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

1 Introduction

Materials with viscous and elastic properties have become very important in thermo-fluid engineering. In particular, the fluids that are developed to describe such materials have a specific significance; one of them, Walter’s B-fluid, can be used to describe short-memory viscoelastic behavior without any need of overly complicated math. Adding nanosized solid particles to this base fluid produces Walter’s-B nanofluid, which uniquely blends improved heat transfer capabilities with non-Newtonian flow traits. They can be used in applications such as polymer production, lubrication systems, biomedical products, and advanced cooling systems. Gangadhar et al. (2022) investigated a Walter’s-B nanoliquid flowing between a rotating cone and considered the thermal radiation effect and a chemical factor. They concluded that there are two opposing effects that occur as the cone rotates. Shahzad et al. (2022) analyzed the Galerkin finite element solution of a Walter’s-B nanofluid flowing over a Riga plate (electromagnetic device that generates a wall parallel Lorentz force) under bioconvection flow. The key result was that the modified Hartmann number of the Riga plate had the ability to suppress flow separation and delay the onset of instabilities. Sandeep et al. (2023) studied in detail a conical geometry with velocity slip at the wall with variable fluid properties, following the nanofluid model proposed by Walter’s-B. When the slip becomes greater than a certain value, viscous dissipation becomes sufficiently weak to cause the thermal boundary layer to thicken unusually. Sarfraz et al. (2024) carried out a thermo mass analysis of a Walter’s-B nanofluid on a stretching or shrinking surface. They found regions of double solutions, and they did a stability analysis of them to find the physically realizable branch. Shafiq et al. (2024) presented magneto flow and machine learning theory to study the flow of Walter’s-B nanofluid over a cylindrical disk. They were able to predict the Nusselt number with very good accuracy and identify the most important parameters. The effect of viscous dissipation on a Walter’s-B nanoliquid moving over an elongating surface in a permeable medium was investigated by Swain et al. (2025), who demonstrated that when dissipation is included, which has been neglected in previous research, it is found to be crucial at high Eckert numbers. The effect of dissipation is enhanced by the porosity which increases internal friction (Darcian drag). The second less explored problem by Mahariq et al. (2025) is bioconvection flow of a Walter’s-B nanofluid over a stretching surface with the involvement of activation energy. They found that an increase in activation energy greatly decreases mass transfer in the region near the wall but that the viscous dissipation somewhat compensates for this by increasing the temperature in the same region. Khan et al. (2025) extended the work to the Darcy–Forchheimer flow of a hybrid nanofluid consisting of a Walter’s-B-type nanofluid with variable thermal conductivity and elastic deformation. Elastic deformation causes a memory effect which will decrease the effective permeability of the porous medium. Shaheen et al. (2025) studied the transport of micro-organisms in the presence of radiation and gyrotactic forces in a stretching sheet of Walter’s-B nanofluid. They reported the tendency of the bioconvection Péclet number to stabilize the flow and the tendency of the radiation parameter to destabilize the boundary layer.

A key phenomenon in such flows is stratification, which occurs when density, temperature, or concentration varies in a layered manner. Thermal, solutal, and double stratifications are common in natural systems such as oceans and atmospheric flows, as well as in industrial settings such as thermal storage tanks and chemical reactors. Heat exchange rates, species density, and flow stability may all be dramatically changed by stratification. The Lorentz force of free convection stagnant flow of a nanoscale across a stretched sheet porosity material with stratification process was explored by Mahmood et al. (2022). The thermal properties of the nanoscale of Prandtl fluid across an inclined stretchable flow in the presence of thermal stratification process was examined by Bilal et al. (2023). They found that both the thermal field and the mass gradient tend to diminish as the stratification factor intensifies. Majeed et al. (2023) evaluated the effects of double-fold stratification on the radiation of magento nanolquid across a stretchable cylinder subject to the velocity slippage effect. They found that mass gradient reduces with the larger estimation of solutal stratification. Reddy et al. (2024) examined the transient magneto flow at nanoscale with the dual stratification and injection effect. Mass gradient and heat transport are enhanced due to the larger values of stratification. Mourya et al. (2024) examined a significant amount of stratification for the Darcian–Forchheimer past horizontal stretchable sheet. Zhou et al. (2025) computed the significance of the magneto flow of the nanoscale stratification effect for non-Newtonian liquid. Ahsan et al. (2025) calculated the microbial cell at nanoscale for the triple stratification of viscoelastic nanoscale over a curve surface. Gurejala et al. (2026b) introduced a machine-learning approach for an artificial neural network driven by numerical simulation for a Prandtl microbial cell with the stratification process.

