Physics-informed neural networks for two-dimensional wall-reactive solute dispersion in canonical shear flows

Nanda Poddar; Subham Dhar

REVIEW 4 major objections 4 minor 68 references

Physics-informed neural networks for two-dimensional wall-reactive solute dispersion in canonical shear flows

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A physics-informed neural network trained on the convection-diffusion equation and Robin wall conditions reproduces the spatiotemporal concentration field of wall-reactive solute dispersion in Couette, Poiseuille, and Couette–Poiseuille…

desk verdict Competent but incremental PINN application; the PINN-ADI agreement likely reflects a shared artificial-boundary error rather than true accuracy, so the validation needs a domain-window check and a convergence study before the central claim can stand. read the letter →

arxiv 2608.00856 v1 pith:VHFLKYOX submitted 2026-08-01 physics.flu-dyn math-phmath.MPphysics.comp-phstat.ML

Nanda Poddar , Subham Dhar This is my paper · ORCID

classification physics.flu-dynmath-phmath.MPphysics.comp-phstat.ML

keywords physics-informedneuralnetworkssolutedispersionshearflowsRobinboundaryconditionswalluptakeconvection-diffusionequationalternating-directionimplicitmethodPécletnumber

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether a physics-informed neural network can replace a grid-based solver for two-dimensional dispersion of a solute that reacts at the channel walls. It claims yes: with the convection-diffusion equation and first-order wall-absorption conditions folded into one loss, the network reproduces the concentration field for Couette, Poiseuille, and Couette–Poiseuille flows at Pe=10, matching an alternating-direction implicit finite-difference benchmark. Because the network is differentiable, the same trained field yields transport diagnostics such as the apparent axial dispersion coefficient and cumulative wall-removal rates without discrete differentiation. The physical payoff is a precise picture of how shear determines where along the wall solute is removed and how unequal wall reactivities break transverse symmetry.

What carries the argument

The load-bearing object is the composite physics-informed loss $$\mathcal{L}_{\rm tot} = \mathcal{L}_{\rm PDE} + \lambda_{\rm IC}\mathcal{L}_{\rm IC} + \lambda_{\rm B}(\mathcal{L}_{x\rm B}+\mathcal{L}_{y\rm B}),$$ minimized by a four-hidden-layer, 64-neuron tanh network. A PINN is a neural network trained to minimize the residual of the governing partial differential equation together with its boundary conditions. The PDE residual is evaluated by automatic differentiation, and the wall residuals enforce first-order uptake at the two plates. The three mean-centred shear profiles $u^{(\rm C)}=y/2$, $u^{(\rm Po)}=\tfrac12(1-y^2)-\tfrac13$, and $u^{(\rm CP)}=u^{(\rm C)}+u^{(\rm Po)}$ are what distinguish the flow cases. This loss is what lets a mesh-free function approximator enforce the transport physics, and the trained network's differentiability is what turns the solution into a source of flux and moment diagnostics.

What would settle it

Refine the ADI grid (for example, double $N_x$ and $N_y$ and halve $\Delta t$) or compare against an independent spectral or high-order solution and an analytical reactive-dispersion solution; if the PINN–ADI pair remains close to each other but both shift when the streamwise window is widened beyond $[-5,5]$ or when the grid is refined, the claim that the PINN reconstructs the true concentration field is refuted.

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Extended reading notes

Core claim

The central discovery is that a single feed-forward network, constrained only by the governing PDE and boundary residuals, can serve as an accurate and differentiable surrogate for wall-reactive solute dispersion. The network output $C_ heta(x,y,t)$ minimizes a composite loss containing the dimensionless convection-diffusion residual $\partial_t C + \mathrm{Pe}\,u^{(s)}(y)\partial_x C - \partial_{xx} C - \partial_{yy} C$, the smooth Gaussian initial source, zero-gradient axial conditions at $x=\pm 5$, and Robin wall conditions $\partial_y C-\beta_2 C=0$ at $y=-1$ and $-\partial_y C-\beta_1 C=0$ at $y=1$. Over $t\in[0,1]$, the learned field stays close to the ADI benchmark for both line-like and point-like sources across non-reactive, symmetric, and asymmetric wall-reactivity regimes, with final cumulative wall-uptake relative errors below about 4%. The differentiable representation then makes wall-resolved diagnostics, such as the apparent dispersion coefficient, cumulative uptake, and flux-weighted centroids, directly computable. The paper's physical claim is that the imposed shear profile controls the streamwise organization of reactive uptake, while unequal wall reactivities control which wall dominates removal.

