What if making a lubricating polymer layer more elastic eventually makes it less useful? This study finds that drag reduction rises toward roughly twenty-four percent, then largely stalls because the polymers suppress the turbulence that stretches them.

Drag reduction is a long-standing goal in both fundamental fluid mechanics and engineering practice. Two especially effective strategies are thin, less viscous lubricating films near the wall and dilute polymer solutions that modify near-wall turbulence.

The lubricating film carries most of the shear and shields the bulk fluid from direct contact with the solid boundary, while dilute polymers can produce dramatic reductions of frictional losses. A previous study combined these mechanisms with a thin non-Newtonian layer, but it used only one Weissenberg number, leaving open how elasticity controls drag-reduction performance.

The central question is whether a viscoelastic lubricating layer has an upper bound on drag reduction, analogous to the bounded drag reduction known for single-phase polymer solutions. Despite their practical importance, multiphase flows involving non-Newtonian fluids have received relatively limited attention, and far fewer studies have addressed multiphase systems than single-phase non-Newtonian flows.

Most existing studies examine drops and bubbles in non-Newtonian fluids, with only a small number addressing turbulent conditions. A major reason is the numerical challenge of solving the conformation tensor equation while keeping the conformation tensor symmetric positive definite, especially when deformable interfaces create sharp jumps in fluid properties and viscoelastic parameters.

The earlier simulations compared Newtonian, shear-thinning, combined shear-thinning and viscoelastic, and purely viscoelastic models for the lubricating layer. That work found that surface tension provides a substantial contribution to drag reduction, while viscoelasticity is equally important: both FENE-P and FENE-CR layers doubled the drag-reduction performance relative to the Newtonian-lubricating-layer case.

Shear-thinning effects produced only marginal gains, motivating a new campaign of high-resolution simulations focused on elasticity. The simulated channel contains two immiscible fluid layers, one above the other: a main Newtonian layer with nominal thickness one point seven h and a thin purely viscoelastic layer with nominal thickness zero point three h.

The channel dimensions are four pi h by two pi h by two h, with x, y and z denoting the streamwise, spanwise and wall-normal directions. The method uses direct numerical simulations of the incompressible Navier–Stokes equations coupled with a diffuse-interface phase-field method based on a Cahn–Hilliard equation.

When the lubricating layer is viscoelastic, the calculation also solves a transport equation for the polymer conformation tensor. Figure one compares instantaneous lubricated turbulent flows for a Newtonian layer, a weakly elastic layer with Wi sub tau equal to fifty, and a highly elastic layer with Wi sub tau equal to one hundred.

Grayscale shows streamwise velocity, while yellow-to-blue shading shows elastic potential energy, or EPE, linked directly to polymer stretching. The close-ups mark the mean interface in red and show that increasing polymer relaxation time produces a more homogeneous EPE distribution and smoother interfacial waves.

The deformable interface is represented by a phase-field order parameter: phi equals one in the main Newtonian layer and minus one in the viscoelastic lubricating layer. The phase-field variable changes smoothly across a thin interfacial region between those two bulk values.

Its evolution is governed by the Cahn–Hilliard equation, with the Peclet number quantifying the ratio between convective and diffusive interface time scales. The two-phase hydrodynamics couples the Cahn–Hilliard equation with the incompressible Navier–Stokes equations in a one-fluid formulation.

A single set of governing equations is solved across the whole domain, while a capillary term accounts for the deformable interface and its surface-tension forces. The resulting equations include mass conservation and momentum equations in dimensionless form.

The polymer conformation tensor describes the polymer state, and the shear Weissenberg number is defined using the polymer relaxation time. Finite extensibility effects are included through an extensibility limit, and the conformation tensor evolves according to its transport equation.

The governing equations are solved using FLOW36, an open-source, GPU-ready pseudo-spectral solver. Spatial discretisation uses Fourier expansions in the two homogeneous directions and Chebyshev polynomials in the wall-normal direction. The Navier–Stokes equations use a wall-normal velocity–vorticity formulation, while the Cahn–Hilliard equation is reformulated as two coupled second-order equations.

Time integration uses an implicit–explicit scheme, with the linear and nonlinear contributions advanced using the stated Crank–Nicolson, backward Euler and Adams–Bashforth treatments. The simulations use a nominal friction Reynolds number of three hundred and a fixed Weber number of zero point five.

The lubricating layer is tested at shear Weissenberg numbers twenty-five, fifty, seventy-five and one hundred. The Cahn number is zero point zero two, and the Peclet number is one hundred fifty to ensure convergence toward the sharp-interface limit. The zero-shear viscosity of the lubricating layer is matched to that of the Newtonian main layer, and the polymer extensibility parameter is thirty.

The single-phase and Newtonian–Newtonian simulations use a grid of five hundred twelve by two hundred fifty-six by two hundred fifty-seven points. All viscoelastic lubricated simulations use a refined grid of five hundred twelve by five hundred twelve by five hundred thirteen points.