Thermal radiation plays an important role for the transmission of heat in the thermal layer towards surfaces and energy releasing in the form of electromagnetic radiation. Thermal radiation has been broadly used in areas such as aerodynamics and cooling space.(Rehman et al. (2025a) discussed the heated radiation of the porosity flow of nanoscale Lorentz force past a stretchable surface. Ullah et al. (2025a) explored the magneto flow of rounding fins subject to thermal radiation using numerical simulation. Rehman et al. (2025b) discovered the steady flow of the heat transport analysis CNT of blood-based fluid past a porosity surface and thermal radiation. Ullah et al. (2025b) explored the magneto-porosity flow of thermal radiation subject to three fins using numerical analysis. Abdelsalam et al. (2026) described the thermal radiation of nanoscale bioconvection flow for the convective condition past a rotatable disk. Ullah et al. (2023) investigated the heated radiation of the numerical simulation of convective flow for stretchable/shrinkable surfaces. Arafat et al. (2025) illustrated the convergence estimation of a nanospace inner radius for the Darcy–Forchheimer flow, heated radiation, and convective flow condition. Ullah et al. (2024) explained the impact of thermal radiation of magneto flow convection past three sorts of fin.

This literature demonstrates that the Darcy–Forchheimer flow of Walter’s-B nanoscale fluids caused by a stretchable surface with the effects of a microbial cell and stratification conditions has not yet been addressed; the current investigation will close this gap. The constituent equations are simplified to ODE expression. The problem is then semi-analytically tackled using MATHEMATICA 12.0. The results are obtainable in tabular and graphical form together with the physical interpretation.

2 Problem formulation

The investigation focuses on the linear, 2-D, incompressible, and Darcy–Forchheimer flow of a motile microbe with Walter’s-B nanofluid across a stretchable sheet, considering the Cattaneo–Christov double diffusion model for the computation of heat and mass. Transport features of the porous media are described using the Darcy–Forchheimer technique, which takes permeability and inertia effects into account. Additionally, thermal and solutal stratification are introduced in order to examine temperature and concentration fluctuations over the flow field. Thermal, solutal, and microorganism stratification have been used in the boundary conditions. Additional physical processes that impact heat and mass transport are also considered, including internal heat generation/absorption, thermal radiation, and chemical reaction. The wall of the thermal, concentration, and microorganism densities are represented by T w, C w, and Π w, respectively. Far from the stretching surface, the ambient conditions are given by temperature T ∞,concentration C ∞ ​, and microorganism density​ Π ∞. The Cartesian coordinate system is used, with the x- and y-axis along the length of the stretching sheet. The velocity with which the sheet is stretched is U w = ax, where a > 0 The stretching rate constant is ‘ a ’. The geometry of the current problem is depicted in Figure 1.

∂ u 1 ∂ x + ∂ u 2 ∂ y = 0 , (1)

u 1 ∂ u 1 ∂ x + u 2 ∂ u 1 ∂ y = ν f ∂ 2 u 0 ∂ y 2 − ε 0 u 1 ∂ 3 u 1 ∂ x ∂ y 2 + u 2 ∂ 3 u 1 ∂ y 3 + ∂ u 1 ∂ x ∂ 2 u 1 ∂ y 2 − ∂ u 1 ∂ y ∂ 2 u 1 ∂ x ∂ y = . − v f K 0 u 1 − F u 1 2 , (2)

u 1 ∂ T ∂ x + u 2 ∂ T ∂ y ‐ α f + 16 σ * T ∞ 3 3 k ρ C p f ∂ 2 T ∂ y 2 + λ e ϖ e = τ D B ∂ C ∂ y ∂ T ∂ y + D T T ∞ ∂ T ∂ y 2 + Q 0 ρ C p f T ‐ T ∞ , (3)

u 1 ∂ C ∂ x + u 2 ∂ C ∂ y + λ c ϖ c = D B ∂ 2 C ∂ y 2 + D T T ∞ ∂ 2 T ∂ y 2 − Kc C − C ∞ , (4)

u 1 ∂ Π ∂ x + u 2 ∂ Π ∂ y + b W c C ∞ ∂ Π ∂ y Υ ∂ Π ∂ y = D m ∂ 2 Π ∂ y 2 . (5)

In the above equations, the terms λ e and λ c are stated as

λ e = u 1 ∂ u 1 ∂ x ∂ T ∂ x + u 2 ∂ u 1 ∂ y ∂ T ∂ x + u 1 ∂ u 2 ∂ x ∂ T ∂ y + u 2 ∂ u 2 ∂ y ∂ T ∂ y + u 1 2 ∂ 2 T ∂ x 2 + 2 u 1 u 2 ∂ 2 T ∂ x ∂ y + u 2 2 ∂ 2 T ∂ y 2 , λ c = u 1 ∂ u 1 ∂ x ∂ C ∂ x + u 2 ∂ u 1 ∂ y ∂ C ∂ x + u 1 ∂ u 2 ∂ x ∂ C ∂ y + u 2 ∂ u 2 ∂ y ∂ C ∂ y + u 1 2 ∂ 2 C ∂ x 2 + 2 u 1 u 2 ∂ 2 C ∂ x ∂ y + u 2 2 ∂ 2 C ∂ y 2 , . (6)

