**Title : The generalized Onsager model for a binary gas mixture with swirling feed**

**Author(s) : Dr. Sahadev Pradhan and Prof. Viswanathan Kumaran**

* **Affiliation : Department of Chemical Engineering, Indian Institute of Science, Bangalore-560 012, India*

Abstract :

The Onsager model for the secondary flow field in a high speed rotating cylinder is extended for a binary gas mixture to incorporate the effect of the angular momentum in the feed gas in a high speed rotating cylinder. The base flow is an isothermal solid body rotation in which there is a balance between the radial pressure gradient and the centrifugal force density for each species. For the secondary flow, the mass, momentum and energy equations in axisymmetric coordinates are expanded in an asymptotic series in a parameter ε = (Δm/m_{av} ), where Δm is the difference in the molecular masses of the two species, and the average molecular mass m_{av} is defined as m_{av} = ((ρ_{w1} m_{1} + ρ_{w2} m_{2} )/ρ_{w} ), where ρ_{w1} and ρ_{w2} are the mass densities of the two species at the wall, and ρ_{w} = ρ_{w1} + ρ_{w2} . The equation for the master potential and the boundary conditions are derived correct to O(ε^{2} ). The leading order equation for the master potential contains a self-adjoint sixth order operator in the radial direction which is different from the generalized Onsager model (Pradhan & Kumaran (JFM-2011), Kumaran & Pradhan (JFM-2014) ), since the species mass difference is included in the computation of the density, viscosity and thermal conductivity in the base state. This is solved, subject to boundary conditions, to obtain the *O(1)* approximation for the secondary flow. The O(ε) and O(ε^{2} ) equations contain inhomogeneous terms which depend on the lower order solutions, and these are solved in a hierarchal manner to obtain the O(ε) and O(ε^{2}) corrections to the master potential. A similar hierarchial procedure is used for the Carrier-Maslen model for the end-cap boundary layers.

The results of the Onsager hierarchy, up to O(ε^{2} ), are compared with the results of direct simulation Monte Carlo (DSMC) simulations for a binary hard-sphere gas mixture for secondary flow due to a combination of wall temperature gradient, as well as mass and momentum sources in the flow with angular momentum in the feed gas. There is excellent agreement between the solutions for the secondary flow correct to O(ε^{2} ) and the simulations, to within 10 %, for the stratification parameter *A = *4.633, the Reynolds number *Re* = 10^{4} , and the aspect ratio (length/diameter) of the cylinder is as low as 2, and the secondary flow velocity is as high as 0.2 times the maximum base flow velocity and the ratio (2Δm/(m_{1} + m_{2} )) is 0.0085, with the angular momemtum of the feed gas F_{θ*} = F_{θ} /(ΩR) = 0.2. Here, the Reynolds number *Re* = (ρ_{w} Ω R^{2} ) / μ, the stratification parameter *A = (m Ω ^{2} R^{2} )/(2 k_{B} T ),* R and Ω are the cylinder radius and angular velocity, m is the molecular mass, ρ

_{w}is the wall density, μ is the viscosity and T is the temperature. The leading order solutions do capture the qualitative trends, but are not in quantitative agreement.

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