The increased resolution is required for numerical stability in the conformation-tensor equation and for resolving sharp polymer-stress gradients near the wall and interface. Because FENE-CR represents a purely viscoelastic fluid with constant viscosity, the refinement resolves polymer-stress fluctuations rather than shear-dependent viscosity variations.

The results begin with mean-flow behaviour, and the cases are distinguished by a colour scheme: black for single-phase flow, blue for a Newtonian lubricating layer, and purple through yellow for increasing viscoelasticity. Figure two panel a shows the wall-normal behaviour of the mean streamwise velocity.

Drag reduction is measured as the relative increase in total flow rate compared with the single-phase case, because every simulation uses a constant pressure gradient. A Newtonian lubricating layer already increases the total flow rate by twelve point two percent.

With viscoelasticity, the increase reaches eighteen point nine percent at a Weissenberg number of twenty-five and twenty-one point seven percent at fifty. At seventy-five and one hundred, the increases are twenty-three point six and twenty-four point one percent, with progressively smaller gains as elasticity rises.

The main-layer flow rate follows the same monotonic trend, increasing from one point zero zero six in the Newtonian lubricated case to one point one one eight at a Weissenberg number of one hundred. Figure 2 shows how increasing elasticity in the lubricating layer changes both the mean streamwise velocity profile and the normalized flow rates.

In panel (a), the colored profiles are compared with the black single-phase turbulent profile, with the nominal interface marked at z over h equals negative zero point seven. Panel (b) separates the total flow rate from the contribution of the main Newtonian layer as functions of the Weissenberg number, making the macroscopic effects of lubrication and elasticity directly visible.

The next analysis examines how the lubricating layer modifies turbulence statistics using the wall-normal behaviour of the Reynolds stress tensor components. The figures show the three normal components and the Reynolds shear stress, normalised by the squared friction velocity of the single-phase case and reported in outer units.

The liquid–liquid interface is located at minus zero point seven h, or zero point three h from the bottom wall. In the single-phase case, the streamwise normal stress has the classical near-wall peak near both walls, with a maximum in the buffer layer at approximately fifteen to twenty wall units.

Adding a Newtonian lubricating layer breaks the profile symmetry: the peak beside the lubricating wall is reduced, while streamwise normal stress increases toward the interface. In the right-hand column, especially panels b and c, the colored profiles separate most clearly from the black single-phase curve around the dashed interface and through the lubricating layer.

Panel d shows the corresponding shear-stress profiles becoming progressively closer together as they move across the channel, with the colored curves remaining distinct from the black reference on both sides. As the Weissenberg number increases, the near-wall peak at the lubricating wall becomes stronger and shifts slightly away from the wall, while fluctuation intensity in the intermediate lubricating layer decreases.

The changes between Weissenberg numbers seventy-five and one hundred become comparatively small, suggesting the onset of a saturation regime. This trend mirrors single-phase viscoelastic turbulent flows, where polymer stretching reduces turbulent kinetic energy production until a maximum drag-reduction state is approached.

In the main Newtonian layer, the near-wall peak stays at approximately the same wall-normal position, but its magnitude increases moderately with increasing Weissenberg number. The spanwise and wall-normal Reynolds stress components are more strongly affected by the interface and viscoelasticity than the streamwise component.

Compared with single-phase flow, both components are suppressed near the interface and within the lubricating layer, with stronger reduction at higher Weissenberg number and reduced wall-normal momentum transport across the interface. To explain drag-reduction saturation, the analysis examines how a viscoelastic lubricating layer changes energy fluxes in the two-phase system.

The energy-box framework recasts kinetic-energy fluxes as sources, sinks and inter-exchanges and represents them graphically. Within this framework, all terms in the energy budgets are evaluated independently from their definitions, and total kinetic energy is split into mean kinetic energy and turbulent kinetic energy.

For viscoelastic cases, the energy box is extended to include elastic potential energy associated with polymer stretching, because the momentum equation contains an additional polymeric stress contribution. The extension is developed first with an ensemble-averaged procedure and then with a phase-averaged procedure, so the resulting energy box can be compared directly with the earlier cases.

The phase-averaged viscoelastic energy boxes cover the considered Weissenberg numbers, increasing from twenty-five to one hundred to show how the energy fluxes evolve toward drag-reduction saturation. Viscoelasticity primarily affects the lubricating-layer dynamics and the interfacial exchange term, while changes in main-layer fluxes follow directly from the changed flow rate.

The analysis therefore focuses on the bottom part of the energy boxes, where polymer-induced reorganisation of energy pathways is most evident. At the lower elasticity, a polymer energy storage and dissipation pathway emerges: mean and turbulent flow stretch the polymers, increasing elastic energy in the lubricating layer, which is then dissipated through polymer relaxation.

This additional pathway changes the energy balance relative to the Newtonian case. Figure five maps the phase-averaged energy pathways for weakly elastic lubricating layers at Wi sub tau equal to twenty-five and fifty. Compared with the Newtonian energy box, the diagram adds a polymer box, showing energy storage as an additional pathway and sink for kinetic energy.

