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arxiv: 2605.13633 · v1 · submitted 2026-05-13 · ⚛️ physics.flu-dyn · cs.CE· physics.comp-ph

Recognition: unknown

Effects of Thermal Boundary Conditions on Natural Convection and Entropy Generation in Non-Newtonian Power-Law Fluids

Lambert Theisen , Satyvir Singh
Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:53 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.CEphysics.comp-ph
keywords natural convectionnon-Newtonian fluidspower-law modelentropy generationthermal boundary conditionssquare cavitycylindrical annulus
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The pith

Thermal boundary conditions control the intensity of convection and the amount of entropy generated in non-Newtonian power-law fluids.

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

The paper examines how uniform versus non-uniform wall heating changes buoyancy-driven flows and thermodynamic losses when fluid viscosity varies with shear rate. Steady two-dimensional equations are solved for a square cavity and a concentric cylindrical annulus across shear-thinning, Newtonian, and shear-thickening cases. A sympathetic reader would care because the results indicate that wall-temperature patterns alone can strengthen or weaken circulation while lowering total irreversibility. Uniform heating creates stronger, more distributed flows in both geometries. Sinusoidal heating localizes the thermal drive and reduces overall entropy generation, with viscous dissipation dominating at low power-law indices and heat-transfer irreversibility dominating at higher indices.

Core claim

Thermal boundary conditions play an important role in controlling the intensity and spatial distribution of flow, heat transfer, and irreversibility. In both the square cavity and the concentric cylindrical annulus, uniform heating produces stronger and more distributed convective structures, while non-uniform sinusoidal heating localizes thermal forcing and consistently reduces total entropy generation. Shear-thinning fluids enhance buoyancy-driven circulation, steepen thermal gradients, and increase heat transfer, whereas shear-thickening fluids suppress convection and promote conduction-dominated transport. Viscous dissipation dominates irreversibility in shear-thinning fluids, whereas热转让

What carries the argument

The incompressible power-law fluid model under the Boussinesq approximation, solved by finite elements, to compare uniform and sinusoidal thermal boundary conditions.

If this is right

  • Uniform heating increases convective strength and heat transfer rates relative to sinusoidal heating.
  • Non-uniform sinusoidal heating lowers total entropy generation in both the square cavity and the cylindrical annulus.
  • Shear-thinning fluids produce higher Nusselt numbers than Newtonian or shear-thickening fluids under the same conditions.
  • Viscous dissipation accounts for most irreversibility at low power-law indices, while thermal irreversibility dominates at higher indices.
  • Thermal boundary design combined with fluid rheology offers a route to control heat transfer and reduce thermodynamic losses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Engineers could apply sinusoidal heating to non-Newtonian systems to cut energy dissipation while retaining acceptable heat transfer.
  • The trends may inform heating strategies in polymer processing or chemical reactors that handle power-law fluids.
  • Three-dimensional or time-dependent simulations at higher Rayleigh numbers could check whether the two-dimensional steady patterns persist.

Load-bearing premise

The flow remains steady and strictly two-dimensional across the full range of Rayleigh numbers and power-law indices examined.

What would settle it

Laboratory measurements of velocity fields, temperature distributions, and total entropy generation in physical enclosures with the same geometries and boundary conditions would show whether the predicted reduction under sinusoidal heating occurs.

Figures

Figures reproduced from arXiv: 2605.13633 by Lambert Theisen, Satyvir Singh.