Load-bearing premise

The load-bearing premise is that the ADI finite-difference solution on the chosen grid ($N_x=101$, $N_y=51$, $\Delta t=10^{-4}$) over the finite window $x\in[-5,5]$ is an accurate proxy for the true solution, since no grid-convergence study or comparison with an independent analytical reactive-dispersion solution is reported.

Editorial extensions

If this is right

At Pe=10, the trained PINN reproduces the ADI concentration field for line-like and point-like Gaussian sources in Couette, Poiseuille, and Couette–Poiseuille flows, in non-reactive, symmetric, and asymmetric wall-reactivity regimes.
The same differentiable network yields $D_a(t)=\frac{1}{2}\frac{d\nu_2}{dt}$, cumulative wall uptake $U_{\rm tot}(t)$, and wall-resolved fluxes $J_\pm(x,t)$ without grid-based derivative approximations.
Unequal wall reactivities break transverse symmetry of the plume: for $(\beta_1,\beta_2)=(0.2,2)$ the lower wall dominates removal, and reversing the pair reverses the dominance, as quantified by $\Phi_-(T)$ and $D_w(t)$.
The shear profile controls the streamwise organization of uptake: Couette shear separates upper- and lower-wall uptake centroids, whereas Poiseuille shear keeps them overlapping, so total removal alone does not characterize reactive dispersion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The forward machinery could be inverted: a PINN trained on noisy concentration or flux observations could infer unknown wall reactivities $\beta_1,\beta_2$; the paper builds the differentiable surrogate but does not attempt the inverse problem.
The smooth Gaussian sources are a practical bridge to point releases: weak-form or distributional PINN formulations could extend the same framework to Dirac-delta initial conditions, making classical point-release dispersion benchmarks directly accessible.
The flux-weighted centroid and spread diagnostics are natural design targets for microfluidic or physiological transport, for example selecting a shear profile that delivers solute to a prescribed streamwise uptake zone; this application lies outside the paper.
The Pe=10 results do not automatically extend to advection-dominated regimes; at higher Pe the network faces stiffer residuals, so the claimed mesh-free advantage should be re-tested where advection is stronger.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a solid but incremental PINN application, and the main validation claim is undercut by a domain-truncation problem. What is new: it treats the initial release as a smooth Gaussian (mollified delta) and uses the differentiable network to extract wall-resolved diagnostics - uptake centroids, spreads, dominance index - that go beyond just matching concentration fields. The formulation is clean, and the reported PINN-ADI errors are mostly a few percent across three shear profiles and symmetric/asymmetric reactivities.

The load-bearing soft spot is the finite window. Both solvers run on x in [-5,5] with zero-gradient axial conditions. At Pe=10 and t=1, a Taylor-Aris estimate for Couette gives an r.m.s. plume width around 3, so the concentration at x=±5 is a nontrivial fraction of the peak. The zero-gradient boundary reflects mass and suppresses variance growth. The paper asserts the cloud stays away from the boundaries but gives no quantitative check. Therefore the PINN-ADI agreement shows the two methods solve the same artificially bounded problem; it does not establish accuracy for the infinite-domain dispersion problem stated in the abstract. This is a genuine flaw, not a nitpick.

There are two further issues. No grid-convergence study is reported for the ADI benchmark (Nx=101, Ny=51, dt=1e-4), and no comparison is made to the analytical reactive-dispersion solutions the paper cites, so the benchmark is unverified. The error tables are also inconsistent: the non-reactive Poiseuille line-like case has L2 error 0.024 at t=0.001 in Table 1 but 0.0017 in Table 2, and one Couette-Poiseuille point-like entry repeats exactly at t=0.1 and t=0.5. That makes the quantitative claims hard to trust as reported.

