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Catheter-modulated transient solute dispersion in micropolar annular flow with a retentive and absorptive arterial wall

Sohel Ahmed, Nanda Poddar, Kajal Kumar Mondal, Subham Dhar

arXiv:2609.38423v1physics.flu-dyncs.NAmath-phmath.MPmath.NAphysics.bio-phq-bio.TO

TL;DR

Drug transport in catheterized arteries couples shear, diffusion, and wall reaction, while catheter insertion makes the lumen annular. This study couples micropolar annular flow to generalized dispersion and finds strong confinement-related suppression of shear dispersion.

  • Problem

    Arterial drug delivery depends on the interplay of shear, molecular diffusion, and boundary reaction, while catheter insertion changes the vessel lumen into an annulus.

    Source: §1 · p. 2

  • Method

    The analysis applies the Gill–Sankarasubramanian generalized-dispersion framework to transient solute transport in a catheterized annulus with explicit surface concentration.

    Source: §5 · p. 29

  • Results

    Chemically passive shear-induced dispersion falls by a factor of 24 as λ rises from 0.01 to 0.30; narrow-gap shear dispersion follows a sixth-power collapse under fixed pressure gradient.

    Source: §5 · p. 30

  • Takeaways & Limitations

    Under fixed pressure gradient, annular clearance and wall kinetics are separate controls on downstream availability and axial spread, while Φ_m, Φ_s, and Φ_a distinguish mobile, retained, and absorbed solute.

    Source: §5 · p. 31

    The flow model assumes a long, straight, concentric annulus with steady, laminar, axisymmetric, fully developed flow and constant axial pressure gradient.

    Source: §2.1 · p. 4

Abstract

from arXiv · show

We study transient solute dispersion in pressure-driven micropolar flow through a concentric annulus representing a catheterized artery. The catheter is impermeable to solute, whereas the arterial wall combines irreversible removal with reversible surface retention. The exact steady micropolar velocity field is coupled to the Gill-Sankarasubramanian generalized-dispersion framework; the resulting bulk-surface hierarchy is advanced with Rannacher-damped Crank-Nicolson time stepping to determine $K_0(t)$, $K_1(t)$ and $K_2(t)$. The exchange coefficient is independent of the micropolar parameters and satisfies $-K_0(0^+)=2(\beta+\theta Da)/(1-\lambda^2)$, whereas $K_2-Pe^{-2}\sim\sigma_v^2t$ initially, so the leading short-time dispersion is hydrodynamic and independent of wall kinetics. In the weak-reaction regime, micropolarity modifies convection approximately linearly and shear dispersion quadratically through the velocity-amplitude factor. A narrow-gap analysis with $\varepsilon=1-\lambda$ gives $\bar v\sim(2-N_c)\varepsilon^2/6$ and $K_2-Pe^{-2}\sim(2-N_c)^2\varepsilon^6/7560$ for a fixed pressure gradient, revealing a sixth-power suppression of shear dispersion as the catheter approaches the arterial wall. Increasing $\lambda$ from $0.01$ to $0.30$ reduces effective convection by about a factor of $2.45$, while chemically passive shear-induced dispersion falls by a factor of $24$. The coupled formulation also yields the exact partition $\Phi_m+\Phi_s+\Phi_a=1$, separating mobile, reversibly retained and irreversibly absorbed solute. The reconstructed field quantifies reaction-modulated transverse non-uniformity, while the results distinguish hydrodynamic effects of confinement and microrotation from kinetic wall effects.

1. Introduction

Transient solute transport in catheterized arteries depends on the joint effects of flow, diffusion, wall reactions, and annular geometry. This study addresses the gap in understanding their full transient coupling for micropolar annular flow with both retaining and absorbing wall kinetics.

1. Introduction

  • The generalized-dispersion framework provides time-dependent transport coefficients from solute release, extending earlier large-time analyses.Taylor (1953) and Aris (1956) treated large-time dispersion; Gill & Sankarasubramanian (1970) introduced the generalized model.Source: §1 · p. 2
  • Irreversible absorption removes mobile solute, while reversible retention stores and later returns it, producing distinct transient effects including a possible intermediate-time reversal of convection.The cited work reports that retention can create a long trailing tail and reverse the convection coefficient at intermediate times.Source: §1 · p. 2
  • Catheter insertion turns the vessel lumen into an annulus, and the catheter-to-artery radius ratio controls shear and tracer spread.Earlier annular studies report that a larger catheter suppresses tracer spread.Source: §1 · pp. 2–3
  • The study targets the unresolved full transient coupling of micropolar annular flow with an inert catheter and an arterial wall that both retains and irreversibly absorbs solute.Prior studies examined these factors separately or used other fluid rheologies or long-time treatments.Source: §1 · p. 3
  • The study couples an exact steady micropolar velocity field to generalized dispersion to obtain transient coefficients, concentration fields, mass fractions, and narrow-gap scaling.The wall uses irreversible uptake and reversible retention; the analysis also examines transverse non-uniformity and recovers established tube limits.Source: §1 · p. 3

