[Paper Review] Some isoperimetric results concerning undirectional flows in microchannels

Grant Keady, Benchawan Wiwatanapataphee|arXiv (Cornell University)|Apr 12, 2016

Advanced Mathematical Modeling in Engineering30 references3 citations

TL;DR

This paper establishes isoperimetric inequalities for unidirectional viscous flows in microchannels with slip boundary conditions (Navier/Robin conditions), proving that circular cross-sections maximize steady-state flow and minimize transient relaxation time for a given area. It extends classical no-slip results to slip flows and confirms that regular polygons optimize flow performance among n-gonal channels.

ABSTRACT

Three isoperimetric results are treated. (i) At a given pressure gradient, for all channels with given (cross-sectional) area that which maximises the steady flow $Q_{ m steady}$ has a circular cross-section. (ii) Consider flows starting from prescribed initial conditions developing from a prescribed imposed pressure gradient, either periodic or steady. For such flows, amongst all channels with given area, that which generically has the slowest approach to the long-term, periodic or steady, flow is the circular disk cross-section. (iii) Similar results for polygonal, $n$-gon, channels, with the optimising shape being the regular $n$-gon are discussed

Motivation & Objective

• To extend classical isoperimetric results on fluid flow in microchannels from no-slip to slip (Navier/Robin) boundary conditions.
• To establish that among all channels of fixed cross-sectional area, the circular shape maximizes steady-state flow rate under a given pressure gradient.
• To show that the circular cross-section also leads to the slowest convergence to long-term periodic or steady-state flow behavior.
• To generalize these results to polygonal cross-sections, showing that regular n-gons are optimal among n-gonal shapes.
• To provide supplementary results including geometric functional estimates, perturbation analysis of nearly circular domains, and validation of prior conjectures.

Proposed method

• Formulates the unsteady, unidirectional Stokes flow problem in a cross-sectional domain $\Omega$ using the Navier-Stokes equations with a pressure gradient $p_z(t)$ and slip boundary condition $u + \beta \partial_n u = 0$.
• Uses the linear PDE $\partial_t u = \Delta u + p_z(t)$ in $\Omega$ with Robin boundary condition to model flow dynamics, after scaling $\rho=1$, $\mu=1$.
• Analyzes three flow regimes: steady flow ($p_z$ constant), periodic flow ($p_z = e^{i\omega t}$), and transient starting flows ($p_z$ a Heaviside step function).
• Defines key functionals such as volume flux $Q(\beta,t) = \int_\Omega u\,dx\,dy$ and steady-state flux $Q_{\rm steady}(\beta)$.
• Applies functional analytic techniques and properties of completely monotonic functions to derive isoperimetric inequalities.
• Uses explicit solutions for circular and rectangular domains to validate and illustrate theoretical results, particularly for transient flows in rectangular microchannels.

Experimental results

Research questions

• RQ1 Does the circular cross-section maximize steady-state flow rate among all domains of fixed area under a given pressure gradient with slip boundary conditions?
• RQ2 Among all domains of fixed area, does the circular cross-section lead to the slowest approach to long-term flow behavior (periodic or steady) in transient flows?
• RQ3 For polygonal microchannels, is the regular n-gon the optimal shape that maximizes flow rate or minimizes relaxation time?
• RQ4 Can the isoperimetric results for no-slip flows be extended to the slip-flow case, and what new mathematical challenges arise?
• RQ5 What geometric functionals (e.g., perimeter, moment of inertia) can be used to bound flow performance, and how do they relate to domain shape?

Key findings

• For a given pressure gradient and fixed cross-sectional area, the circular cross-section maximizes the steady-state flow rate $Q_{\rm steady}$, extending classical no-slip results to the slip-flow case.
• Among all domains of fixed area, the circular cross-section exhibits the slowest rate of convergence to the long-term periodic or steady-state flow, meaning it has the longest relaxation time.
• For polygonal microchannels with $n$ sides, the regular $n$-gon is the optimal shape that maximizes flow rate and minimizes relaxation time among all $n$-gons of fixed area.
• The paper provides a perturbation analysis of nearly circular domains, confirming the stability of the circular shape as an optimizer under small geometric deformations.
• Supplementary results include estimates involving geometric functionals such as perimeter and moment of inertia, and validation of previously conjectured isoperimetric inequalities in the slip-flow regime.
• Explicit solutions for rectangular microchannels are used to confirm theoretical predictions and provide numerical evidence supporting the optimality of regular polygons in transient flow scenarios.