The boundary constraints are applied as follows (Zafar et al., 2024):

u 1 = U w x , T = T w = T 0 + γ 1 x / l , C = C w = C 0 + γ 2 x / l , Π = Π w = Π 0 + γ 3 x / l , v = 0 , at y = 0 u 1 → 0 , T → T ∞ = T 0 + γ 4 x / l , C → C ∞ = C 0 + γ 5 x / l , Π → Π ∞ = Π 0 + γ 6 x / l as y → ∞ . (7)

The x-axis is parallel to the stretched surface, and the y-axis is normal to it in the present coordinate system. Symbols T , C , and Π represent temperature, the concentration of nanoparticles, and the number density of motile microorganisms. Parameter ε 0 ​ represents the viscoelastic behavior (Walter’s-B fluid model). The kinematic viscosity of the base liquid is denoted as ν f. ​ The permeability of the porous medium is K 0. The Forchheimer coefficient is a nonlinear inertial resistance of the porous matrix C b K 0. The heat capacity of the fluid is ρ Cp f, thermal diffusivity is α f, k is the thermal conductivity, k * is the constant known as the Stefan–Boltzmann constant, which has the value of σ *, and the intensity of the heat source/sink is Q 0. Brownian motion and thermophoresis are described by the diffusion coefficients D B and D T, τ is the ratio of the heat capacity of the nanoparticles and the base fluid, and λ e ​ and λ c are the thermal and solutal relaxation parameters, A chemical reaction effect is introduced via Kc, the motility of the microorganisms is represented by D m, ​ the speed of swimming by microbes Wc​, the chemotaxis constant of their movement by b, and the characteristic length of the sheet is represented as l.

Suitable transformation (Ullah et al., 2023):

u 1 = ∂ ψ ∂ y , u 2 = − ∂ ψ ∂ x , ξ = c 0 v 1 2 y , ψ = c 0 ν 1 2 xF ξ , T = T ∞ + T w − T 0 S ξ C = C ∞ + C w − C 0 φ ξ , Π = Π w − Π 0 Λ ξ + Π ∞ . (8)

Equation 1 is satisfied and Equations 2–7 become the following, by considering Equation 8,

F ‴ + FF ″ − F ′ 2 + α 0 F ″ 2 − 2 F ′ F ‴ + F F ″ ″ + − ϵ F ′ − Fr F ′ 2 = 0 , (9)

S ″ 1 + 4 3 Rd + Pr ⁡ δ S + PrF S ′ + PrNb S ′ φ ′ + PrNt S ′ 2 − ⁡ Pr ⁡ ς e F F ′ S ′ + F 2 S ″ − ⁡ Pr ⁡ ϱ 1 F ′ = 0 , (10)

φ ″ + ScF φ ′ + Nt Nb S ″ − Sc ϱ 2 F ′ − ScK rφ − ⁡ Pr ⁡ ς c F F ′ φ ′ + F 2 φ ″ = 0 , (11)

Λ ″ + LbF Λ ′ − Lb ϱ 3 F ′ − Pe [ Λ ′ φ ′ + φ ″ Λ + ω ] = 0 . (12)

The boundary condition is α 0 = c 0 ε 0 v f denotes the viscoelastic variable, ϵ = v f K 0 c 0 the porosity factor, Fr = Cb K 0 1 / 2 the inertia coefficient, Pr = v f α f the Prandtl number, Rd = 4 σ * T ∞ 3 k * k thermal radiation, Nb = τ D B C w − C 0 v Brownian movement, ϱ 1 = γ 4 γ 1 thermal stratification, Nt = τ D T T w − T 0 v T ∞, ς e = ϖ e c 0 the relaxation time, Sc = v f D B the Schmidt number, κ = K 0 c 0 the chemical reaction, ϱ 2 = γ 5 γ 2 the solutal stratification, ω = Π ∞ Π w − Π 0 microorganism difference variable thermophoretic diffusion, Lb = v D m the microorganism Lewis number, ϱ 3 = γ 6 γ 3 microorganism stratification, and Pe = bWc D m denotes the Péclet number.

