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In Bull. Amer. Math. Soc. \textbf{36} (1930), 762--772, R. Vaidyanatha- swamy established a remarkable identity valid for any multiplicative arithmetic function and involving Dirichlet convolution. D. H. Lehmer (Trans. Amer. Math. Soc. \textbf{33} (1931), 945--952) introduced a very general class of arithmetical convolutions, called $\psi$-products, which include the well-known Dirichlet products, Eckford Cohen's unitary convolutions, and in fact Narkiewicz's so-called regular $A$-convolutions. In this paper, we establish an identical equation valid for multiplicative arithmetic functions and Lehmer's $\psi$-convolutions which yields, as special cases, all known identical equations valid for the Dirichlet and unitary convolutions, besides establishing identical equations for several new convolutions. [ABSTRACT FROM AUTHOR]

Rarefied gas flows in microchannels from slip to free-molecular regimes were studied experimentally and numerically with higher order slip boundary model. The second order slip model was derived and implemented in FLUENT® to simulate the gas flow for a wide range of Knudsen numbers. An in-depth computational study to determine the flow characteristics such as, mass flow rate variation, centreline pressure distribution and streamwise and normal velocity distribution were performed for Knudsen numbers between 0.1 and 10. The model was validated with accurate and high-resolution experimental mass flow rate data generated using microchannels by passing nitrogen. The channels were fabricated in silicon wafers using photolithography and etching techniques. For Knudsen numbers (Kn)  ⩽  1, the measured mass flow rate values are in good agreement with simulated data for the slip coefficients C1 as 1.31 and C2 as 0.11 with an average deviation of less than 2%. For higher Knudsen numbers (Kn  >  1), the simulated normalized volumetric flow rate values agree with experimental data available in the literature with an average deviation of 3.3%, whereas, other models deviate from 10% to 21%. Also, the present model was able to capture Knudsen minimum effect and the normalized volumetric flow rate plot showed that the present experimental and numerical studies predict accurately the characteristics of higher Knudsen number flows. [ABSTRACT FROM AUTHOR]