What if adding a tiny amount of polymer could turn an ordinary channel flow into a heat-transfer machine? This study finds that the answer depends on whether the polymer stress forms the right kind of hook. Modern electrical devices demand efficient cooling systems to dissipate heat and maintain optimal operating conditions.

The typical heat generated by modern high-performance electronics requires cooling heat fluxes exceeding ten to the third watts per square centimetre. The present primary cooling system remains air based, but the next generation focuses on fluid-flow-based systems, making hydrodynamics important in contemporary thermal management.

In practical cooling configurations, a working fluid passes through a channel where one wall is heated and the opposite wall is much cooler, producing a thermally stratified layer. The Rayleigh–Bénard–Poiseuille flow captures this interplay of forced and buoyancy-driven motions: a pressure-driven stream in a horizontal channel is subjected to a transverse temperature gradient.

Instabilities arise through shear mechanisms, which depend on the Reynolds number and provoke Tollmien–Schlichting waves, or buoyancy-driven modes, governed by the Rayleigh or Richardson number and producing convective rolls. In microscale channels, the Rayleigh number can fall below the critical threshold for buoyancy-driven instability, leaving the flow laminar and eliminating buoyancy-induced mixing.

One potential strategy to overcome the suppression of instability and mixing is adding long-chain polymers to the working fluid. Stretching and relaxation of these polymer chains give rise to elastic stresses, which can fundamentally modify flow stability. In curved geometries, hoop stress polymer instability can lead to elastic turbulence, and the literature on elasticity-induced heat-transfer enhancement has largely focused on this phenomenon.

In a serpentine channel, heat transfer increased by up to two hundred percent for low polymer concentrations and up to three hundred and eighty percent for higher concentrations compared with equivalent Newtonian flows. In straight-channel flows, some studies report reduced heat transfer, while others observe both enhancement and suppression, with reported enhancement remaining modest: approximately two percent or approximately ten percent.

The centre mode and polymer diffusive instability can be triggered in a straight channel. However, the effectiveness of the resulting motions for heat flux has yet to be explored. This study focuses specifically on cases where the centre mode is excited.

The study uses linear stability analysis and direct numerical simulations to investigate heat-transfer enhancement in viscoelastic, unstably stratified Rayleigh–Bénard–Poiseuille flow and stably stratified shear flow. Previous studies had focused on stability analysis, but heat-transfer enhancement had not been investigated because assessing it requires a finite-amplitude manifestation of the instability.

Linear stability analysis had found that viscoelasticity can destabilise the flow at low and high Reynolds numbers but stabilise it at moderate Reynolds numbers. The system is pressure-driven flow of an incompressible viscoelastic fluid in a channel with two walls at positive and negative H.

The dimensionless equations include continuity, Navier–Stokes, Oldroyd-B, and density transport equations under the Boussinesq approximation. Figure one schematically shows Poiseuille flow between an upper wall at z-star equals H and a lower wall at z-star equals minus H, giving a channel height of two H.

The curved profile represents the base flow, with velocity arrows increasing toward the centreline and the centreline speed marked U-star. This geometry matters because the paper uses the channel half-height H and centreline speed U-star to nondimensionalise the velocity and governing equations.

The Reynolds number is taken to be of order ten squared to match applications in microchannels, and a negative Richardson number indicates unstable stratification. The Prandtl number is fixed to seven to represent water at room temperature, while the Weissenberg number describes the polymer-related parameter in the model.

The imposed pressure gradient is fixed in time for a given Reynolds number, whereas the mass flux varies with the exact flow state. The Oldroyd-B model represents the polymer dynamics because of its relative mathematical simplicity and the small number of extra parameters it introduces.

The model can also be derived from a microscopic description in which each polymer molecule is represented as two masses connected by a Hookean spring, or dumbbell. A known deficiency is that the dumbbells can be infinitely extended, so repeating the computations with a more sophisticated model would be valuable.

Linear stability analysis provides a preliminary overview of the unstable modes in the chosen parameter space. Two unstable modes are identified in the Weissenberg-number and Richardson-number parameter space: a convective mode and an elastic centre mode.

The analysis examines how the Weissenberg number affects the convective mode and how the Richardson number affects the elastic centre mode. The convective mode, corresponding to transverse rolls, appears only in the unstably stratified region once the Richardson number falls below a threshold.

The elastic centre mode is present in both stable and unstable stratification at large Weissenberg number, and it can trigger instability even when the corresponding Newtonian state is linearly stable. In both panels, the black dashed boundary traces the handoff between the lower-left convective region and the higher-Weissenberg centre-mode region.

The labelled points make this contrast concrete: A through D sit in the convective area, while E and F lie in the centre-mode area, with F positioned on the stably stratified side of the vertical zero-Richardson line. Two-dimensional direct numerical simulations investigate heat transfer associated with the nonlinear states.

The centre mode produces only weak heat-transfer enhancement, whereas the convective mode yields substantial enhancement and produces travelling waves or periodic orbits with a hook-shaped polymer stress. A survey of Reynolds number and Weissenberg number identifies significant enhancement at sufficiently large Reynolds number and within a certain range of Weissenberg number.

