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Perturbative modelling of Arrhenius reactive transport in non-isothermal Gordon– Schowalter Couette–Poiseuille flow

This paper presents a perturbative analysis of coupled heat and mass transfer in non-isothermal Gordon–Schowalter viscoelastic Couette–Poiseuille flow with Arrhenius kinetics, establishing that the convected-derivative parameter significantly influences thermal and reactive responses through competing advective and thermal-reaction mechanisms governed by a single crossover number that dictates reactor sizing.

Original authors: Leonardo Damián Soria Rodríguez, Anthony Albert Harrup Gutiérrez

Published 2026-08-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Leonardo Damián Soria Rodríguez, Anthony Albert Harrup Gutiérrez

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where fluids don't just flow like water; they stretch, bounce, and remember their shape, like a very slow, sticky piece of chewing gum. This is the realm of viscoelastic fluids, which include things like polymer melts, ketchup, and even some biological fluids. When these fluids are squeezed through narrow channels, they generate heat just by the friction of moving past each other, a bit like how rubbing your hands together makes them warm. This heat is dangerous for chemical engineers because many chemical reactions are incredibly sensitive to temperature: a tiny bit of extra warmth can make a reaction explode in speed, while a tiny bit of cooling can stall it completely.

Now, imagine trying to bake a cake in a long, narrow tube where the batter is constantly moving, generating its own heat, and reacting to turn into a cake. If you don't understand how the "stretchy" nature of the batter changes the heat and the speed of the reaction, your cake might be undercooked in the middle or burnt on the edges. This is the puzzle of reactive transport: figuring out how heat, fluid motion, and chemical reactions dance together. Engineers need to know exactly how long a tube needs to be to get the reaction just right. If they guess wrong, they waste money on a tube that's too long, or worse, they get a product that isn't fully made because the tube was too short.

This paper dives into that exact problem, but with a twist. The researchers looked at a specific type of stretchy fluid (called a Gordon–Schowalter fluid) flowing between two moving plates. They wanted to know: Does the fluid's stretchiness change how long the reaction tube needs to be?

The answer is a fascinating "yes, but it's complicated." The team found that the stretchiness of the fluid creates a tug-of-war with two very different effects.

First, there is the Traffic Effect. Because the fluid is stretchy, it actually flows faster through the tube when pushed by the same pressure (it "thins out" under stress). Imagine a crowd of people walking through a hallway; if they suddenly start moving more efficiently, they get to the end of the hall faster. In the chemical tube, this means the reacting material zooms through the tube more quickly, giving it less time to react. This effect tries to make the tube need to be longer to finish the job.

Second, there is the Hot-Box Effect. Because the fluid is moving faster and stretching, it generates more friction heat. Since the chemical reaction loves heat, this extra warmth makes the reaction speed up dramatically. It's like turning up the oven temperature; the cake bakes much faster. This effect tries to make the tube need to be shorter because the reaction finishes sooner.

The paper's main discovery is that these two effects are fighting each other, and which one wins depends on a specific "crossover number" (which the authors call K).

If the reaction isn't too sensitive to heat (a low K), the Traffic Effect wins. The fluid moves so fast that the reaction doesn't finish in time, and you need a longer tube than you would for normal water.
If the reaction is very sensitive to heat (a high K), the Hot-Box Effect wins. The extra heat makes the reaction so fast that you actually need a shorter tube than you would for normal water.

The researchers calculated exactly where this switch happens. They found that for many common situations, the "Traffic Effect" is stronger, meaning engineers might be building tubes that are slightly too short if they ignore the fluid's stretchiness. However, for very hot, fast-reacting systems, the opposite is true.

One of the most interesting findings is about the type of stretchiness. The fluid's behavior depends on a parameter called the "convected-derivative parameter" (represented by the letter a). The paper proves mathematically that if the fluid behaves like a "perfect" upper- or lower-convected Maxwell fluid (where a is exactly +1 or -1), this whole tug-of-war disappears, and the fluid acts just like normal water. But for any other type of stretchy fluid (where a is between -1 and +1), this tug-of-war is real and measurable. The effect is strongest when the fluid is "corotational" (a = 0), which is a specific way the fluid spins and stretches.

The authors didn't just guess this; they built a detailed mathematical model and checked it against powerful computer simulations. They found their math was incredibly accurate, with errors less than 2% in their predictions. They even provided a "design map" (a visual chart) that engineers can use to look up their specific conditions and see if they need a longer or shorter tube.

In short, this paper gives engineers a new rulebook for designing chemical reactors with stretchy fluids. It shows that you can't just treat these fluids like water; their internal "elasticity" creates a hidden battle between speed and heat that determines whether your chemical process needs more space or less. By understanding this battle, engineers can build better, more efficient reactors for making everything from plastic coatings to life-saving medicines.

