The numerical optimisation is now performed for symmetric branching of rarefied gas flows. For dilute gases, the slip length can be related to the mean free pathλvia
η=Cλ, whereC= 1.11 is the first-order slip coefficient for the hard-sphere model of gases with purely diffuse molecular reflection at walls [Hadjiconstantinou, 2003]. The mass-flow-rate data is obtained from a variety of sources, including the author’s own simulations, computed using a version of low-variance deviational simulation Monte Carlo (LVDSMC) [Radtke et al., 2011]1. In Fig. 5.5, additional mass-flow-rate data is procured from the solution of the linearised Boltzmann equation [Loyalka and Hamoodi, 1990, 1991] and the S-model [Sharipov, 1996] to verify the accuracy of the LVDSMC results.
The results of the analytical and numerical optimisations, for circular cross- sections, are presented in Fig. 5.5. To demonstrate that the generalised law is valid for any number of daughter branches (N ≥ 2), Γ is plotted for N = 2, N = 3 and N = 5 and, for clarity, is normalised with respect to the continuum-flow limit (N−2/3). Again, the agreement between the numerical optimisation and the plug-
flow limit of the generalised law is excellent for each case considered. It is perhaps unexpected that a slip solution to the generalised law should converge to the same result as that of kinetic theory and LVDSMC at the free-molecular limit, given the approximate nature of slip boundary condition at such scales. However, as
˜
Rp → 0, Γ in equation (5.37) becomes independent of the slip length η, and is
thus unaffected by any inaccuracy in the slip model. Due to computational cost, molecular simulations are not performed for sufficiently large areas to see the solution meet the continuum-flow limit; but, since the results from kinetic theory converge to the solution of the slip model, agreement at the continuum-flow limit is also expected. It is well known that when the Knudsen number Kn = λ/L is much greater than∼0.1, slip solutions become inaccurate, explaining the departure in Γ between the limits. The kinetic-theory and LVDSMC results all show a minimum in Γ beneath the plug-flow/free-molecular limit. This is possibly a manifestation of the Knudsen minimum [Knudsen, 1909], a rarefied gas phenomenon that occurs when the diffusive flux starts to dominate the convective flux as length scale decreases [Mitra and Chakraborty, 2012].
1
The author’s LVDSMC simulations used periodic boundary conditions in the streamwise direc- tion, 10 deviational particles per cell, a cell size of ∆x=λ/5, and a time-step of ∆t= ∆x/c¯, where ¯
10−10 10−5 100 105 1010 0.8 0.85 0.9 0.95 1 non-dimensional area, A/2 norm ali se d optim al ar ea rat io, ΓN 2/3 N =2 N =3 N =5 generalised law plug-flow limit continuum-flow limit numerical optimisation LVDSMC Boltzmann eqn. S-model
Figure 5.5: Normalised optimal daughter-parent area ratio against non-dimensional parent area for a symmetrically branching network of channels with a circular cross section. Comparison of the analytical slip solution to the generalised law (equation (5.37)) and the numerical optimisation using data from kinetic theory [Loyalka and Hamoodi, 1990; Sharipov, 1996] and LVDSMC. Plotted for N = 2, N = 3 and
N = 5.
Similar results are also found in networks with rectangular geometries, as shown in Fig. 5.6. Again, the LVDSMC numerical optimisation agrees with the slip solution at the plug-flow limit for multiple numbers of daughter branches, and separates from the slip solution between the length-scale limits. Minimum values of Γ, below that of the plug-flow limit, are also exhibited. This minimum occurs at a larger area for the higher aspect ratio network, likely due to relative size of the characteristic length. The departure from the slip solution at the minimum increases with aspect ratio. This provides further evidence for a link between the Γ minimum and the Knudsen minimum, as the depth of the Knudsen minimum is also noted to increase with aspect ratio [Mitra and Chakraborty, 2012].
Note, although equations (5.37) and (5.39) are only approximate between the scale limits, the precise result of the numerical optimisation can be reclaimed by expressing the kinetic theory mass flow rate data in terms of flow resistance per unit
10−10 10−5 100 105 1010 0.8 0.85 0.9 0.95 1 norm ali se d optim al ar ea rat io, ΓN 2/3 non-dimensional area, A/ 2 N =2 N =5 plug-flow limit continuum-flow limit generalised law numerical optimisation square rectangle, = 10
Figure 5.6: Normalised optimal daughter-parent area ratio against non-dimensional parent area for a symmetrically branching network of channels with a rectangular cross section. Comparison of the analytical slip solution to the generalised law (equation (5.37)) and the numerical optimisation using data from kinetic theory [Loyalka and Hamoodi, 1990; Sharipov, 1996] and LVDSMC. Plotted for squares and rectangles of aspect ratioα= 10 atN = 2 and N = 5.
length,k(A). By interpolating between data points, the (symmetric) generalised law in equation (5.10) can be evaluated, but this does not afford an analytical relation.
5.3
Summary
In this chapter, a generalised optimisation principle has been derived that leads to analytical expressions for the optimum daughter-parent area ratio Γ for any shape, at any length scale, and for any number of daughter branches.
Analytical solutions have been verified with a numerical optimisation and shown that, for symmetric branching at the continuum-flow limit, this is equivalent to Murray’s law, where Γ = N−2/3. However, when applied to an asymmetrically
channels (or the daughter channels do not have equal pressure gradients), it has been shown that Murray’s law is sub-optimal. This is because, for asymmetric branching, the global optimisation of the entire network is not equal to the local optimisation of each individual channel, which Murray’s law presumes.
Unlike the generalised law presented, Murray’s law is also sub-optimal for slip flows and plug flows that occur at smaller length scales, where the optimal daughter- parent area ratio converges to Γ =N−4/5. The new optimal design relation proposed
can be used as a biomimetic design principle to be applied to a variety of micro and nanofluidic networks that require non-circular geometry, due to manufacturing constraints, and are designed for increasingly smaller scales in order to achieve a greater degree of control, functionality, and analytical and economic efficiency.