Abstract
An analysis of the Hagen-Poiseuille flow for a rarefied gas is presented using Grad’s equations and regularized equations in the 13 moment approximation, which provide a correction for the solution in the hydrodynamic regime. Slip boundary conditions are obtained through a simple wall model in which we take into account diffuse and specular wall-particle interactions. From the solution of the regularized equations, we recover as a special case, the velocity profile in Grad’s approximation and also an equivalent expression in the hydrodynamic regime. In addition, other relevant variables are calculated.
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References
Brodkey, R., Hershey, H.: Transport Phenomena: A Unified Approach. Brodkey Publishing, Columbus (2003)
Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, New York (1988)
Cercignani, C., Sernagiotto, F.: Cylindrical Poiseuille flow of a rarefied gas. Phys. Fluids 9(1), 40–44 (1966). doi:10.1063/1.1761530
Davis, E., Schweiger, G.: The Airborne Microparticle. Springer, Berlin (2002)
Ferziger, J., Kaper, H.: Mathematical Theory of Transport Processes in Gases. North-Holland, Amsterdam (1972)
Ferziger, J.H.: Flow of a rarefied gas through a cylindrical tube. Phys. Fluids 10(7), 1448–1453 (1967). doi:10.1063/1.1762304
Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (2007). doi:10.1002/cpa.3160020403
Loyalka, S.K.: Kinetic theory of thermal transpiration and mechanocaloric effect. II. J. Chem. Phys. 63(9), 4054–4060 (1975). doi:10.1063/1.431847
Loyalka, S.K., Hamoodi, S.A.: Poiseuille flow of a rarefied gas in a cylindrical tube: Solution of linearized Boltzmann equation. Phys. Fluids A, Fluid Dyn. 2(11), 2061–2065 (1990). doi:10.1063/1.857681. http://link.aip.org/link/?PFA/2/2061/1
Mortensen, N.A., Bruus, H.: Universal dynamics in the onset of a Hagen-Poiseuille flow. Phys. Rev. E 74(1), 017301 (2006). doi:10.1103/PhysRevE.74.017301
Niven, E.W.: The Scientific Papers of James Clerk Maxwell on Stresses in Rarified Gases Arising from Inequalities of Temperature. Dover, New York (1965)
Shah, R.K., London, A.L.: Laminar Flow Forced Convection in Ducts. Academic Press, New York (1978)
Shen, C.: Rarefied Gas Dynamics. Fundamentals, Simulations and Micro Flows. Springer, Berlin (2005)
Struchtrup, H.: Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory. Springer, Berlin (2005)
Struchtrup, H., Torrilhon, M.: Regularization of grad’s 13 moment equations: Derivation and linear analysis. Phys. Fluids 15(9), 2668–2680 (2003). doi:10.1063/1.1597472
Thatcher, T., Zheng, Y., Struchtrup, H.: Boundary conditions for grads 13 moment equations. Prog. Comput. Fluid Dyn. 8(15), 69–83 (2008). doi:10.1504/PCFD.2008.018080
Valougeorgis, D., Thomas, J.R.: Exact numerical results for Poiseuille and thermal creep flow in a cylindrical tube. Phys. Fluids 29(2), 423–429 (1986). doi:10.1063/1.865725
Wang, H., Wang, Y.: Influence of three-dimensional wall roughness on the laminar flow in microtube. Int. J. Heat Fluid Flow 28(2), 220–228 (2007). doi:10.1016/j.ijheatfluidflow.2006.08.005
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Martínez, M.A. Hagen-Poiseuille Flow Solutions in Grad-Type Equations. J Stat Phys 142, 710–725 (2011). https://doi.org/10.1007/s10955-011-0142-x
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DOI: https://doi.org/10.1007/s10955-011-0142-x