Credit where due: the authors are honest about limitations - Section 5 concedes the dispersion coefficient is prone to local approximation errors - and the supplementary material is thorough. The paper is not a waste of time; it is a competent draft for the computational-transport audience. I would not cite it in its current form, and the revision bar is high. A serious editor could send it to review, but the authors should first show the diagnostics are unchanged when Lx is doubled, add an ADI convergence study, and fix the error tables.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a physics-informed neural network (PINN) for two-dimensional advection–diffusion transport of a reactive solute in Couette, Poiseuille, and Couette–Poiseuille shear flows between absorbing walls. The PINN is trained to satisfy the convection–diffusion equation, Robin wall conditions, and localized Gaussian initial sources on a finite streamwise domain, and the results are compared with an alternating-direction implicit (ADI) finite-difference benchmark at Pe=10 for non-reactive, symmetric, and asymmetric wall reactivities. The authors further exploit the differentiability of the network to extract diagnostics such as the apparent axial dispersion coefficient, total surviving mass, cumulative wall uptake, wall-dominance index, and flux-weighted uptake centroids. The central claim is that the PINN reconstructs the spatiotemporal concentration field and that the extracted wall-resolved diagnostics are validated by close agreement with the ADI solution.

Significance. If the central claim is established, the paper offers a useful differentiable surrogate for boundary-coupled reactive transport in canonical shear flows, and the attempt to validate not only concentrations but also derived integral diagnostics is commendable. The ADI comparison is an independent numerical benchmark rather than a circular check, and the range of flow and reaction regimes considered is appropriate for a methods paper. However, the validation rests on two unverified assumptions: that the finite streamwise domain is large enough not to perturb the plume, and that the ADI grid is converged. In addition, the quantitative error reporting is internally inconsistent. These are load-bearing gaps because both the PINN and the ADI solver impose the same artificial truncation, so agreement between them does not by itself establish accuracy against the infinite-domain physical problem.

major comments (4)

[§2.6, §3.7] The truncation of the infinite streamwise domain to x∈[-5,5] with zero-gradient conditions (Eq. 2.14) is not shown to be harmless. For the Couette case at Pe=10, the mean-centred shear u^{(C)}=y/2 gives a Taylor–Aris estimate D_eff≈1+Pe^2/15≈7.7, so the streamwise standard deviation at t=1 is σ≈sqrt(2D_eff)≈3.9, implying that the concentration at x=±5 is not negligible. The assertion in §2.6 that the cloud 'remains away from the artificial axial boundaries' is therefore not quantitatively credible, and the sentence in §3.7 stating that the concentration 'may stay non-zero at the artificial longitudinal boundaries throughout the simulation' further undercuts it. Because the PINN loss (§3.4) and the ADI benchmark (§3.8) impose exactly the same zero-gradient condition, the agreement reported in Tables 1–3 and Figures 2–7 may only demonstrate consistency between two solvers on the same artificially bounded problem. Please add a domain-size study (e.g., L_x=20 or 40) or compare against an analytical infinite-domain solution to show that truncation does not affect the reported fields and diagnostics.
[§3.8] The ADI finite-difference solution is used as the reference, but no grid-convergence study is reported: the manuscript fixes Nx=101, Ny=51, and Δt=10^-4 and does not refine the grid. The reader cannot tell whether the small PINN–ADI differences in Tables 2 and 3 reflect approximation errors of the PINN or discretization errors of the benchmark. Please add a convergence test with at least two refined grids and, for at least one reactive case, a comparison with a published analytical solution (e.g., the reactive dispersion solutions cited in the introduction, such as Jiang et al. 2022). Without this, the ADI benchmark is not independently verified as ground truth.
[§4.1, Tables 1–3] The quantitative validation is internally inconsistent. For identical non-reactive line-like Couette cases at t=1, Table 1 reports an L2 error of 0.0415 while Table 2 reports 0.0161; for line-like Couette–Poiseuille at t=1 the values are 0.0721 and 0.0199, respectively. Tables 2 and 3 do not define whether the listed numbers are relative or absolute, L1 or L2, and Table 1 gives no definition of the norm either. This makes it impossible to assess the accuracy claims quantitatively. Please define the error metrics precisely, ensure consistency across all tables, and correct the duplicate entry for the Couette–Poiseuille point-like case in Table 1.
[§4.3, §5] The time-dependent dispersion coefficient D_a(t) (Eq. 4.5) is presented as a central diagnostic, yet the manuscript's own conclusion states that it is 'prone to local approximation errors.' The validation figures 11–15 show the discrepancy E_D(t) but provide no quantitative summary, so the reader cannot judge whether the errors are a few percent or substantial. Please report the maximum and mean of E_D(t) over t for the cases shown and discuss how this uncertainty propagates into the physical conclusions in Sections 4.7–4.10.