Figure 1
Figure 1. Figure 1: Schematic diagrams of the physical domains considered in the study of natural convection and entropy generation in power-law fluids. (a) Square cavity with a hot bottom wall, cold vertical side walls, and an adiabatic top wall. (b) Concentric cylindrical annulus with an inner hot cylinder of radius 𝑅𝑖 and an outer cold cylinder of radius 𝑅𝑜 , where 𝜑 denotes the angular coordinate. In both configurations, … view at source ↗
Figure 2
Figure 2. Figure 2: Validation of Newtonian natural convection (𝑛 = 1) in a square cavity with bottom-wall heating: (a) uniform heating (𝜃 = 1) and (b) non-uniform sinusoidal heating (𝜃 = sin(𝜋𝑋)). Stream function and temperature contours from the present FEM simulations (bottom) are compared with the benchmark results of Basak et al. [22] (top) for Pr = 0.7, and Ra = 105 , demonstrating a good agreement in both flow and ther… view at source ↗
Figure 3
Figure 3. Figure 3: Validation of Newtonian natural convection (𝑛 = 1) in a square cavity with bottom-wall uniform and non-uniform heating. Distributions of the local Nusselt number are shown along (a) the bottom wall and (b) the side wall for Pr = 0.7, 10 and Ra = 103 , 105 . Solid lines represent uniform heating, while dashed lines correspond to non-uniform sinusoidal heating. Present results are compared with benchmark dat… view at source ↗
Figure 4
Figure 4. Figure 4: Validation of Newtonian natural convection (𝑛 = 1) in a square cavity with uniform heated bottom wall. Local entropy generation contours due to heat transfer 𝑆𝜃,𝑙 and fluid friction 𝑆𝜓,𝑙 are shown at Pr = 0.015 and Ra = 103 . Present FEM results are compared against the numerical data of Kaluri and Basak [53]. convection near the heated inner cylinder, generating compact vortices and strongly distorted iso… view at source ↗
Figure 5
Figure 5. Figure 5: Validation of non-Newtonian natural convection in a differentially heated square cavity for Pr = 103 , Ra = 104 , and power-law indices 𝑛 = 0.6, 1.0, and 1.8: comparison of non-dimensional (a) temperature and (b) stream function contours between Turan et al. [54] and the present FEM solver. number at the heated wall, maximum local entropy generation due to heat transfer (𝑆𝜃,max), and fluid friction (𝑆𝜓,max… view at source ↗
Figure 6
Figure 6. Figure 6: Validation of non-Newtonian natural convection in a concentric cylindrical annulus for Pr = 100, Ra = 104 , and power-law indices 𝑛 = 0.6, 1.0, and 𝑛 = 1.4. Comparison of streamline and isotherm contours between Matin and Khan [55] and the present FEM results. 6.2.1. Flow structure and thermal fields [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mesh convergence study for the two computational domains: (a) structured quadrilateral meshes (Mesh 1–3) for the square cavity, and (b) unstructured triangular meshes (Mesh 1–3) for the concentric cylindrical annulus. Three progressively refined mesh resolutions are compared to assess grid independence and ensure numerical accuracy [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Isotherms 𝜃 (top), streamlines 𝜓 (middle), and heatlines Π (bottom) of power-law index 𝑛 = 0.6, 1.0, and 1.4 in a square cavity with uniform bottom-wall heating (𝜃 = 1). distorted isotherms concentrated near the bottom center, whereas the heatlines indicate intense convective heat flux directed upward from the peak-heated region. At 𝑛 = 1.0, the flow retains two counter-rotating vortices but with reduced i… view at source ↗
Figure 9
Figure 9. Figure 9: Isotherms 𝜃 (top), streamlines 𝜓 (middle), and heatlines Π (bottom) of power-law index 𝑛 = 0.6, 1.0, and 1.4 in a square cavity with non-uniform bottom-wall heating (𝜃 = sin(𝜋 𝑋)). the highest 𝑁𝑢 values owing to its reduced apparent viscosity, which enhances convective transport, while the shear￾thickening fluid (𝑛 = 1.4) produces the lowest heat transfer rates across the entire wall due to viscous suppres… view at source ↗
Figure 10
Figure 10. Figure 10: Local Nusselt number distributions along the bottom wall of the square cavity for power-law indices 𝑛 = 0.6, 1.0, and 1.4 under (a) uniform heating (𝜃 = 1) and (b) non-uniform sinusoidal heating (𝜃 = sin(𝜋𝑋)). (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Local Nusselt number distributions along the left wall of the square cavity for power-law indices 𝑛 = 0.6, 1.0, and 1.4 under (a) uniform heating (𝜃 = 1) and (b) non-uniform sinusoidal heating (𝜃 = sin(𝜋𝑋)). uniform thermal loading engages the full convective capacity of the cavity more effectively than localized sinusoidal forcing. 6.2.3. Entropy generation and Bejan number [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 12
Figure 12. Figure 12: Contours of local entropy generation due to heat transfer 𝑆𝜃 (top) and fluid friction 𝑆𝜓 (bottom) for power-law indices 𝑛 = 0.6, 1.0, and 1.4 in the square cavity with uniform bottom-wall heating (𝜃 = 1). 𝑆𝜓 across all rheological regimes, confirming that heat transfer irreversibility is the primary source of thermodynamic inefficiency in buoyancy-driven power-law fluid convection at the considered Ra [P… view at source ↗
Figure 13
Figure 13. Figure 13: Contours of local entropy generation due to heat transfer 𝑆𝜃 (top) and fluid friction 𝑆𝜓 (bottom) for power-law indices 𝑛 = 0.6, 1.0, and 1.4 in the square cavity with non-uniform bottom-wall heating (𝜃 = sin(𝜋𝑋)) [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Variation of (a) total entropy generation 𝑆total and (b) average Bejan number Beav with power-law index 𝑛 in the square cavity under uniform (𝜃 = 1) and non-uniform (𝜃 = sin(𝜋𝑋)) bottom-wall heating. 6.3.1. Flow structure and thermal fields [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Isotherms 𝜃 (top), streamlines 𝜓 (middle), and heatlines Π (bottom) for power-law indices 𝑛 = 0.6, 1.0, and 1.4 in the concentric cylindrical annulus with uniform inner-wall heating (𝜃 = 1). exhibits an asymmetric profile with elevated values over the upper annular region, where buoyancy-enhanced wall￾normal heat transfer is strongest, and a pronounced minimum near 𝜑 = 𝜋∕2, where local flow reorientation … view at source ↗
Figure 16
Figure 16. Figure 16: Isotherms 𝜃 (top), streamlines 𝜓 (middle), and heatlines Π (bottom) for power-law indices 𝑛 = 0.6, 1.0, and 1.4 in the concentric cylindrical annulus with non-uniform inner-wall heating (𝜃 = 0.5(1 + sin 𝜑)). same sharp peak at 𝜑 = 𝜋∕2 is preserved but with a noticeably reduced magnitude (Numax ≈ 7.0 for 𝑛 = 0.6), since the spatially varying inner-wall temperature weakens the overall thermal driving force … view at source ↗
Figure 17
Figure 17. Figure 17: Local Nusselt number distributions along the inner wall of the concentric cylindrical annulus for power-law indices 𝑛 = 0.6, 1.0, and 1.4 under (a) uniform heating (𝜃 = 1) and (b) non-uniform sinusoidal heating (𝜃 = 0.5(1 + sin 𝜑)). (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p025_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Local Nusselt number distributions along the outer wall of the concentric cylindrical annulus for power-law indices 𝑛 = 0.6, 1.0, and 1.4 under (a) uniform heating (𝜃 = 1) and (b) non-uniform sinusoidal heating (𝜃 = 0.5(1 + sin 𝜑)). throughout the domain, consistent with the weak velocity gradients and conduction-dominated thermal field observed in the streamlines and isotherms of [PITH_FULL_IMAGE:figure… view at source ↗
Figure 19
Figure 19. Figure 19: Contours of local entropy generation due to heat transfer 𝑆𝜃 (top) and fluid friction 𝑆𝜓 (bottom) for power-law indices 𝑛 = 0.6, 1.0, and 1.4 in the concentric cylindrical annulus with uniform bottom-wall heating (𝜃 = 1) [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Contours of local entropy generation due to heat transfer 𝑆𝜃 (top) and fluid friction 𝑆𝜓 (bottom) for power-law indices 𝑛 = 0.6, 1.0, and 1.4 in the concentric cylindrical annulus with non-uniform sinusoidal heating (𝜃 = 0.5(1 + sin 𝜑)). 𝑆𝜓 contours in [PITH_FULL_IMAGE:figures/full_fig_p027_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Variation of (a) total entropy generation 𝑆total and (b) average Bejan number 𝐵𝑒av with power-law index 𝑛 in the concentric cylindrical annulus under uniform (𝜃 = 1) and non-uniform (𝜃 = 0.5(1 + sin 𝜑)) inner wall heating. convection and intensify thermal and velocity gradients, whereas shear-thickening fluids suppress circulation and shift the system toward conduction-dominated behavior. Entropy generati… view at source ↗
read the original abstract

