Nanoliquid with microbes and double diffusion solved for stratified stretching flow

A team of researchers has presented a comprehensive mathematical analysis of heat, mass, and microorganism transport in a Walter’s-B nanoliquid flowing over a stretching sheet, incorporating the Cattaneo-Christov double diffusion model alongside thermal radiation and gyrotactic microbes. The study, published in Frontiers in Thermal Engineering, provides a semi-analytical solution using the homotopy analysis method, with convergence verified through residual tables and comparison with earlier literature.

The researchers modeled the flow of a non-Newtonian Walter’s-B nanoliquid containing nanoparticles, where Brownian motion and thermophoresis drive the transport of thermal energy and species. The Cattaneo-Christov formulation replaces the classical Fourier and Fick laws, introducing finite relaxation times that account for the memory effects in heat and mass diffusion. Gyrotactic microorganisms, which orient themselves relative to the gravitational field and swim in a directed manner, add a further layer of biological transport to the system.

Thermal radiation was included in the energy equation, contributing an additional radiative flux that augments the overall heat transfer away from the stretching boundary. The sheet was assumed to stretch linearly, and the boundary layer approximation was applied to reduce the governing partial differential equations to a set of nonlinear ordinary differential equations. The homotopy analysis method was then employed to obtain semi-analytical solutions, with the auxiliary convergence parameters tuned to ensure rapid and stable convergence of the series.

The results demonstrate that increasing the radiation parameter substantially enhances the local Nusselt number, reflecting the additional radiative flux that contributes to the thermal energy transport. Brownian motion also improves heat transfer performance by augmenting the thermal diffusion of nanoparticles. Conversely, higher values of the thermophoresis parameter reduce the Nusselt number, as the directed migration of particles away from the boundary diminishes the effective thermal gradient.

For mass transport, the Sherwood number increases gradually with the Schmidt number, indicating that the molecular diffusivity of species plays a key role in the concentration boundary layer. The Cattaneo-Christov relaxation times introduce memory effects that modify the classical concentration profiles, and the study quantifies how these finite relaxation times alter the species transport relative to the Fickian limit.

The transport of gyrotactic microorganisms was analyzed through the motile microorganism density number, which increases with both the bioconvection Lewis number and the bioconvection Péclet number. The Lewis number represents the ratio of thermal to microorganism diffusivity, and higher values augment the density gradient near the wall. The Péclet number captures the directed swimming speed of the microbes relative to their random diffusion, and increasing values enhance the accumulation of motile species near the boundary.

The study also examined the effects of the Walter’s-B fluid parameters on the skin friction coefficient, demonstrating that the non-Newtonian characteristics of the nanoliquid modify the shear stress at the stretching surface. The homotopy analysis solutions were validated against previously published results for limiting cases, with excellent agreement observed across the parameter ranges examined.

This comprehensive analysis provides a detailed framework for understanding the coupled transport of thermal energy, species, and microorganisms in non-Newtonian nanoliquids with finite relaxation times. The findings are relevant to applications in biomedical engineering, biotechnology, and materials processing, where the directed transport of biological species in complex fluids plays a critical role. The semi-analytical solutions offer a reliable foundation for future studies exploring more complex geometries and boundary conditions.

The porous medium through which the Walter’s-B nanoliquid flows is described by the Darcy-Forchheimer framework, which accounts for two distinct sources of hydraulic resistance. The classical Darcy term captures the linear viscous drag exerted by the solid matrix on the percolating fluid, while the Forchheimer term adds a quadratic contribution that becomes significant at higher pore velocities, when inertial effects within the pores can no longer be neglected. In the present formulation, increasing either the porosity parameter or the Forchheimer inertia coefficient raises the effective resistance offered by the medium, and the computed velocity profiles respond by flattening and thinning near the stretching surface. This behavior is physically intuitive: a more restrictive pore structure dissipates more of the momentum imparted by the moving boundary, confining the motion to a narrower region adjacent to the sheet.

The viscoelastic character of the base fluid acts in the opposite direction. Walter’s-B fluids possess a short memory of past deformation, and the associated viscoelastic parameter enhances the effective momentum diffusivity of the liquid. As this parameter grows, the fluid transmits the stretching action of the sheet more efficiently into the bulk, accelerating the flow and broadening the momentum boundary layer. The interplay between this elastic momentum transfer and the porous-medium drag therefore provides a tunable balance: engineers working with polymer solutions or lubricants in permeable or fibrous media can, in principle, select operating conditions that either promote or suppress boundary-layer growth depending on the desired transport outcome.