minor comments (4)

[Figure 23] The x-axes of the local wall-flux profiles span -10 to 10, whereas the computational domain stated in §3.7 is x∈[-5,5]; please reconcile the axes with the domain actually used.
[Table 1] In the Couette–Poiseuille point-like row, the error at t=0.5 is identical to the error at t=0.1 (0.0170186), which appears to be a typographical duplication and should be corrected.
[§3.7, Data availability] The code availability statement says that code is available 'upon reasonable request'; for a computational methods paper, depositing the code in an open repository would substantially improve reproducibility.
[§2.6] The sentence in §2.6 asserting that the cloud 'remains away from the artificial axial boundaries' should be replaced by a quantitative statement once the truncation check requested in the major comments is performed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: PINN solves the forward IBVP and is checked against an independent ADI discretization of the same IBVP.

full rationale

The PINN is trained by minimizing the PDE residual (3.1), initial-condition loss (3.3), and boundary losses (3.4)-(3.7); the ADI solution never enters the loss function and is used only as a post-hoc comparison. All extracted diagnostics (D_a, M_0, nu_2, U_tot, J_plus, J_minus, centroids, spreads) are integrals or derivatives of the trained concentration field, not fitted parameters. Self-citations (Poddar et al., Dhar et al.) are used only as background for Robin conditions, shear-flow context, and transient-effect remarks; no uniqueness theorem, ansatz, or fitted quantity is imported from the authors' prior work. The skeptical concern that PINN and ADI share the same truncated domain and zero-gradient axial conditions (Section 2.6, Eq. 2.14) and that the ADI grid (Section 3.7: Nx=101, Ny=51, dt=1e-4) is not grid-converged is a validity and accuracy threat, not a circularity: the two solvers are still independent numerical approximations of the same IBVP, and their agreement does not reduce to a fitted or self-referential quantity. Section 5 itself concedes that the longitudinal dispersion coefficient is 'prone to local approximation errors,' an honest limitation rather than a circular step. Therefore no prediction in the paper is equivalent to its inputs by construction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the assumed validity of the continuum transport model and on the unverified accuracy of the ADI reference solution. The PINN hyperparameters (loss weights, epochs, collocation counts, architecture) are hand-chosen and not systematically varied. No new physical entities are introduced.

free parameters (5)

Loss weights lambda_IC = 30, lambda_B = 30 = 30, 30
Chosen by hand to emphasize initial and boundary constraints; reported accuracy depends on these values (Section 3.7).
Training epochs = 5000 (up to 20000)
Stopping criterion not tied to a tolerance; Table 4 shows composite loss still around 1e-3 to 1e-2 at epoch 20000.
Collocation point counts = N_f=8000, N_0=1000, N_xB=800, N_yB=400
Chosen by hand per epoch (Section 3.7); no sensitivity study.
ADI grid resolution = Nx=101, Ny=51, dt=1e-4
Reference solution grid; no convergence study, so its accuracy as ground truth is unverified (Section 3.8).
Network architecture = 4 hidden layers, 64 neurons, tanh
Chosen by hand; only one alternative activation (Gaussian) is partially explored (Section 3.1, Figure 9).

assumptions (5)

domain assumption Convection-diffusion equation with Fickian diffusion and prescribed unidirectional shear flow (Eq. 2.13)
The continuum transport model is assumed valid; no molecular dynamics or experimental verification.
domain assumption First-order Robin boundary conditions model wall uptake (Eq. 2.5)
The paper cites prior usage, but treats irreversible first-order kinetics as the reaction model.
ad hoc to paper Finite window x in [-5,5] with homogeneous Neumann axial conditions does not affect plume statistics over t in [0,1] (Eq. 2.14)
Load-bearing truncation assumption; no test that concentration is negligible at the axial boundaries.
domain assumption ADI scheme on Nx=101, Ny=51, dt=1e-4 is a converged reference solution
No grid-refinement study is reported; all PINN accuracy is measured against this single grid.
ad hoc to paper Gaussian initial sources approximate point and line releases (Eqs. 2.7-2.8)
Chosen to bypass Dirac delta incompatibility; the paper states singular sources are left for future work.