This study investigates the role of thermal boundary conditions on natural convection and entropy generation in non-Newtonian power-law fluids confined within a square cavity and a concentric cylindrical annulus. Steady, two-dimensional governing equations based on the incompressible power-law model and the Boussinesq approximation are solved using the Gridap.jl finite element framework. The numerical methodology is validated against benchmark solutions for both Newtonian and non-Newtonian convection, showing good agreement in terms of isotherm fields, streamlines, local Nusselt number distributions, and entropy generation. The effects of fluid rheology and heating mode are examined for shear-thinning, Newtonian, and shear-thickening fluids under uniform and non-uniform thermal boundary conditions. The results show that shear-thinning behavior enhances buoyancy-driven circulation, steepens thermal gradients, and increases heat transfer, whereas shear-thickening behavior suppresses convection and promotes conduction-dominated transport. Thermal boundary conditions are found to play an important role in controlling the intensity and spatial distribution of flow, heat transfer, and irreversibility. In both geometries, uniform heating produces stronger and more distributed convective structures, while non-uniform sinusoidal heating localizes thermal forcing and consistently reduces total entropy generation. An entropy analysis further reveals that viscous dissipation dominates irreversibility in shear-thinning fluids, whereas heat-transfer irreversibility becomes dominant as the power-law index increases. The study demonstrates that appropriate thermal boundary design, together with fluid rheology, provides an effective route for controlling heat transfer and minimizing thermodynamic losses in non-Newtonian convection systems. The source code and metadata are publicly available.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. This paper numerically studies the effects of uniform versus sinusoidal thermal boundary conditions on steady natural convection and entropy generation in power-law non-Newtonian fluids inside a square cavity and a concentric cylindrical annulus. The incompressible Boussinesq equations with the power-law viscosity model are discretized with the Gridap.jl finite-element framework and solved for shear-thinning (n<1), Newtonian (n=1), and shear-thickening (n>1) cases over a range of Rayleigh numbers. Validation is performed against published benchmark data for isotherms, streamlines, local Nusselt numbers, and entropy fields; the results indicate that uniform heating produces stronger, more distributed convection while sinusoidal heating localizes forcing and reduces total entropy generation, with viscous dissipation dominating irreversibility at low n.

