Hybrid nanofluid blood flow modeled in ciliated porous concentric tubes

Cilia are tiny, whip-like filaments that line many of the body’s internal passages, and when they beat in coordinated waves they can push fluid along without any external pump. Now, a team of researchers has built one of the most complete mathematical models yet of how such cilia-driven pumping moves blood laced with engineered nanoparticles through the narrow annular gap surrounding a moving endoscope—a scenario that mirrors real minimally invasive medical procedures. The study, published in Results in Engineering, simultaneously captures magnetic forces, porous tissue drag, thermal radiation, and the nonlinear rheology of blood, filling a gap that previous single-effect models could not.

The work, led by Sabah Noor with Nabeela Kousar, Khalil Ur Rehman and Wasfi Shatanawi, models blood as a so-called special third-grade fluid, a class of viscoelastic liquid whose stress response includes higher-order nonlinear terms. That choice matters. At low shear rates blood does not behave like water; its resistance to flow changes as it is continuously deformed by beating cilia and the sliding motion of an endoscope. Simpler non-Newtonian models such as Casson, Maxwell, Jeffrey or second-grade fluids can each capture one aspect of this behavior—yield stress, relaxation, or linear elasticity—but none account for the additional nonlinear elastic response that emerges under the complex, sustained deformation that characterizes cilia-driven endoscopic transport. The third-grade model, the authors argue, is the smallest framework that can represent these effects realistically.

The geometry of the problem is deliberately faithful to physiology. Two coaxial cylinders form the flow domain: the outer tube carries cilia on its inner surface, and these cilia beat asynchronously with a continuous phase lag between neighbors, producing a metachronal wave that travels along the wall. Depending on whether the wave and the effective stroke travel in the same or opposite directions, the beating is classified as symplectic or antiplectic—the former seen in the protozoan Opalina, the latter in Paramecium. Inside sits the endoscope, a solid rigid tube that moves axially at a fixed velocity, sweeping fluid along with it through the no-slip condition at its wall. Fluid occupies the gap between the two, bounded by constant-temperature walls that create a thermal gradient driving heat transfer.

On top of this mechanical picture, the researchers superimposed several physical effects that previous studies had treated only one at a time. An external radial magnetic field acts on the electrically conductive blood, generating Lorentz forces that oppose motion—the essence of magnetohydrodynamics, a field first described by Hannes Alfvén. The outer wall is treated as a porous medium, mimicking the permeability of biological tissue, which matters for predicting how fluid and suspended nanoparticles are absorbed into surrounding structures. Thermal radiation is included through the Roseland approximation, which is relevant for laser and thermal therapies where internal heat management is critical. And blood itself is transformed into a hybrid nanofluid by suspending two kinds of nanoparticles: zinc oxide (ZnO) and aluminum oxide (Al₂O₃).

The choice of nanoparticles is not arbitrary. ZnO offers chemical stability, low toxicity, antimicrobial activity, and proven biocompatibility, and it is already investigated for drug delivery, biosensing, and antiviral or anticancer applications. Al₂O₃ brings high thermal conductivity, corrosion resistance, and the ability to target specific tissues, improving treatment efficiency while limiting side effects. Combined at low volume fractions—no more than about three percent to avoid agglomeration and excessive viscosity—the two particles enhance heat transport while keeping the suspension stable enough for biomedical use. Blood’s ionic solution serves as the carrier fluid, with effective properties such as viscosity, density, thermal conductivity, and electrical conductivity computed through established hybrid-nanofluid mixing rules.

Perhaps the most significant departure from earlier work is the treatment of inertia. Most prior models of ciliary and peristaltic transport assumed creeping flow—negligible Reynolds number—which strips the governing equations of inertial terms. Here, the authors retained a moderate Reynolds number, between roughly 5 and 15, which is realistic for blood flow in larger vessels and keeps the momentum equations closer to their full form. Combined with the nonlinear shear-thickening rheology of the third-grade model, this produces a coupling among momentum transport, magnetic damping, porous drag, and heat transfer that simply cannot appear in creeping-flow formulations using conventional fluids. The cilia length parameter, wave number, and eccentricity of the elliptical ciliary stroke all enter directly through the boundary conditions at the outer wall, anchoring the mathematics to the actual beating motion of the biological structures.