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Cite this review

Pith. "Pith review of Physics-informed neural networks for two-dimensional wall-reactive solute dispersion in canonical shear flows." pith.science (2026). https://pith.science/paper/VHFLKYOX

@misc{pith2026260800856,
  author       = {Pith},
  title        = {Pith review of: Physics-informed neural networks for two-dimensional wall-reactive solute dispersion in canonical shear flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VHFLKYOX}},
  note         = {Machine review of arXiv:2608.00856}
}

abstract

The dispersion of reactive solutes in shear flows is governed by the interplay between advective stretching, transverse diffusion, and boundary exchange kinetics. While classical analytical methods and grid-based numerical solvers have extensively characterised these transport mechanisms, accurately resolving the spatiotemporal evolution of solute plumes in asymmetric reactive environments remains computationally demanding. In this study, we introduce a physics-informed neural network (PINN) framework to simulate two-dimensional wall-reactive solute dispersion in canonical shear flows (Couette, Poiseuille, and Couette-Poiseuille) bounded by absorbing walls. By embedding the governing convection-diffusion equation and Robin boundary conditions into a unified loss function, the mesh-free PINN reconstructs the spatiotemporal concentration field. The network predictions are validated against an alternating-direction implicit (ADI) finite-difference benchmark, showing close agreement across non-reactive, symmetric, and asymmetric reactive regimes. The computations are carried out at $\mathrm{Pe}=10$ for impermeable walls, symmetric absorption $(\beta_1,\beta_2)=(1,1)$, and tenfold asymmetric wall-reactivity contrasts $(\beta_1,\beta_2)=(0.2,2)$ and (2,0.2). Leveraging the differentiable nature of the trained PINN, we extract wall-resolved transport diagnostics, including the apparent axial dispersion coefficient, cumulative wall-removal dynamics, and localised uptake fluxes. The results show that the imposed shear profile governs the streamwise organisation of reactive uptake, while unequal wall reactivities induce transverse asymmetry that modifies the macroscopic spreading rate. Overall, this framework establishes PINNs as an interpretable mesh-free tool for analysing boundary-coupled reactive transport in shear flows.

Figures

Figures reproduced from arXiv: 2608.00856 by the authors.

**Figure 2.** Blockwise absolute error 𝐵err = |𝐶PINN − 𝐶ADI| for the point-like source in Couette flow with symmetric wall absorption (𝛽1, 𝛽2) = (1, 1) [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Blockwise absolute error 𝐵err = |𝐶PINN − 𝐶ADI| for the point-like source in Poiseuille flow with lower-wall-dominated absorption (𝛽1, 𝛽2) = (0.2, 2). Additional blockwise-error results for the line-like reactive cases are provided in the supplementary material. These results complement the representative point-like reactive validation shown in the main manuscript while avoiding excessive repetition of similar contou… view at source ↗

Figures from the paper (40 more)

**Figure 4.** Figure 4: Blockwise absolute error 𝐵err = |𝐶PINN − 𝐶ADI| for the point-like source in Couette–Poiseuille flow with upper-wall-dominated absorption (𝛽1, 𝛽2) = (2, 0.2) [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

**Figure 6.** Figure 6: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

**Figure 8.** Figure 8: Training loss histories for the line-like source [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

**Figure 9.** Figure 9: Training loss histories for the point-like source [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

**Figure 10.** Figure 10: Training-loss histories for asymmetric reactive cases in Couette–Poiseuille [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

**Figure 11.** Figure 11: Validation of the effective axial dispersion coefficient for Couette flow with the [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]

**Figure 12.** Figure 12: Validation of the effective axial dispersion coefficient for non-reactive [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]

**Figure 13.** Figure 13: Validation of the effective axial dispersion coefficient for symmetric wall [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]

**Figure 14.** Figure 14: Validation of the effective axial dispersion coefficient for lower-wall-dominated [PITH_FULL_IMAGE:figures/full_fig_p026_14.png]