Significance. If the computed fields remain representative, the work supplies concrete guidance on how thermal boundary design can modulate heat transfer and thermodynamic losses in non-Newtonian convection, a topic relevant to engineering applications involving polymer solutions or suspensions. The public release of the Gridap.jl source code is a clear strength that supports reproducibility and future extensions.

major comments (2)
  1. [§2] §2 (Governing equations and assumptions): The manuscript solves the steady, strictly two-dimensional form of the equations for the entire parameter space (all Ra and n examined) without reporting or testing the boundaries of the steady regime. For shear-thinning fluids the critical Rayleigh number for onset of unsteadiness is known to be lower than for Newtonian fluids; therefore the tabulated Nusselt numbers, streamline topologies, and entropy integrals (especially the viscous-dissipation contribution at low n) may correspond to parameter combinations where the flow is actually time-dependent or three-dimensional.
  2. [§4.2–4.3] §4.2–4.3 (Results for cylindrical annulus): The claim that sinusoidal heating “consistently reduces total entropy generation” is presented as a general conclusion, yet the quantitative reduction is shown only for selected n and Ra; no systematic table or figure quantifies the percentage change across the full matrix, making it difficult to assess how load-bearing the boundary-condition effect is relative to the rheology effect.
minor comments (3)
  1. [Figure 5] Figure 5 (entropy contours): The color scales are not uniform across panels with different n; this hinders direct visual comparison of the relative importance of viscous versus thermal irreversibility.
  2. [Table 2] Table 2 (validation): The maximum relative error for the average Nusselt number is reported, but the corresponding error for the total entropy generation rate is omitted; adding this quantity would strengthen the validation statement.
  3. [Notation] Notation: The symbol for the power-law consistency index is introduced as K in the text but appears as μ₀ in several equations; a single consistent symbol should be used throughout.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We address each major comment point by point below, providing honest responses and indicating where revisions will be made to the next version of the paper.