Solving the resulting system of coupled nonlinear partial differential equations required a regular perturbation approach, carried out in two stages with the help of Mathematica. First, the equations were expanded in powers of the small wave number; then, because the axial momentum equation remained nonlinear even at zeroth order, a second perturbation was applied in the small third-grade parameter. The validity ranges—wave number and third-grade parameter both between 0.01 and 0.03—are small enough that neglected higher-order terms stay negligible. Residual analysis confirms convergence: maximum absolute errors were on the order of 10⁻⁶ to 10⁻⁸ for radial velocity, axial velocity, temperature, and pressure, indicating that the truncated series accurately approximates the true solution within its regime of validity.

The results reveal a rich interplay of competing mechanisms. Radial velocity falls as the radius ratio increases—the annular gap narrows, geometric confinement suppresses ciliary motion, and mass conservation forces the radial component to shrink—but rises dramatically with wave number and with cilia length, because longer cilia and shorter-wavelength metachronal waves transfer more momentum from wall to fluid. Doubling the wave number increased radial velocity by roughly 166 percent, and lengthening cilia by a modest fraction boosted it by nearly 87 percent. A faster-moving endoscope, by contrast, pushes fluid axially and weakens radial displacement.

Axial velocity behaves differently. It increases with radius ratio—momentum transfer from the moving endoscope intensifies in the confined space—and climbs steeply with wave number, roughly 200 percent over the studied range. Higher permeability of the porous wall reduces Darcy resistance and lets the fluid move more freely, raising axial speed by about a third, while a stronger magnetic field cuts axial velocity by roughly 30 percent as Lorentz forces impose electromagnetic drag. Notably, increasing the third-grade parameter raised axial velocity by about 60 percent: the nonlinear elastic stress terms let the fluid store deformation energy during ciliary pumping and endoscope motion and then release it back into the flow, facilitating axial momentum transport.

Temperature shows non-monotonic behavior in several cases. Increasing the third-grade parameter roughly doubles fluid temperature because enhanced axial motion strengthens convective heat transport from the hot wall. Permeability first raises temperature and then lowers it beyond a threshold, as convective retention of heat within the fluid gives way to radiative transfer toward the cooler wall. Magnetic field strength first cools the fluid by about 100 percent—Lorentz damping slows circulation—and then warms it again as heat accumulates in a stagnating fluid. Faster endoscope motion and higher Reynolds number both raise temperature by strengthening convection, while increasing the radiation parameter lowers it by distributing heat more uniformly.

Pressure responds as expected from the momentum balance. Stronger magnetic fields and larger third-grade parameters demand steeper pressure gradients to sustain flow—pressure rose 2.6 percent and 66 percent respectively across the tested ranges—while greater permeability and Reynolds number reduce the pressure needed. The team also examined bolus entrapment, the formation of recirculating fluid pockets trapped inside closed streamlines that travel with the metachronal wave. Bolus size grows with permeability and third-grade elasticity, because faster, more elastic fluid circulates more within each closed streamline, but shrinks under magnetic damping, which suppresses the overall velocity and traps less volume per pumping cycle.

The authors are candid about the model’s boundaries: velocity-slip and thermal-slip at the walls are neglected, the analysis is axisymmetric, and thermophysical properties are held constant. The perturbation solution is valid only for small wave numbers and weak third-grade effects, and stronger flows would require numerical methods. Even so, by uniting endoscope motion, magnetohydrodynamics, porous media, thermal radiation, ciliary propulsion, nonlinear blood rheology, and moderate inertia in a single analytically tractable framework, the study offers a physiologically grounded toolkit for designing magnetically controlled drug delivery, optimizing thermal therapies during endoscopy, and predicting how engineered nanoparticles will behave in the body’s ciliated passages. In an era when targeted nanomedicine is moving from bench to bedside, understanding these coupled transport physics is not an academic luxury—it is a prerequisite for safe and effective treatment.