Data-Driven Design Optimization of Streaming-Potential-Mediated Electrokinetic Transport of Viscoelastic Fluids in Microchannels

Ankan Basu; Sumanta Banerjee

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Data-Driven Design Optimization of Streaming-Potential-Mediated Electrokinetic Transport of Viscoelastic Fluids in Microchannels

Ankan Basu, Sumanta Banerjee

arXiv:2608.29939v1Aug 30, 2026physics.flu-dyncs.LG

TL;DR

Existing models do not systematically search the full four-dimensional design space or formally assess the relationship between flow rate and efficiency. This study uses a surrogate-assisted framework with NSGA-II to accelerate exploration and identify operating conditions for streaming-potential-mediated sPTT transport.

Problem
Existing models rely on partial parameter sweeps and iterative electroneutrality solutions, leaving joint global optimization of flow rate and efficiency unresolved.
Method
A machine-learning surrogate predicts streaming potential across four design variables, and NSGA-II searches operating conditions while flow rate and efficiency are computed from the predicted potential.
Results
The surrogate achieved validation L1 Loss of 3.86×10−6 and 0.68% error, while optimization identified a compact Pareto front concentrated at κ →10, Du = 0, ϵsWi2 k = 2, and ζ ≈6.
Takeaways & Limitations
Flow rate and energy conversion efficiency are positively correlated, so the same operating conditions maximize both objectives without a fundamental throughput-efficiency trade-off.
Takeaways & Limitations
Future work should add objectives such as minimizing polymeric stresses and accounting for thermal effects.

Abstract

from arXiv · show

Streaming-potential-mediated transport of viscoelastic fluids has attracted research attention owing to its applications in electrokinetic energy conversion and microfluidic transport. Existing analytical and semi-analytical models in published literature provide valuable physical insights, but require repeated numerical evaluations for exploring large design spaces and identifying the optimal operating conditions. In this work, a surrogate-assisted framework is developed for rapid design optimization of pressure-driven electrokinetic transport of simplified Phan-Thien-Tanner fluids in a slit microchannel. A high-fidelity numerical database is generated over a broad range of governing dimensionless parameters, which includes the zeta potential, the Debye parameter, the Dukhin number, and the viscoelastic parameter. A Machine Learning surrogate model is subsequently trained to accurately approximate the nonlinear relationship between the governing parameters and the streaming potential, while the volumetric flow rate and hydroelectric energy conversion efficiency were calculated from closed form equation by using the streaming potential predicted by the surrogate. This is coupled with a multi-objective optimization strategy to identify operating conditions that simultaneously maximize energy conversion efficiency and volumetric flow rate. The proposed methodology can significantly accelerate parametric exploration compared with repeated numerical simulations across different parameters and provides practical design guidelines for electrokinetic microfluidic devices. The study demonstrates the potential of combining computational fluid mechanics with data-driven surrogate modeling for efficient engineering design and optimization.

1 Introduction

The paper addresses the lack of a global, computationally practical optimization method for streaming-potential-mediated transport of viscoelastic fluids. It replaces repeated implicit numerical evaluations with a neural-network surrogate coupled to multi-objective optimization.

Existing analytical and semi-analytical sPTT models provide physical insight across electro-osmotic, pressure-driven, high-zeta, and non-isothermal conditions.
Prior studies did not jointly search the four-dimensional space of zeta potential, Dukhin number, Debye parameter, and viscoelastic parameter.
The governing pipeline is costly because streaming potential is obtained iteratively and flow-rate expressions require demanding polylogarithm-series quadrature, especially at large Debye parameter.
A feed-forward neural network learns the mapping from four dimensionless parameters to streaming potential, replacing only the expensive implicit solution while retaining closed-form flow-rate and efficiency calculations.
NSGA-II jointly maximizes volumetric flow rate and hydroelectric conversion efficiency, producing a Pareto-optimal design front across the full parameter range.

2 Literature Review

The literature provides analytical foundations and emerging machine-learning tools, but has not addressed global optimization of streaming-potential-mediated sPTT transport across all governing parameters. This work combines a neural surrogate with NSGA-II to search jointly for flow-rate and efficiency designs.

Prior analytical and semi-analytical work: Analytical research progressed from Debye–Hückel solutions toward broader viscoelastic models and exact high-zeta-potential PTT electro-osmotic solutions.
Machine learning for electrokinetic and viscoelastic flow modeling: Existing neural-surrogate and PINN studies accompany semi-analytical models but do not target the electroneutrality condition determining streaming potential from four dimensionless groups.
Multi-objective design optimization: Surrogate-assisted NSGA-II is established for multi-objective engineering design, yet this combination had not been applied to electrokinetic–viscoelastic streaming-potential transport.
Gaps and present contribution: Prior studies explored the four-dimensional parameter space through fixed-partial sweeps, yielding qualitative local optima rather than a joint global search.
Gaps and present contribution: The implicit electroneutrality solve and polylogarithm-heavy quadrature make direct integration of the semi-analytical model into many-evaluation optimizers impractical.
Gaps and present contribution: The proposed framework trains a feed-forward neural network on iteratively generated data and couples it with closed-form flow and efficiency expressions inside NSGA-II.

