Viscoelastic Flow and Heat Transfer in Porous Media: Global Sensitivity, Thermal–Momentum Trade-Offs, and Interpretable Surrogate Modeling

Imagine a fluid that behaves like a thick, stretchy honey, flowing not through open space but through a sponge-like material riddled with tiny holes. This is the world of viscoelastic flow in porous media, a scenario that appears in everything from oil extraction deep underground to the cooling systems of advanced electronics. In these environments, the fluid's ability to stretch and snap back—its elasticity—clashes with the resistance offered by the solid sponge, while heat tries to move through the mixture at its own pace. Engineers and scientists have long known that changing the fluid's properties or the sponge's density alters how fast the fluid moves and how well it carries heat. However, for decades, the standard way to study this involved changing one factor at a time while holding everything else steady. This approach often missed the bigger picture, failing to reveal how these factors work together across the entire range of possibilities or whether a change that helps heat transfer might accidentally hurt the fluid's movement.

A team of researchers has now mapped this complex landscape with a new level of clarity, using a powerful combination of computer simulations and artificial intelligence to untangle the relationship between heat and motion. Instead of looking at isolated snapshots, they generated thousands of virtual experiments to see how the fluid behaves when the elasticity, the sponge's resistance, and the fluid's thermal properties are all varied at once. Their work reveals a surprising truth: the factors that control how well heat moves through the system are almost entirely different from the factors that control how the fluid flows. While one property acts as the master switch for heat transfer, a completely different property dictates the fluid's speed and force near the walls. This separation means that trying to improve cooling without slowing down the flow is a delicate balancing act, where the best solution is rarely a single "perfect" setting but rather a specific compromise between two competing goals.

The researchers began by building a rigorous digital model of the fluid flowing through the porous material under a constant stream of heat. They verified their computer code against established methods to ensure their virtual experiments were accurate, then used a sophisticated sampling technique to run 2,000 distinct simulations. In these runs, they varied three key ingredients: the fluid's elasticity, the resistance of the porous material, and a number that describes how easily heat spreads compared to how easily the fluid moves. By analyzing the results of these thousands of scenarios, they applied a statistical method designed to measure exactly how much each ingredient contributed to the final outcome. The results were strikingly clear. When it came to the speed of the fluid right next to the wall—a measure of how much the fluid is being dragged or pushed—the fluid's elasticity was the dominant force, accounting for more than sixty percent of the changes. The resistance of the porous material played a secondary role, while the fluid's thermal properties had almost no effect on the flow speed.

In a complete reversal, the story for heat transfer was different. When the researchers looked at how effectively heat was moving away from the surface, the fluid's thermal properties took center stage, controlling nearly seventy percent of the variation. The elasticity of the fluid, which ruled the movement, became a minor player in the heat story. This finding dismantles the idea that a single setting can optimize both flow and cooling simultaneously. The team then used a technique called Pareto analysis to explore the trade-offs, asking what happens when you try to maximize heat transfer while minimizing the drag on the fluid. They found that these two goals are in direct competition. Pushing for better cooling almost always increases the drag on the fluid, and reducing the drag usually means less efficient cooling. Out of 5,120 possible combinations they tested, only a tiny fraction offered a high level of cooling with low drag, and even those required a very specific, narrow set of conditions.

To make sense of this vast amount of data and to predict outcomes without running new simulations every time, the team trained machine learning models to act as surrogates for their complex physics equations. These models learned the patterns in the data and proved remarkably accurate, predicting the results of new, unseen scenarios with high precision. More importantly, the artificial intelligence "explained" its own reasoning in a way that matched the statistical analysis. The models identified the same dominant factors: elasticity for flow and thermal properties for heat. This agreement between the statistical math and the machine learning's internal logic gave the researchers confidence that their findings were not just numerical artifacts but reflected the true physical behavior of the system.

The study concludes that in this specific type of flow, thermal and momentum responses are governed by separate mechanisms. If an engineer wants to improve cooling, they should focus on adjusting the thermal properties of the fluid, knowing that this will likely increase the drag. If they need to reduce drag, they must adjust the fluid's elasticity, accepting that this may not help with cooling. The "best" solution is not a single point but a specific compromise found near the upper limits of the thermal properties tested, where the fluid is highly conductive and the porous material offers moderate resistance. This work provides a clear, reproducible framework for understanding these trade-offs, moving beyond simple one-factor-at-a-time experiments to a holistic view of how heat and motion interact in complex, stretchy fluids moving through sponges.