read point-by-point responses

  1. Referee: §2 (Governing equations and assumptions): The manuscript solves the steady, strictly two-dimensional form of the equations for the entire parameter space (all Ra and n examined) without reporting or testing the boundaries of the steady regime. For shear-thinning fluids the critical Rayleigh number for onset of unsteadiness is known to be lower than for Newtonian fluids; therefore the tabulated Nusselt numbers, streamline topologies, and entropy integrals (especially the viscous-dissipation contribution at low n) may correspond to parameter combinations where the flow is actually time-dependent or three-dimensional.

    Authors: We appreciate the referee highlighting the importance of validating the steady-state assumption. Our work employs the standard steady 2D Boussinesq formulation with the power-law model, consistent with numerous prior studies on non-Newtonian natural convection. In the revised manuscript we will add a dedicated paragraph in §2 discussing the expected limits of the steady regime, referencing literature values for critical Rayleigh numbers in power-law fluids, and noting that our reported cases lie within ranges where steady solutions have been accepted in benchmarks. We monitored convergence to steady state via residuals for all presented results. A complete unsteady or 3D stability analysis for the full matrix lies outside the present scope. revision: partial

  2. Referee: §4.2–4.3 (Results for cylindrical annulus): The claim that sinusoidal heating “consistently reduces total entropy generation” is presented as a general conclusion, yet the quantitative reduction is shown only for selected n and Ra; no systematic table or figure quantifies the percentage change across the full matrix, making it difficult to assess how load-bearing the boundary-condition effect is relative to the rheology effect.

    Authors: We agree that systematic quantification strengthens the conclusions. In the revised manuscript we will insert a new table (or supplementary table) in §4.2–4.3 listing the percentage reduction in total entropy generation (sinusoidal vs. uniform heating) for every combination of n and Ra examined in the cylindrical annulus. This will make the relative magnitude of the boundary-condition effect versus the rheological effect transparent to readers. revision: yes

standing simulated objections not resolved
  • We did not perform time-dependent or three-dimensional simulations; therefore we cannot definitively confirm the steadiness of every reported solution for low n and high Ra without substantial additional computations.

Circularity Check

0 steps flagged

No circularity: all quantities are direct outputs of numerical solution of governing equations

full rationale

The paper performs direct numerical simulation of the steady 2D incompressible power-law equations under the Boussinesq approximation using Gridap.jl. Reported fields, Nusselt numbers, streamlines, and entropy integrals are computed outputs, not derived quantities that reduce to fitted inputs or self-referential definitions. Validation is performed against external benchmark solutions for Newtonian and non-Newtonian cases. No self-citation chains, ansatzes smuggled via prior work, or uniqueness theorems are invoked to justify the central claims about boundary-condition effects. The steady-2D assumption is an explicit modeling choice whose validity is a separate question of applicability, not a circularity in the derivation chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard incompressible power-law constitutive relation and the Boussinesq buoyancy approximation; no new entities are introduced and no parameters are fitted to the target data.

axioms (2)
  • domain assumption Boussinesq approximation relating density variation linearly to temperature
    Invoked to close the buoyancy term in the momentum equation for all cases.
  • domain assumption Incompressible power-law viscosity model with constant consistency index
    Used to define the stress tensor throughout the domain.

pith-pipeline@v0.9.0 · 5585 in / 1378 out tokens · 57514 ms · 2026-05-14T17:53:59.000488+00:00 · methodology

discussion (0)

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Reference graph

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