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Pressure-Free Surface-Induced Flow by Geometric Rectification

Zheng Li

Abstract

Pressure-driven flow collapses when confined ($u\propto r^{2}$). Asymmetry rectifies surface activity (exchange or slip gradients) into axial flux at $ΔP=0$ despite zero net exchange. Lorentz reciprocity yields a projection law: throughput is the inner product of source with a geometry kernel. Signatures include inverted ``narrower-is-faster'' scaling ($u\propto r^{-1}$), leading-order viscosity independence, length amplification ($Q\propto L$), and linear superposition, defining surface-induced flow as a pressure-free Stokes-transport mode from microfluidics to physiology.

Pressure-Free Surface-Induced Flow by Geometric Rectification

Abstract

Pressure-driven flow collapses when confined (). Asymmetry rectifies surface activity (exchange or slip gradients) into axial flux at despite zero net exchange. Lorentz reciprocity yields a projection law: throughput is the inner product of source with a geometry kernel. Signatures include inverted ``narrower-is-faster'' scaling (), leading-order viscosity independence, length amplification (), and linear superposition, defining surface-induced flow as a pressure-free Stokes-transport mode from microfluidics to physiology.

Paper Structure

This paper contains 1 section, 18 equations, 4 figures.

Table of Contents

  1. Data Availability Statement

Figures (4)

  • Figure 1: Surface-induced transport model (illustrated here for wall-normal exchange). (a) Schematic of confined channel with varying width. Walls exhibit distributed injection (green, $s>0$) and removal (red, $s<0$). (b) Geometry-dependent sensitivity profile $\phi(x)$ (downstream fraction of total hydraulic resistance), weighting local wall exchange contribution to net axial flux. In a widening channel (solid), $\phi(x)$ is biased toward the narrow end, so upstream exchange contributes more strongly than downstream exchange.
  • Figure 2: Validation of lubrication theory against 2D finite-element simulations. Relative error between 2D FEM and 1D lubrication prediction versus slenderness $\varepsilon$, consistent with expected $O(\varepsilon^2)$ accuracy.
  • Figure 3: Mechanisms of hydrodynamic rectification. (I) Symmetric null: in a uniform channel, mirror-symmetric exchange profiles cancel ($Q=0$). (II) Exchange rectification: broken symmetry in $s(x)$ yields $Q\neq 0$ even for uniform channels. (III) Geometric rectification: broken symmetry in $\kappa(x)$ yields $Q\neq 0$ even for symmetric $s(x)$, via asymmetric sensitivity $\phi(x)$.
  • Figure 4: Optimization and scaling. (a) In a uniform channel, throughput is maximized by opposing injection and removal across the midpoint. (b) Poiseuille flow weakens in confinement ($u \propto r^2$). Surface-induced flow has a geometric baseline $u_{\rm ex}\propto r^{-1}$ when $J_0$ is $r$ independent (for wall-normal exchange, $J_0=v_w$), and dominates below $r_c$ defined by $\Gamma=1$.