2. Mathematical formulation

The formulation combines steady micropolar flow in a concentric annulus with bulk solute transport and parallel irreversible uptake and reversible retention at the outer wall. A generalized-dispersion hierarchy yields transient coefficients, reconstructed concentrations, and an exact partition of injected mass.

2.1. Micropolar flow in the catheterized annulus

  • The carrier is modeled as an incompressible, steady, laminar, axisymmetric, fully developed micropolar fluid in a long, straight annulus under a constant pressure gradient.Micropolarity is a phenomenological representation of rotational microstructural effects, not a literal rigid-particle model of erythrocytes.Source: §2.1 · p. 4
  • The steady flow model solves coupled linear- and angular-momentum equations with no slip and no spin at both solid boundaries.The no-spin condition represents strong anchoring, arresting microrotation at the catheter and arterial surfaces.Source: §2.1 · p. 4
  • The pressure gradient, and thus the reference velocity scale, remains fixed across parameter studies so changes in catheter ratio can be interpreted consistently.The coupling number measures linear–angular motion coupling, while the micropolar parameter compares arterial and characteristic micropolar length scales.Source: §2.1 · p. 6

2.2. Reactive solute transport

  • A cross-sectionally uniform point injection is normalized to the prescribed total mobile mass, with the annular area factor ensuring that the initial concentration integrates to that mass.The axial far-field conditions accompany the initial-value problem.Source: §2.2 · p. 6
  • The catheter is inert and impermeable to solute, while the arterial wall applies irreversible uptake and reversible retention in parallel.The model tracks mobile concentration in the fluid and retained solute per unit area of the outer wall.Source: §2.2 · p. 6

2.3. Generalized dispersion formulation

  • The generalized-dispersion formulation expands mobile and retained concentrations in axial derivatives of the sectional mean and advances the hierarchy through K2.The retained-phase expansion follows from its time evolution, and the local concentration is reconstructed consistently to second order.Source: §2.3 · pp. 7–9
  • The formulation partitions injected mass into mobile, reversibly retained, and irreversibly absorbed fractions that sum to one.The absorbed fraction is non-decreasing, whereas the retained fraction may rise during loading and fall as desorption returns solute to the mobile phase.Source: §2.3 · p. 9
  • The reduced bulk–surface hierarchy jointly represents annular micropolar flow, reversible retention, irreversible uptake, and transient transport coefficients.With the stated sign convention, negative K0 indicates mobile-phase depletion, negative K1 indicates downstream advection, and K2 − Pe^-2 is the shear-induced dispersion.Source: §2.3 · pp. 8–9

3. Numerical scheme

A radial finite-difference scheme advances the coupled generalized-dispersion hierarchy with damped startup followed by Crank–Nicolson time stepping. Refinement tests show stable transport coefficients and a decreasing independent mass-conservation residual.

3. Numerical scheme

  • Four backward-Euler half-steps on a fine startup step damp the initial incompatibility between the uniform f0 profile and the reactive-wall flux condition before Crank–Nicolson evolution.The startup addresses high-frequency oscillations that Crank–Nicolson can produce for nonsmooth initial data.Source: §3 · p. 9
  • Centered second-order boundary formulas eliminate ghost nodes while preserving a tridiagonal system, and the hierarchy is advanced sequentially through n = 0, 1, 2.The retained-phase coefficients use the same time weighting; Picard iteration is limited to 200 iterations per time step.Source: §3 · p. 10
  • Projection enforces the exact normalization identities for the bulk coefficients, preventing small discretization errors from contaminating K1 and K2 when the mobile fraction is small.The time integrals for mass fractions and diagnostics use the same backward-Euler and trapezoidal weights as the evolution scheme.Source: §3 · p. 11
  • The coarsest-to-finest relative changes in −K0, −K1, and K2 − Pe^-2 are 2.94×10^-5, 9.30×10^-5, and 6.47×10^-5, respectively.Between the two finest calculations, K2 − Pe^-2 is unchanged to the digits shown; the maximum global mass residual reaches 1.762×10^-10 on the finest grid.Source: §3 · pp. 11–12, Table 1 · p. 11

4. Results and discussion

The section presents validation-supported results on how rheology, catheter confinement, and wall kinetics shape transient solute transport. It examines their effects on shear dispersion, mobile-solute transport, and transverse segregation through irreversible loss, reversible retention, and exchange.