Technical Summary: Perturbative Modelling of Arrhenius Reactive Transport in Non-Isothermal Gordon–Schowalter Couette–Poiseuille Flow

Problem Statement
This study addresses the coupled heat and mass transfer problem in a plane Couette–Poiseuille flow of a viscoelastic fluid undergoing a first-order homogeneous Arrhenius reaction. The specific physical system involves a Gordon–Schowalter fluid flowing between two parallel plates separated by a distance $H_0$ . The plates move with identical velocity $V$ (eliminating net Couette shear in the plate frame), and the flow is driven entirely by an external pressure gradient. The walls are isothermal at temperature $T_0$ and perfectly reactive (Dirichlet boundary condition). The fluid exhibits viscous dissipation, which acts as a distributed heat source, and the reaction rate is exponentially sensitive to temperature. The analysis is conducted in the regime of small Brinkman number ( $Br \ll 1$ ) and weakly viscoelastic flow ($We < 1$), focusing on the "triple coupling" of viscoelasticity, thermal dissipation, and chemical kinetics—a scenario not previously addressed in the literature.

Methodology
The authors employ a perturbation method based on a composite parameter $\varepsilon \equiv (1-a^2)We^2$ , where $a$ is the convected-derivative parameter ( $a \in [-1, 1]$ ) and $We$ is the Weissenberg number.

Constitutive Modeling: The Gordon–Schowalter constitutive equation is used, which admits a closed-form shear stress solution in steady simple shear. The shear viscosity is rate-thinning for $|a| < 1$ but constant for the upper- and lower-convected Maxwell limits ( $a = \pm 1$ ).
Perturbative Hierarchy: The velocity, temperature, and concentration fields are expanded in powers of $\varepsilon$ $ε$ and $Br$. The analysis isolates three distinct physical contributions:
- The base Newtonian reactive Graetz field.
- The Newtonian thermal–reactive correction (viscous heating in a Newtonian fluid).
- The mixed viscoelastic correction, which requires the simultaneous presence of elasticity, dissipation, and kinetic sensitivity.
Analytical Solution: The velocity correction is derived by inverting the constitutive law at fixed stress. The temperature field is obtained via closed-form integration of the energy balance. The reactive field is treated as a Sturm–Liouville eigenvalue problem, where the viscoelastic perturbation induces a shift in the fundamental eigenvalue ( $\lambda_1^2$ ), governing the axial decay rate and conversion length.
Numerical Validation: The analytical results are validated against direct numerical solutions of the energy balance and the full reactive transport equation using shooting methods and finite-difference discretization.

Key Contributions and Results

The Mixed Group Signature: The leading viscoelastic correction to the thermal–reactive response is proportional to the mixed group $Br \cdot We^2 (1-a^2)$ . This term vanishes identically for the upper-convected Maxwell ( $a=1$ ) and lower-convected Maxwell ( $a=-1$ ) limits, where the shear viscosity remains constant. The effect is maximal for the corotational case ( $a=0$ ). Consequently, the convected-derivative parameter $a$ acquires a measurable signature in the temperature rise, axial decay rate, and conversion length, not merely in the stress field.
Dual Competing Routes: The viscoelastic perturbation influences the concentration field through two additive, first-order routes of opposite sign:
- Advective Route: At a fixed pressure gradient, the rate-thinning response increases the through-flow velocity. This carries the reactant farther before consumption, increasing the conversion length (negative shift in eigenvalue).
- Thermal–Reactive Route: The same rate-thinning increases the shear rate and thus viscous dissipation at fixed driving stress. This raises the temperature, accelerating the Arrhenius kinetics and decreasing the conversion length (positive shift in eigenvalue).
The Crossover Number ( $K$ ): The competition between these routes is governed by a single dimensionless crossover number $K$ .
- If $K < 1$ , the advective route dominates: viscoelasticity increases the required reactor length compared to a Newtonian design.
- If $K > 1$ , the thermal–reactive route dominates: viscous heating accelerates the reaction sufficiently to shorten the required reactor length.
- The sign of the net viscoelastic effect on reactor sizing reverses exactly at $K=1$ .
Quantitative Accuracy: Closed-form expressions for velocity and temperature corrections are derived. The eigenvalue shift predicted by first-order perturbation theory matches direct numerical solutions with a relative error better than 1.9%. The additivity of the two routes is verified to within 0.13%.

Significance and Claims
The paper claims to provide the first structural analysis of how viscoelasticity signatures manifest in reactive transport when thermal coupling is present.

Structural Insight: It demonstrates that standard Maxwell models ( $a=\pm 1$ ) are "structurally blind" to this specific viscoelastic signature in rectilinear shear flows because normal stresses perform no work in this kinematics. Only fluids with mixed convected derivatives ( $|a| < 1$ ) exhibit this effect.
Engineering Utility: The study offers a compact design rule for reactive coating, polymer processing, and channel-reactor applications. The derived crossover number $K$ allows engineers to predict whether a Newtonian-based design will under-predict or over-predict the required reactor length when dealing with viscoelastic fluids.
Design Map: A parametric map in the $(Da_{II}, Br \gamma_A)$ plane is presented to delineate regions where the viscoelastic correction is negligible, advection-dominated, or kinetics-dominated.

The authors conclude that their model serves as a fully reproducible benchmark for systems where viscous heating, reaction kinetics, and elasticity interact, noting that the sign reversal of the net effect is a robust feature even beyond the strict linearization limits of the Arrhenius term.

Technical Summary: Perturbative Modelling of Arrhenius Reactive Transport in Non-Isothermal Gordon–Schowalter Couette–Poiseuille Flow

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