| Details of the generalized model of the parallel ion viscous stress tensor, pi∥, is presented in this work. Kinetic-based derivation of pi∥, employing a Chapman-Enskog-like (CEL) expansion of the plasma particle distribution function, is part of a broad research effort aimed at incorporating suitable kinetic physics into the physical modeling of tenuous, high-temperature (fusion-grade) plasmas. Often, this goal is achieved through the use of generalized, integral closures in the evolution equations of fluid quantities, which correspond to low-order velocity-space moments of the particle distribution functions. The primary analytical task in the formulation of pi∥ is the derivation of a drift kinetic equation (DKE) from the plasma kinetic equation (via appropriate gyro-averaging and ordering schemes). Next, the time-dependent DKE is solved for the kinetic distortion by reducing it to a system of coupled, linear equations, that results from an expansion in Legendre polynomials, and the correct exploitation of their orthogonality properties. The tensor, pi∥ , is calculated in the final step as a second-order velocity-space moment of the kinetic distortion term in the CEL expansion. This is a steady-state version of pi∥, which is valid for arbitrary collision and transit frequencies. The upgraded theory reproduces Braginskii's pi ∥ in the regime of high collisionality, and agrees with Chang and Callen's results in the nearly collisionless regime. Subsequently, a time-dependent version of pi∥ (incorporating an exact form of linearized Coulomb collision operator) is formulated, as an enhancement to the steady-state model. Numerical simulations of three known physical phenomena in plasmas, incorporating finite effects of the steady-state, generalized pi∥, are executed in slab geometry, using the NIMROD simulation code. Specifically, ion acoustic wave propagation and dissipation, stress-induced ion heating, and parallel ion momentum (or flow) flattening across magnetic islands are analyzed and modeled. Analogous simulations involving the Braginskii (local or diffusive) form of pi∥ are also executed using similar plasma parameters and magnetic configurations. Comparative studies of the simulation results are made, with the view of verifying and validating the physical capabilities and merits of the newly formulated pi ∥ as a robust and accurate closure that should be incorporated into the modeling of plasma systems of all degrees of collisionality. In the simulations, the parallel viscosity diagnosed for 1 keV plasma ≈159246.15 m 2s-1, and the parallel flow profile flattening observed when the ratio of parallel to perpendicular diffusivity, nu∥/nu⊥≈10 6, is very large. Planned future refinement of this work, in scope, theory, and applications, is also discussed. |
