Microfluidics and Slip flow

Microfluidic devices are characterised by small length-scales L that become comparable with the molecular mean free path λ. The relevance of the Navier-Stokes equations, derived as an asymptotic expansion for small Knudsen number, Kn = λ/L ≪ 1, thus becomes questionable for describing flows in these devices. Indeed, the Navier-Stokes equations cannot capture rarefied, micro-scale effects such as the Knudsen layer at boundaries.

Microdevices typically operate in the isothermal slip-flow regime, characterised by Knudsen numbers in the range 0.01<Kn<0.1, and a Mach number Ma≪ 1. The Navier– Stokes equations remain valid in the bulk of the flow in this parameter regime, but they must be supplemented by suitable slip boundary conditions.

kramer
The Knudsen layer and velocity in Kramer’s problem (a), and the velocity defect function (b). Taken from Reis and Dellar 2012)

The application of the LBM to rarefied flows has received a lot of attention, presumably because it approximates the continuous Boltzmann equation by restricting the particle velocities to a finite discrete set. However, the standard 9-veclocity LBM is restricted to capturing just the first few moments of the solutions of the true Boltzmann equation. Thus one is essentially computing solutions to the Navier–Stokes equations, but with a Knudsen number appreciably larger than zero.  The applicability of the LBM to the slip-flow regime is made more complicated by the existence of a purely numerical slip that exists for integer lattices.

slipflow
Streamwise velocity, spanwise velocity, and the centreline pressure deviation for flow in a microchannel using LBM moment-based boundary implementations. Taken from Reis and Dellar (2012).

Realising that the D2Q9 LBM cannot capture Knudsen layers (or any other kinetic effects in the velocity field not found in the Navier-Stokes equations with slip boundary conditions), we decided to impose Navier-Maxwell conditions directly and precisely at grid points using the moment-based method.  This allowed us to capture the nonlinear streamwise pressure variation and the cross-channel velocity component, as well as the streamwise velocity and volume flux. The former effects are both absent from almost all previous work that approximated the pressure difference using a uniform body force. Furthermore,  the velocity components converged towards their asymptotic solutions for long micro channels with second-order accuracy.

The absence of any deviation from the Navier–Stokes parabolic profile in the streamwise velocity does not completely preclude kinetic effects. The Burnett equations and the Grad 13 moment equations are two PDE systems that include corrections to the Navier–Stokes equations at order Kn².  Neither system shows deviations from a linear profile in Couette flow, but the Burnett equations show Knudsen layer behaviour in higher moments that do not contribute to the velocity field. If you plot the tangential component of the deviatoric stress as predicted by the LBM in planar channel flow then you will see a quadratic profile that fits the order  Kn² Burnett solution perfectly, provided boundary conditions that are consistent with the underlying PDE at Burnett order are imposed. If inconsistent conditions are imposed (that do not consider the Burnett contribution) then spurious oscillations are generated at the boundary nodes.  Note that that the tangential Newtonian stress (in the Navier-Stokes equations) is zero in planar channel flow.

stress
Plot of the tangential stress in planar channel flow using the LBM with moment–based boundary conditions (“Numerical”) compared with the solution from the discrete Boltzmann equation (“Analytical”). The plot on the left uses the original Navier-Stokes stress condition and the plot on the right the Burnet stress condition. Note that the profile would be flat if the stress were Newtonian. From Reis (2019).

This inspired us to analyse deviatoric stress embedded with the LBM. We solved the LBM analytically for the stress field and found it does indeed contain an order Kn² Burnett contribution. Furthermore, the analysis shows us the finite difference stencil used by the LBM, which in term gives us an interpretation of optimal relaxation times for TRT/MRT models. Some additional information can be found in the section on moment-based boundary conditions. My former PhD student, Seemaa Mohammed is studying moment-based boundary conditions with slip in her PhD thesis and has applied the method to cavity flows. Another doctoral student, Zainab Bu Sinnah, has shown that the approach can be used to study pulsatile flows with slip, too.

PulsatileKn
Simulated and exact solutions of the velocity in pulsatile flow with slip. The non-dimensional slip lengths (analogous to the Knudsen number) from left the right: 0.388, 0.194, 0,0194. Taken from Sinnah, Graham, and Reis (2018).

 

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