Flow-field analysis links this behaviour to a wall-attached hook-shaped structure, while perturbation kinetic energy budgets show that buoyancy and polymer stresses compete to sustain the states. The simulations use the open-source pseudospectral code Dedalus to evolve the nonlinear perturbation away from the base state.

A small global diffusion term acts on the possibly large perturbation to stabilise the direct numerical simulation. The Schmidt number is fixed at five hundred for the convective mode and one thousand for the centre mode. Heat transfer is quantified using a Nusselt number normalised by the conductive heat flux.

The temporal evolution of six representative cases is used to characterise the nonlinear states and their associated heat transfer. The Newtonian convective mode A provides the baseline for viscoelastic convective modes B and C, while only one convective mode at Reynolds number eight hundred is represented because of the computational cost of direct numerical simulation.

Figure six tracks how the four convective modes develop: panels a and b show perturbation kinetic energy and Nusselt number over time, while panels c through f visualize their resulting flow or stress fields. Mode C exhibits a periodic orbit, with its stress field captured when the Nusselt number peaks; modes A, B, and D settle into travelling-wave states, stationary in their co-moving frames.

The measured energy-growth rates agree closely with the linear-stability predictions, linking the early evolution to the observed nonlinear states. The centre-mode cases evolve into travelling-wave solutions with steady Nusselt number and a nonlinear arrowhead structure in the polymer-stress field, localised near the channel centreline.

The associated vertical velocity perturbation is too weak to convect heat efficiently across the channel, producing only an approximately zero point zero three zero percent increase in the mean Nusselt number relative to the laminar Newtonian counterpart. The centre-mode cases and their arrowhead structure are therefore ineffective for heat-transfer enhancement in this flow.

The mean Nusselt number increases monotonically with Reynolds number and undergoes a sudden transition near Reynolds number two hundred to a region with substantially high mean Nusselt number. A maximum mean Nusselt number of ten point four seven six is achieved at Reynolds number two hundred and fifty with Weissenberg number twelve point five.

That value is approximately seven hundred percent more than its Newtonian counterpart, whose mean Nusselt number is one point three two three. Figure seven maps the mean Nusselt number across Reynolds number and wave parameters for fixed beta and Richardson number, marking the locations of convective modes A, B, and C.

The white dashed line at W equals twelve point five identifies the slice examined in panel b, where the authors compare Reynolds-number variations across Schmidt numbers and with the Newtonian counterpart. This matters because it connects parameter-space structure to the paper’s broader finding that elasticity promotes heat transfer.

The simulations at higher Reynolds number use fixed Weissenberg number twelve point five, and the results are insensitive to Schmidt number. All of these flows are travelling waves, and the monotonic increase of mean Nusselt number continues toward Reynolds number five hundred. The maximum mean Nusselt number is fifteen point eight zero seven at Reynolds number five hundred and Schmidt number two hundred and fifty, representing approximately eleven hundred percent enhancement over the Newtonian case, whose mean Nusselt number is one point two nine one.

Figure eight links the flow structures to contours of the mean Nusselt number for fixed beta equal to zero point nine eight and Richardson number minus zero point eight. Panels (a) through (d) show travelling waves in co-moving frames, while panel (e) captures a periodic orbit when Nusselt number is maximal; the figure also contrasts these streamlines with laboratory-frame flow rates and the parabolic base profile.

The authors use the progressively stronger hook-like structures to connect elasticity with enhanced heat transfer. At sufficiently large Reynolds number and within a certain range of Weissenberg number, the hook structure attaches to the wall. The corresponding mean Nusselt number experiences a sharp increase, producing the high-enhancement region.

The boundary interaction drives strong flow from the wall toward the channel centre and forces the fluid into pairs of strong counter-rotating rolls. Those rolls drastically enhance vertical heat transport, but they also strongly reduce the flow rate.

Figure eleven maps Reynolds number against W, overlaying contours of the mean Nusselt number with the energy-budget ratio T over B. The dashed boundaries divide Newtonian-like travelling waves and periodic orbits from detached and attached travelling-wave regimes, while the cyan contour marks where T over B changes sign and the green curve marks zero HTE.

The visual matters because it connects regime transitions and heat-transfer behavior with whether polymer stresses contribute to sustaining the flow. A region with thermal performance factor greater than one is identified as attractive for heat-transport applications such as cooling systems.

In this regime, the nonlinear state is a wall-detached hook and may be either a travelling wave or a periodic orbit. The maximum thermal performance factor is one point three eight five at Reynolds number one hundred and seventy-five and Weissenberg number twelve point five, with mean Nusselt number three point four five two, approximately one hundred and fifty percent above the Newtonian value of one point three five one.

The corresponding Stanton number is zero point zero one zero five five, approximately six times the Newtonian value of zero point zero zero one six four eight. Figure thirteen maps the thermal performance factor eta and Stanton number across Reynolds number and W, with black dashed lines marking the flow-regime boundaries.

The upper blue eta region is about zero point five, while the upper green Stanton-number region is about zero point six; both align with the strong heat-transfer region and wall-attached hook structure identified elsewhere. A red area with eta above one marks a wall-detached-hook regime relevant to heat-transport applications, including cooling systems.

The key result is that convective states with polymer-stress hooks can raise heat transfer dramatically, but the most intense enhancement comes with a major flow-rate penalty. Detached hooks offer a more efficient compromise.