regime where Ri b = O(1) in which the flow fully localises in all three directions at the minimal Kolmogorov scale which then corresponds to the Osmidov scale. Implications for the laminar-turbulent boundary in the (Re-Ri b ) plane are briefly discussed.
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