The authors highlight this viscoelastic modification through the lubricating-layer fluxes and the interface exchange, providing the energy-budget context for the reported reduction in turbulent kinetic-energy production. At Weissenberg number twenty-five, turbulent kinetic energy production in the lubricating layer decreases by almost fifty percent because elasticity adds an energy-storage pathway and a further sink for kinetic energy.

The capillary energy exchange decreases from three point two percent to two point two percent as smaller lubricating-layer velocity fluctuations reduce the exchanged energy. These changes coincide with an initial flow-rate increase of approximately eighteen point nine percent, showing that even weak polymer elasticity can produce drag reduction.

At a Weissenberg number of fifty, turbulent kinetic energy production in the lubricating layer decreases further to seven point four percent. Meanwhile, elastic potential energy production in the lubricating layer remains similar to the previous case at four point three percent.

Figure six presents phase-averaged energy boxes for highly elastic lubricating layers at Weissenberg numbers seventy-five and one hundred. It tracks energy through the main and lubricating regions, the interface, and an added polymer box, with purple arrows representing polymer effects on lubricating-layer fluxes.

The authors use these diagrams to show how increasing elasticity modifies lubricating-layer turbulence and interfacial exchange, helping explain the reported evolution and eventual saturation of drag reduction. As the Weissenberg number rises from twenty-five to one hundred, interfacial work, turbulent kinetic energy production in the lubricating layer and polymer-relaxation dissipation all decrease.

The decline in capillary work has a kinematic origin: capillary work depends on wall-normal velocity fluctuations, which viscoelastic stresses strongly damp. The interface becomes flatter as elasticity increases, because the probability-density function of interface elevation narrows around its mean and extreme elevations become less likely.

Wave damping also reflects the stabilising action of normal stresses, which modify the normal-stress balance across the interface and damp interfacial deformations. Polymeric dissipation decreases with increasing Weissenberg number, indicating that less elastic energy is stored and subsequently dissipated as wall-normal fluctuations are suppressed.

Figure seven summarizes how the energy budget and interface fluctuations change with the Weissenberg number. Panel a plots normalized surface-tension energy flux, lubricating-layer TKE production, and polymer-related dissipation, with an inset separating mean and turbulent polymer-energy transfers.

Panel b shows probability distributions of interface elevation fluctuations, ζ over h, for Newtonian and polymer cases, together with root-mean-square bounds, linking the energy-box measurements to interfacial behavior. As elasticity increases, wall-normal fluctuations are progressively suppressed, reducing both elastic-potential-energy storage through polymer stretching and the absorption of mean and turbulent kinetic energy through interfacial deformation.

The result is a competition: modest elasticity initially enhances polymer action and drag reduction, but stronger viscoelastic stresses reduce the turbulent kinetic energy available in the lubricating layer. Stretching efficiency then decreases, elastic potential energy saturates, and the energy extracted from the flow by polymers becomes progressively smaller.

In short, the polymers progressively suppress the turbulent motions responsible for stretching them, limiting their own effectiveness and producing smaller incremental drag-reduction gains. Although the macroscopic upper bound resembles limits in other polymeric flows, the driving mechanism here comes from energy transfer across a deformable interface between Newtonian and non-Newtonian turbulence.

The simulations combine a phase-field description of the deformable interface with a FENE-CR model for the viscoelastic lubricating layer, varying the shear Weissenberg number from zero to one hundred. The Newtonian lubricating layer produces twelve point two percent drag reduction, rising to eighteen point nine percent at twenty-five and only marginally further to twenty-four point one percent at one hundred.

The extended energy-box analysis shows that higher elasticity significantly reduces the turbulent kinetic energy absorbed by the lubricating layer, making polymer action progressively less effective. The system approaches a regime in which further elasticity produces only marginal gains because elasticity suppresses wall-normal turbulent fluctuations, reducing both capillary-work transfer and turbulent kinetic energy production within the layer.

Reduced turbulent kinetic energy limits further polymer stretching and elastic potential energy, which reach an upper bound. The system therefore enters a self-limiting regime where further increases in elasticity no longer produce additional drag reduction. The saturation level may depend on geometric parameters such as layer thickness, while the mechanism may persist when turbulent transport across the deformable interface controls energy flux to and from the lubricating layer.

The mechanism is expected to weaken as surface tension vanishes, and its possible persistence with shear-thinning effects requires further investigation. The results support combining lubrication and elasticity, but the gains cannot be treated as a straightforward sum because polymer elasticity and interfacial dynamics restrict the achievable improvement.

The driving physics differs from single-phase polymeric drag reduction because it originates from coupling interfacial dynamics with non-Newtonian turbulence. The key result is a self-limiting feedback: elasticity suppresses wall-normal turbulence, reducing both interfacial energy transfer and polymer stretching, so additional elasticity brings only marginal drag reduction.

The reported gain rises from twelve point two percent to twenty-four point one percent.