3.1 Governing Equations

The model describes nonlinear electrokinetic transport of an sPTT fluid through a slit channel, coupling the electric double layer, streaming-potential body force, and viscoelastic stress. The streaming potential is determined implicitly, while flow rate and energy-conversion efficiency follow from the resulting velocity field.

Channel and assumptions: The configuration is a steady, unidirectional, incompressible sPTT flow through a slit channel with thin, non-overlapping electrical double layers.The plates are located at Y = ±a, and the channel length is L with W ≪L.
Electrostatic potential: The nonlinear Poisson–Boltzmann equation determines the dimensionless EDL potential using the zeta potential ζ and Debye parameter κ.The full nonlinear solution is retained rather than using the Debye–Hückel approximation and is later used to generate surrogate training data.
Flow and constitutive equations: The Cauchy momentum equation includes an electrokinetic body force from the streaming-potential field, while the polymeric stress follows the sPTT constitutive model.The solvent viscosity is set to zero relative to the polymeric contribution, and the body force couples electrostatics to momentum balance.
Velocity field: The specialized constitutive and momentum equations reduce the velocity problem to a nonlinear dimensionless ODE with a closed-form solution involving polylogarithms.The solution satisfies no-slip wall conditions and centreline symmetry, but its repeated evaluation is computationally expensive.
Streaming potential: The streaming potential E is obtained iteratively by balancing streaming, bulk conduction, and Stern-layer conduction currents.Because the velocity profile depends on E, the electroneutrality equation is nonlinear and must be solved separately throughout the four-dimensional design space.
Performance measures: Once E is known, integrating the velocity profile gives volumetric flow rate q, while a closed-form ratio defines hydroelectric energy-conversion efficiency η.These quantities form the two objectives used in the subsequent multi-objective optimization.

3.2 Surrogate Architecture

A feed-forward neural network surrogate approximates the nondimensional streaming potential from four governing parameters. It is trained on numerically generated samples so that closed-form expressions for downstream quantities can be evaluated without repeatedly solving the implicit streaming-potential problem.

Network architecture: The surrogate is a three-hidden-layer neural network that maps κ, ϵsWi2_k, ζ, and Du to the predicted nondimensional streaming potential E.The architecture is presented in Fig. 2.
Training database: 2000 parameter combinations were generated by Latin Hypercube Sampling over κ ∈ [10, 40], ϵsWi2_k ∈ [0, 2], ζ ∈ [1, 10], and Du ∈ [0, 25].Each sample used the governing equations to calculate the nondimensional streaming potential.
Data split: 1800 data points were used for training and the remaining 10% for validation to detect overfitting.The validation set compares performance on held-out data against the training set.
Training and evaluation: L1 loss was used for training, with percentage error as an additional evaluation metric.The percentage-error formulation averages before division to avoid division by zero when the true value is zero.

3.3 Optimization Problem Formulation

The design problem jointly maximizes dimensionless flow rate and hydroelectric efficiency over four bounded dimensionless parameters. NSGA-II searches for the Pareto-optimal designs while using the surrogate to avoid expensive repeated evaluations of the implicit governing model.

Optimization objectives: The optimization simultaneously maximizes q and η over the four-dimensional design vector (ζ, κ, ϵsWi2_ik, Du) within physically relevant bounds.The objectives are not generally maximized by the same design.
Pareto formulation: The target is the Pareto-optimal set, where no objective can improve without degrading the other.This avoids forcing the two objectives into a single weighted scalar before optimization.
NSGA-II procedure: NSGA-II encodes each candidate as a chromosome of four decision variables and evolves an initial population of 200 candidates over 100 generations.The algorithm uses non-dominated sorting and crowding-distance estimation to preserve objective trade-offs.
Surrogate-assisted evaluation: The surrogate supplies the streaming potential for each candidate, after which closed-form expressions evaluate q and η during optimization.This replaces repeated electroneutrality iterations and polylogarithm-heavy velocity evaluations inside the optimizer.

4.1 Surrogate Results

The surrogate reproduces streaming potential with approximately 0.6% error, enabling calculation of flow rate and efficiency across the design space. Results show that increasing Dukhin number reduces E, q, and η, whereas increasing the viscoelastic parameter increases E and q, with stronger effects at lower κ.