**Figure 15.** Figure 15: Validation of the effective axial dispersion coefficient for upper-wall-dominated [PITH_FULL_IMAGE:figures/full_fig_p026_15.png]

**Figure 16.** Figure 16: Validation of the total surviving solute mass [PITH_FULL_IMAGE:figures/full_fig_p027_16.png]

**Figure 17.** Figure 17: Validation of the total surviving solute mass [PITH_FULL_IMAGE:figures/full_fig_p028_17.png]

**Figure 18.** Figure 18: Validation of the axial variance 𝜈2 (𝑡) for point-like releases under upper-wall-dominated reactivity, (𝛽1, 𝛽2) = (2, 0.2). Panel (𝑎) corresponds to Couette flow, while panel (𝑏) corresponds to Couette–Poiseuille flow. In each case, 𝜈 PINN 2 (𝑡) is compared with 𝜈 ADI…

**Figure 19.** Figure 19: ADI–PINN validation of cumulative wall uptake for the point-like Gaussian [PITH_FULL_IMAGE:figures/full_fig_p031_19.png]

**Figure 20.** Figure 20: Wall-resolved reactive removal extracted from the trained PINN solution for [PITH_FULL_IMAGE:figures/full_fig_p033_20.png]

**Figure 21.** Figure 21: Final wall uptake components extracted from the trained PINN solution for the [PITH_FULL_IMAGE:figures/full_fig_p034_21.png]

**Figure 22.** Figure 22: Wall-selective reactive-dispersion diagnostics extracted from the trained PINN [PITH_FULL_IMAGE:figures/full_fig_p034_22.png]

**Figure 23.** Figure 23: Local wall uptake flux profiles for the point-like Gaussian source in the [PITH_FULL_IMAGE:figures/full_fig_p035_23.png]

**Figure 24.** Figure 24: Flux-weighted uptake centroid and spread for the point-like Gaussian source in [PITH_FULL_IMAGE:figures/full_fig_p036_24.png]

**Figure 25.** Figure 25: Concentration contours with superposed mean-centred velocity arrows and [PITH_FULL_IMAGE:figures/full_fig_p037_25.png]

**Figure 1.** Figure 1: Blockwise absolute error for Couette flow with the line-like source [PITH_FULL_IMAGE:figures/full_fig_p043_1.png]

**Figure 2.** Figure 2: Blockwise absolute error for Poiseuille flow with the line-like source [PITH_FULL_IMAGE:figures/full_fig_p044_2.png]

**Figure 3.** Figure 3: Blockwise absolute error for Couette–Poiseuille flow with the line-like source [PITH_FULL_IMAGE:figures/full_fig_p044_3.png]

**Figure 4.** Figure 4: Blockwise absolute error for Couette flow with the line-like source [PITH_FULL_IMAGE:figures/full_fig_p045_4.png]

**Figure 5.** Figure 5: Blockwise absolute error for Poiseuille flow with the line-like source [PITH_FULL_IMAGE:figures/full_fig_p045_5.png]

**Figure 6.** Figure 6: Blockwise absolute error for Couette–Poiseuille flow with the line-like source [PITH_FULL_IMAGE:figures/full_fig_p046_6.png]

**Figure 7.** Figure 7: Blockwise absolute error for Couette flow with the line-like source [PITH_FULL_IMAGE:figures/full_fig_p046_7.png]

**Figure 8.** Figure 8: Blockwise absolute error for Poiseuille flow with the line-like source [PITH_FULL_IMAGE:figures/full_fig_p047_8.png]

**Figure 9.** Figure 9: Blockwise absolute error for Couette–Poiseuille flow with the line-like source [PITH_FULL_IMAGE:figures/full_fig_p047_9.png]

**Figure 10.** Figure 10: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p048_10.png]

**Figure 11.** Figure 11: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p048_11.png]

**Figure 12.** Figure 12: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p049_12.png]

**Figure 13.** Figure 13: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p049_13.png]

**Figure 14.** Figure 14: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p050_14.png]

**Figure 15.** Figure 15: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p050_15.png]

**Figure 16.** Figure 16: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p051_16.png]

**Figure 17.** Figure 17: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p051_17.png]

**Figure 18.** Figure 18: Cross-sectionally averaged concentration comparison and mean-concentration [PITH_FULL_IMAGE:figures/full_fig_p052_18.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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