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A Boundary Layer Approach in the Modelling Flows in Microscale

In this paper we discuss a gradient model as an extension of classical fluid mechanics with a higher order of spatial and time derivatives of velocities, pressure and acceleration. Size effects, characterized by a mean free path, or the size of a microstructural lattice are captured by this model making feasible its applications in micro and nano-scales. One of the challenges of gradient models is to keep the number of additional constitutive parameters to a minimum. In contrast to the number of gradient models, requiring a large quantity of phenomenological constants, the present model requires a one additional constant, linked to the microscale characteristic length for the steady state flow, and two additional parameters in a non-steady counterpart. In order to account for the micro structure the higher order continuum approximation is built based on a continualization strategy inside microstructural volume and a relating variational principle. This model does not require introduction of a gyration vector, or coupled stresses, although these quantities could be deduced by coordinating with the micropolar or another gradient theory. The derived model exactly matches the non-steady linearized micropolar fluid flow model (small gyrations and velocity distributions), giving analytical interpretation of typical micropolar empirical coefficients in terms of a Knudsen number and a time characteristic constant. An asymptotic behaviour of a two dimensional flow through a microchannel is investigated via rigorous singular perturbation analysis with respect to the small parameter, proportional to the Knudsen number. This small parameter is involved in both partial differential equations and the Maxwell’s slip boundary conditions. A singular perturbation technique reveals a thin boundary layer (Knudsen layer) region near the solid boundary described by the inner solution, whereas the outer solution relates to the classical NavierStokes model. For the boundary layer solution is presented in an analytical form, whilst the outer region is modelling using analytical solution in case of a canonical, or a numerical solution for a non-canonical domain respectfully. Calculational results obtained compare favourably with the currently available experimental data.