Dual Solutions Reveal Which Shrinking-Cylinder Nanofluid Flows Can Truly Exist

When a heated cylinder shrinks rather than stretches, the boundary layer wrapped around it can settle into more than one mathematical flow state, and only one of those states survives in the real world. A new computational study published in Discover Nano maps out exactly where that boundary lies for a sophisticated class of engineered fluids, combining non-Newtonian rheology, three kinds of nanoparticles, an applied magnetic field, Joule heating, and cylindrical curvature in a single analysis. The work is entirely numerical: it formulates and solves boundary-value equations rather than measuring anything in a laboratory, and its conclusions should be read as model predictions for an idealized configuration.

The fluid at the center of the study is a tri-hybrid nanofluid, meaning a base liquid, in this case pure water, seeded with three different nanoparticle species simultaneously: copper, aluminum oxide, and titanium dioxide. The authors adopt the Tiwari and Das single-phase formulation, using published thermophysical property tables for the nanoparticles and the base fluid, with mixture relations drawn from earlier literature to compute effective density, viscosity, heat capacity, thermal conductivity, and electrical conductivity. The rationale for stacking three particle types is that metallic particles contribute high thermal conductivity while ceramic particles contribute chemical stability, allowing the effective properties of the working fluid to be tuned in ways that single-particle and two-particle nanofluids cannot fully achieve.

The rheology is handled with the Eyring–Powell model, a constitutive law with a kinetic, rate-process origin that can describe both shear-thinning and shear-thickening behavior while recovering the Newtonian limit in the appropriate low- and high-shear regimes. The researchers consider a two-dimensional, steady, laminar boundary-layer flow of this incompressible fluid over a vertical cylinder of constant radius, whose surface moves axially with a linear velocity profile and can stretch or shrink. The surface is permeable, so fluid can be drawn in through suction or pushed out through injection, and a uniform magnetic field is applied perpendicular to the primary flow direction, with induced magnetic fields neglected as small compared with the applied field.

Using similarity transformations, the governing continuity, momentum, and energy equations are reduced to a coupled system of nonlinear ordinary differential equations that carry the combined influence of the Eyring–Powell fluid parameters, the magnetic parameter, viscous dissipation through the Eckert number, and the curvature parameter associated with the cylindrical geometry. The system is solved numerically with MATLAB’s bvp4c solver, and the authors report residual-error and grid-independence checks to verify the accuracy of the solutions. Two engineering quantities receive particular attention: the local skin friction coefficient, which measures wall shear stress, and the local Nusselt number, which measures the surface heat transfer rate.

Validation against earlier studies anchors the numerical scheme. The heat-transfer results for the stretching case are compared with values reported by Devi and Devi and by Khashi’ie and colleagues, with close agreement reported. A second comparison covers the skin friction coefficient against results from Javed and co-workers for the limiting case of a regular fluid, with the nanoparticle volume fractions, curvature, suction, magnetic, and Eckert parameters all set to zero. The authors also note an interesting asymmetry in that limiting comparison: increasing the Eyring–Powell fluid parameter M decreases the reduced skin friction coefficient, while the parameter B has the contradictory, opposite influence.

The central result concerns the shrinking regime. Over a finite range of shrinking parameters, the similarity equations admit two solutions, an upper branch and a lower branch, which merge at a critical value of the shrinking parameter, computed as approximately minus 5.079891. Beyond that critical point, no valid solution exists, meaning the boundary layer can no longer sustain a physically meaningful flow and heat transfer from the surface effectively breaks down. The upper branch consistently delivers higher Nusselt numbers, reflecting stronger heat transfer at the surface, while the lower branch corresponds to weaker cooling performance. In the skin friction analysis, the first branch remains higher and decreases only slightly with the shrinking parameter, whereas the second branch drops rapidly as the critical value is approached.

Several parameter effects shape this dual-solution landscape. Increasing the Eckert number, which quantifies viscous dissipation, shifts both solution branches of the Nusselt number downward, indicating that viscous heating hinders heat removal by lowering the wall temperature gradient, with the effect more evident in the first solution. Increasing the curvature parameter shifts the critical shrinking point slightly to the right, from minus 5.08306 at zero curvature to minus 5.07934 at a curvature parameter of 0.02, so larger curvature modestly extends the range of shrinking parameters over which dual solutions exist. The authors interpret this as a stabilizing influence of cylindrical geometry, though they emphasize that the effect is relatively small. Suction, meanwhile, is reported to enhance both flow stability and heat transfer.

The decisive test of which branch matters comes from a temporal stability analysis. The steady equations are reformulated in unsteady form, small disturbances are introduced into the velocity and temperature fields in exponential form, and linearization yields an eigenvalue problem solved with the additional far-field condition introduced by Harris and colleagues. The smallest eigenvalue acts as the arbiter: a negative value means the mathematical solution does not represent a physical flow, while a positive value indicates a stable, physically meaningful state. The computed results show that the first solution branch maintains positive smallest eigenvalues throughout the admissible range of the shrinking parameter and is therefore temporally stable, whereas the second branch has negative eigenvalues and is unstable. Only the first branch, despite the second being mathematically valid, can be sustained under real operating conditions.

The parameter ranges chosen for the computations are deliberately constrained. The magnetic parameter is limited to between 0.01 and 0.1 so that the magnetic force affects the flow without completely stopping it, the curvature parameter spans 0.01 to 0.2 to keep the boundary-layer approximation valid, and the Eckert number is restricted to 0.1 to 1 to represent moderate viscous dissipation without excessive thermal instability. Nanoparticle volume fractions are capped at 0.05 to keep the suspension thin and discourage particle agglomeration, the Eyring–Powell parameter B is fixed at 0.01, and the Prandtl number is set to 6.2, the value for water as the base fluid. The suction parameter is held at a positive constant because suction is known to stabilize the boundary layer and encourage the existence of dual solutions.

The authors frame the findings as theoretical support for cylindrical cooling, coating, wire drawing, extrusion, and thermal processing applications, with magnetic-field strength and surface permeability serving as controllable levers for regulating flow stability and heat transfer. They are careful to note the limits of the approach: the study is a simulation of an idealized two-dimensional laminar configuration, and the conclusions rest on the validity of the boundary-layer approximation, the single-phase nanofluid model, and the published thermophysical property data. The authors themselves call for experimental validation and more detailed three-dimensional and transient models as the necessary next steps. Within those limits, the message is clear: in shrinking flows of this engineered fluid, mathematics offers two answers, but stability analysis shows that only one of them describes a flow that nature will actually sustain.

Subject of Research: Stability and heat transfer in magnetohydrodynamic Eyring–Powell tri-hybrid nanofluid flow over a permeable shrinking cylinder

Article Title: Stability and heat transfer of mhd eyring powell tri hybrid nanofluid flow over a permeable shrinking cylinder

Article References: Mohd Kasim, A. R., Mustafa, P. E. S., Aljabali, A., Zainal, N. A., Adeosun, A. T., Nisa Ariffin, N. A., & Tijani, Y. O. (2026). Stability and heat transfer of mhd eyring powell tri hybrid nanofluid flow over a permeable shrinking cylinder. Discover Nano, 21(1), Article 554. https://doi.org/10.1186/s11671-026-05053-z

Image Credits: AI Generated

DOI: 10.1186/s11671-026-05053-z

Keywords: tri-hybrid nanofluid, Eyring-Powell fluid, shrinking cylinder, boundary layer, dual solutions, stability analysis, eigenvalue, Joule heating, magnetic field, heat transfer, bvp4c, curvature parameter