Surrogate accuracy: Approximately 0.6% error was achieved on both training and validation datasets, with close losses suggesting significant overfitting is unlikely.The surrogate learned the relationship between the dimensionless inputs and streaming potential.
Variation with Dukhin number: Increasing Dukhin number Du causes monotonic exponential decay of streaming potential E across all κ values.The magnitude of E also decreases as κ increases because thinner EDLs contain fewer mobile ions to generate streaming current.
Variation with Dukhin number: Increasing Dukhin number Du monotonically decreases volumetric flow rate q, with the strongest reduction at κ = 10.At higher κ, the pressure-driven parabolic core attenuates the influence of streaming potential on the velocity field.
Variation with Dukhin number: Increasing Dukhin number Du exponentially decreases efficiency η, with the largest values at κ = 10 and the smallest at κ = 40.Stern-layer conduction reduces useful electrical power because charge carriers bypass the external load and dissipate energy as heat.
Variation with viscoelastic parameter: Increasing the viscoelastic parameter ϵsWi2_k increases streaming potential E approximately linearly, with the highest E at κ = 10 and lowest at κ = 40.The sPTT shear-thinning response increases velocity gradients and advective counterion transport, requiring a larger induced streaming potential.
Variation with viscoelastic parameter: Increasing the viscoelastic parameter ϵsWi2_k increases volumetric flow rate q linearly, with the strongest influence when κ = 10.A larger EDL fraction allows shear-thinning near the walls to affect the integrated flow rate more substantially.

4.1.6 Variation of Energy Conversion Efficiency with Viscoelastic Parameter

Efficiency increases only weakly with viscoelasticity because electrical and hydraulic power rise at comparable rates, while zeta potential produces bell-shaped responses in streaming potential, flow rate, and efficiency.

Efficiency increases weakly and nearly linearly with the viscoelastic parameter, with the highest value at κ = 10 and the lowest at κ = 40.
Viscoelasticity raises electrical and hydraulic power at roughly comparable rates, causing partial cancellation in the efficiency ratio.The efficiency depends on isE/q, while velocity and flow rate scale similarly with viscoelasticity.
The main benefit of increasing viscoelasticity is higher flow rates and streaming potentials rather than a major improvement in energy conversion efficiency.
Streaming potential, flow rate, and efficiency each show bell-shaped dependence on zeta potential, peaking near ζ ≈8.5–9, with maxima at κ = 10.
The zeta-potential peak reflects a balance between increasing streaming current at higher surface charge and electroviscous retardation beyond the optimum.
Efficiency at κ = 10 is approximately an order of magnitude higher than at κ = 40 because thicker EDLs provide more mobile ions for streaming-current generation.

4.2 Pareto-Optimal Design Space

NSGA-II converges to a compact Pareto-optimal region rather than a broad trade-off surface. The shared favorable settings fix three parameters near their advantageous extremes, leaving zeta potential as the principal compromise variable.

The NSGA-II hypervolume plateaued around generation 16 during a 100-generation search, supporting convergence of the reported non-dominated set.
All non-dominated solutions occur at κ →10, ϵsWi^2k →2, and Du →0, while ζ remains confined to [5.75, 5.90].
The full four-dimensional search sharpens prior qualitative observations by showing that κ and Du act concordantly on both objectives, leaving ζ as the genuine compromise variable.
Three governing groups can be set toward thicker EDLs, stronger viscoelasticity, and negligible Stern-layer conductivity without sacrificing either flow rate or efficiency.

4.3 Global Sensitivity Analysis

Sobol analysis identifies κ as the dominant first-order driver of flow rate and efficiency, while viscoelasticity affects flow rate but not efficiency at first order. The results explain why optimization collapses toward shared κ and Du settings, with ζ retaining the main trade-off.

For efficiency, κ leads with S1 ≈0.29, followed by Du at S1 ≈0.25 and ζ at S1 ≈0.17.
The viscoelastic parameter has S1 ≈0.00 for efficiency despite a non-trivial influence on flow rate.
Viscoelasticity reshapes the velocity profile without proportionally changing the streaming-current-to-flow-rate ratio that determines efficiency.
Because κ and Du affect both objectives in the same direction, ζ is the only parameter requiring a genuine design compromise.

5 Conclusion

The paper develops a neural-network surrogate and NSGA-II framework for rapid four-dimensional optimization of streaming-potential-mediated sPTT transport. It identifies a compact favorable regime and translates it into practical design guidance, while noting extensions needed for additional physics and objectives.

A feed-forward neural network approximates the mapping from (ζ, κ, ϵsWi^2k, Du) to E, replacing iterative electroneutrality solutions.
The surrogate achieves L1 losses of 4.37×10−6 on training and 3.86×10−6 on validation, corresponding to percentage errors of 0.66% and 0.68%.
Sobol analysis identifies κ as dominant for q and η, while viscoelasticity has negligible first-order influence on η despite affecting q.
NSGA-II finds a compact Pareto front at κ →10, ϵsWi^2k →2, and Du →0, with ζ confined to [5.75, 5.90].
The favorable regime corresponds to thicker EDLs, stronger viscoelasticity, minimized Stern-layer conduction, and ζ ≈6.
The study reports positive correlation between q and η and no fundamental throughput-efficiency trade-off in the explored system.
Future work should add objectives or physics such as polymeric-stress minimization, thermal effects, boundary heat transfer, and polymer-depleted layers.