Stratification emerges as one of the most consequential features of the model. In the thermal case, the ambient temperature is assumed to vary linearly with distance from the surface rather than remaining uniform. This layered temperature environment reduces the wall-to-fluid temperature difference that drives conductive and radiative heat exchange, and the computed temperature profiles respond by compressing toward the sheet. The thermal penetration depth shrinks as the thermal stratification parameter increases, which is the mirror image of the widening produced by stronger radiation, Brownian motion, or thermophoresis. An analogous mechanism operates for the solutal field: a stratified ambient concentration diminishes the driving concentration gradient, so nanoparticle concentration distributions contract as solutal stratification intensifies. Chemical reaction effects reinforce this contraction, since reactive consumption of the diffusing species depletes the concentration field more rapidly than diffusion alone would allow.

The bioconvective component of the model introduces a third stratification mechanism acting on the density of motile microorganisms. Gyrotactic algae and similar self-propelled cells accumulate in response to gradients in their own population density, and when the ambient microbial density is itself stratified, the driving gradient for bioconvection is weakened. The study shows that increasing the microbial stratification factor lowers the local population density of motile organisms near the surface. The bioconvection Péclet number, which measures the ratio of directed swimming speed to the random diffusivity of the cells, acts in the same direction: larger Péclet numbers reduce the local density number, indicating that stronger directed swimming disperses the accumulation that would otherwise form near the boundary. These results underscore that bioconvection intensity is not fixed by biology alone but can be steered through the ambient density environment imposed on the system.

The wall transport coefficients tie the field solutions back to quantities of direct engineering interest. The skin-friction coefficient, which quantifies the shear stress exerted on the stretching surface, increases with the Darcy resistance, the Forchheimer inertia, and the viscoelastic parameter. This reflects improved interfacial momentum transfer: a more resistive medium or a more elastically responsive fluid both transmit greater traction to the wall. For heat transfer, the Nusselt number grows with the intensity of radiative effects, consistent with the additional radiative flux leaving the surface, but falls as thermophoresis strengthens, because nanoparticles drifting away from the hot wall carry less thermal energy across the near-wall gradient. The motile microorganism density number rises with the bioconvection Lewis number, confirming that the diffusivity of the microorganisms relative to other transport mechanisms is central to sustaining the bioconvection flux at the boundary.

Methodologically, the reliance on the homotopy analysis method deserves emphasis. Unlike purely numerical schemes, this semi-analytical technique expresses the solution as a convergent series whose structure is controlled by an auxiliary parameter, denoted by the convergence-control symbol h. By plotting h-curves for each governing equation, the authors identified the ranges of this parameter within which the series converges reliably, and residual error tables provided quantitative confirmation that successive approximations were approaching a stable solution. This combination of graphical and tabular convergence verification is particularly valuable for systems of strongly coupled nonlinear equations, where naive series expansions may diverge or converge to spurious branches.

The simultaneous treatment of thermal, solutal, and microbial stratification within a single framework is relatively rare in the literature on stretching-sheet flows, and the present results clarify how the three layering mechanisms interact with the distinct transport physics of a viscoelastic nanofluid. Because stratification suppresses the driving gradients for heat, species, and microorganisms alike, while radiation, Brownian diffusion, and thermophoresis amplify them, the model effectively maps out a competition between gradient-eroding and gradient-enhancing influences. This mapping offers practical guidance for systems such as solar ponds, thermal energy storage tanks, chemical reactors, and bioreactors, where layered density environments are common and where controlling the near-wall transport of heat, dissolved species, and suspended biological cells simultaneously is essential to performance.

Subject of Research: Semi-analytical transport of heat, mass, and gyrotactic microorganisms in a Walter's-B nanoliquid with Cattaneo-Christov double diffusion over a stretching sheet

Article Title: Thermal and mass management of Walter-B nanoliquid with heat radiation and the Cattaneo–Christov double diffusion model incorporating microorganisms: convergence solution for stratified effect

Article References: Ali, F., Hussain, I., Zafar, S. S., & Ahmed, M. F. (2026). Thermal and mass management of Walter-B nanoliquid with heat radiation and the Cattaneo–Christov double diffusion model incorporating microorganisms: convergence solution for stratified effect. Frontiers in Thermal Engineering, 6, Article 1864538. https://doi.org/10.3389/fther.2026.1864538

Image Credits: AI Generated

DOI: 10.3389/fther.2026.1864538

Keywords: Walter's-B nanoliquid, Cattaneo-Christov double diffusion, gyrotactic microorganisms, stretching sheet, homotopy analysis, thermal radiation, Brownian motion, thermophoresis, bioconvection, Nusselt number, Thermal, mass