4.1. Validation

  • The reduced Newtonian, irreversible-absorption calculations agree closely with published benchmark data over 10^-2 ≤ β ≤ 10^2, and inert-wall limits approach K0 = 0, K1 = 1, and K2 = 1/48.The velocity solution also recovers the unobstructed micropolar pipe-flow result and, as Nc → 0, the Newtonian Hagen–Poiseuille profile; agreement in all three benchmark coefficients checks the exchange, convection, and dispersion terms.Source: §4.1 · pp. 12–14, Figure 2 · p. 13

4.2. Effect of the flow parameters

  • In weak reaction, K0 is independent of Nc and mp; at small mp, −K1 ratios are approximately 1 : 0.79 : 0.58, while shear-dispersion ratios are 1 : 0.62 : 0.34 for Nc = 0.1, 0.5, and 0.9.At t = 1, retention dominates the shear-dispersion contribution for Nc = 0.5 and mp = 1: K2 − Pe^-2 = 8.24 × 10^-3, about 6.5 times the inert Taylor value. The centered concentration peak rises 1.69-fold from Nc = 0.1 to 0.9 and falls 9% as mp rises from 1 to 5.Source: §4.2 · pp. 14–15

4.3. Effect of the annular aspect ratio

  • Between λ = 0.01 and 0.3, baseline −K1 falls by a factor of about 2.45, while passive-wall shear dispersion falls by a factor of 24, from 1.26 × 10^-3 to 5.29 × 10^-5.At λ = 0.01 to 0.3, the centered concentration peak rises about 2.6-fold even as mobile mass falls from Φm = 0.576 to 0.546; accumulated dispersion falls by a factor of 7.7.Source: §4.3 · pp. 16, 17, 19, 20, Figure 5 · p. 17, Figure 7 · p. 20
  • The chemically passive narrow-gap shear dispersion follows a sixth-power collapse, with numerically fitted exponents 6.0041, 6.0042, and 6.0045 for Nc = 0.1, 0.5, and 0.9.For fixed pressure gradient, the compensated values approach (2 − Nc)^2/7560, with relative deviations of about 0.10–0.11% at the smallest gap; mean speed vanishes quadratically as the gap closes.Source: §4.3 · p. 18, Figure 6 · p. 19
  • Retention and absorption jointly reshape dispersion: at λ = 0.01, increasing β from 0.01 to 10 raises dual-reaction K2 − Pe^-2 from 8.24 × 10^-3 to 2.97 × 10^-2, while absorption alone lowers it.At t = 1 and λ = 0.01, increasing Da from 0.1 to 1.5 raises dispersion by a factor of 6.6; the β = 10 convection coefficient changes sign near λ ≈ 0.27 when λ increases.Source: §4.3 · pp. 16, 19

4.4. Effect of the reaction parameters

  • The initial −K0 plateaus increase with β, θ, and Da, but the Da curves cross: the Da = 1.5 curve falls to 0.200 by t = 1, below 0.280 and 0.270 for Da = 0.5 and 1.The initial exchange combines irreversible uptake β and reversible forward transfer θDa; the plateau lasts while (β + θDa)^2t ≪ 1. Faster exchange fills the retained store sooner, whereas θ sets capacity and its curves remain distinct over the interval.Source: §4.4 · pp. 20–22, Figure 8 · p. 21
  • For β = 10, −K1 reaches a maximum of 0.409 and falls to 0.215 by t = 1; across Da values, faster exchange builds retained solute sooner and produces earlier retardation.In the baseline case, −K1 is 0.147 at t = 4, compared with mean speed divided by R, 0.148. Released solute re-enters behind the advancing cloud near the slow-moving wall.Source: §4.4 · p. 23
  • Initially, K2−Pe−2 grows with slope 2.35 ×10−2, set by cross-sectional velocity variance and independent of β, θ, and Da.Reaction-dependent curves initially collapse while (β+θDa)^2t≪1; wall exchange is active at t=0+ but affects velocity-weighted spreading beyond the leading O(t) shear-dispersion term.Source: §4.4 · p. 23
  • At t=1, K2−Pe−2 rises with θ (2.89, 8.24, and 11.8 ×10−3 for 0.1, 0.5, and 1) and Da (1.42, 4.42, 8.24, and 9.35 ×10−3 for 0.1, 0.5, 1, and 1.5).For Da=0.1, the second stage has barely begun by t=1; the baseline dispersion rises again to 8.24 ×10−3 at t=1 and reaches about 1.5 ×10−2 near t≈3 as exchange approaches local equilibrium.Source: Figure 10 · p. 24, §4.4 · p. 24
  • At t=0.3, stronger absorption lowers K2−Pe−2 (1.66, 1.55, and 0.53 ×10−3 for β=0.01, 1, and 10), but at t=1 the ordering reverses (8.24 ×10−3, 1.39 ×10−2, and 2.97 ×10−2).Increasing β depletes mobile solute, leaving a larger retained-to-mobile ratio and producing later dispersion through desorption.Source: §4.4 · pp. 24–25

4.5. Partition of mobile, retained and absorbed solute

  • At t=1, Φm=0.576, Φs=0.411, and Φa=0.0135, so reversible storage holds about 41% of the injected solute while irreversible uptake accounts for about 1.35%.For the baseline case, mobile mass decreases while retained and absorbed fractions rise; the independently evaluated fractions satisfy the partition with a maximum residual of 3.4 ×10−9.Source: Figure 12 · p. 26, §4.5 · pp. 26–27

4.6. Spatial structure of the solute cloud

  • The signed dispersive correction has zero cross-sectional mean, and increasing λ from 0.01 to 0.10 suppresses it at each displayed time.Its paired lobes represent local enrichment and depletion; transverse diffusion broadens the structure and reduces peak magnitude over time.Source: §4.6 · p. 27, Figure 13 · p. 28
  • N⊥,max falls from approximately 1.35 at t=0.05 to 0.426 at t=1, while increasing β from 0.01 to 1 raises it from about 0.426 to 0.793 and β=10 exceeds 2.Increasing Nc from 0.1 to 0.9 leaves the peak near 0.426 but contracts the axial extent; increasing θ from 0.1 to 1 raises N⊥,max from approximately 0.345 to 0.484.Source: §4.6 · pp. 28–29, Figure 14 · p. 29

5. Conclusions

The conclusions distinguish wall-kinetic effects on mobile solute from hydrodynamic effects of micropolarity and catheter confinement. They show that wall processes shape transient transport, while narrowing the annulus sharply suppresses shear-driven spreading.

5. Conclusions

  • The formulation couples an impermeable catheter and a reactive, retentive outer wall to transient bulk–surface transport, advanced with Rannacher-damped Crank–Nicolson stepping.Its implications are mechanistic rather than patient-specific because it assumes a straight, rigid, concentric vessel, steady flow, and first-order wall kinetics.Source: §5 · pp. 29–31
  • K0 is independent of micropolar parameters, while K1 and K2 inherit rheology through the velocity field; initially, shear-dispersion growth is set by cross-sectional velocity variance.The initial mobile-phase depletion rate remains close to its starting value only while (β + θDa)^2t ≪ 1; afterward, wall-layer depletion and retained-phase filling reduce it.Source: §5 · p. 30
  • A 24-fold fall in passive-wall shear dispersion accompanies increasing λ from 0.01 to 0.30; under fixed pressure gradient, the narrow-gap limit gives a sixth-power collapse.In the baseline reactive case, the same increase in λ reduces −K1 by about 2.45 and K2 − Pe^-2 by about 6; the numerical cell problem recovers the narrow-gap exponent and prefactor.Source: §5 · p. 30
  • Retention retards the cloud and adds a second stage of spreading; at t = 1, baseline dispersion exceeds the inert Taylor level by a factor of 6.5.Absorption alone lowers dispersion fourfold between β = 0.01 and 10; with retention, β’s effect reverses over time, and strong uptake can make convection and dispersion negative.Source: §5 · p. 30
  • The exact mass partition Φm + Φs + Φa = 1 distinguishes mobile, reversibly retained, and irreversibly absorbed solute.The partition also provides an independent conservation check on surface coupling and time integration.Source: §5 · p. 30
  • The reconstructed dispersive field shows confinement weakening local shear redistribution; transverse non-uniformity peaks early and is amplified more by irreversible uptake than reversible retention.Microrotational coupling primarily changes the heterogeneity’s axial extent rather than its peak magnitude.Source: §5 · pp. 30–31
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