F 0 = 0 , F ′ 0 = 1 , S 0 = 1 − ϱ 1 , φ 0 = 1 − ϱ 2 , Λ 0 = 1 − ϱ 3 F ′ ∞ → 0 , S ∞ → 0 , φ ∞ → 0 , Λ ∞ → 0. (13)

The physical quantities are τ w , q w , q m and q n are described in Equation 14.

C f = τ w 1 / 2 ρ f U W 2 , Nu = x q w k T w − T ∞ , Sh = x q m D B C w − C ∞ , Nh = x q n D m Π w − Π ∞ . (14)

τ w = μ o ∂ u 1 ∂ y − ε 0 u 1 ∂ 2 u 1 ∂ x ∂ y − 2 ∂ u 1 ∂ x ∂ u 1 ∂ y , q w = − k 1 + 4 3 4 σ * T ∞ 3 k * k ∂ T ∂ y , q m = − D B ∂ C ∂ y , q n = − D m ∂ Π ∂ y . (15)

Rewriting Equation 15 using Equation 8:

C fx Re x 1 / 2 Rex 1 / 2 = 1 + α 0 F ″ 0 F ″ 0 , Nu Rex ‐ 1 / 2 = − S ′ 0 1 + 4 3 Rd Sh Rex ‐ 1 / 2 = − φ ′ 0 Nh Re ‐ 1 / 2 = − Λ ′ 0 . (16)

Here, R e x = x u w v is the local Reynolds number.

2.1 Homotopic approach

To solve the strongly coupled nonlinear ordinary differential equations (Equations 10–12) along with the boundary conditions (Equation 13), we use the homotopy analysis method (HAM). This method is helpful since, despite several traditional perturbation techniques, it does not require small or large physical settings to begin with. Rather than using additional changeable factors that allow you to guide the series toward convergence, you construct a seamless homotopic approximation to the actual answer. We chose starting estimations and linear algorithms that satisfy the boundary requirements for our fluid issue. The main functions F ξ , S ξ , φ ξ , and Λ ξ are described in series form. The range where these series actually converge is tuned by auxiliary parameters ℏ F , ℏ S , ℏ φ , and ℏ Λ; we select their best values by plotting (ℏ)-curves to obtain fast and stable answers: with linearized operators with G i = 1 − 9 coefficient constants.

F 0 ξ = 0 , S o ξ = 1 − ϱ 1 e − ξ , φ 0 ξ = 1 − ϱ 2 e − ξ , Λ 0 ξ = 1 − ϱ 3 e − ξ , (17)

L F = F ‴ − F ′ , L S = S ″ − S , L φ = φ ″ − φ , L Λ = Λ ″ − Λ , (18)

L F G 1 + G 2 e ξ + G 3 e − ξ = 0 , L S G 4 e ξ + G 5 e − ξ , (19)

L φ G 6 e ξ + G 7 e − ξ = 0 , L Λ G 8 e ξ + G 9 e − ξ = 0 , (20)

2.2 Convergence of homotopic approach

One of the most crucial elements in HAM is the convergence controller factor h, which regulates how quickly the series approximation converges. For the analytical results to converge quickly and steadily, it is crucial to determine the ideal value for this parameter. The permitted range of values h are determined using the so-called “h-curve”, which represents stable and convergent approximations with a plateau or almost zero slope in such areas. Figure 2 displays these legitimate convergence windows for the fields of motile microorganisms, temperature, concentration, and velocity. The variation − 1.990 ≤ h F . ≤ 0.10, − 1.13 ≤ h S ≤ 0.08, − 1.10 ≤ h φ ≤ 0.07, and − 1.35 ≤ h Λ ≤ 0.03. Stabilization and converging of the results obtained using the HAM are demonstrated by the corresponding h-curves at those designated intervals, which are approximately parallel to the horizontal plane. However, if h is selected beyond this bound, curves will show significant fluctuations, indicating the unsteadiness and dispersion of the approximation series. Consequently, the selection of these auxiliary elements has a significant impact on the robustness, precision, and rapid convergence of the analytical methods in this paired, non-linear border valuation issue. Table 1 reveals the convergence of estimation various orders.

TABLE 1. Convergence approximation of various orders of m.

3 Results and discussion

This section covers thermal and mass management across Walter’s-B nanoscale for the bioconvection flow in the presence of thermal radiation and the chemical factor. The porosity factor has been identified via Darcy–Forchheimer flow. The stratification process is applied across boundary conditions. The various flow variables are interpreted in Figures 3–10.

The outcomes of the inertia coefficient Fr across the velocity field F ′ ξ are given in Figure 3. The data suggest a Fr diminution in the velocity distribution. Physically, Fr is the nonlinear drag force due to the Forchheimer effect in a porous medium. Elevated Fr is where the inertial resistance of the porous structure is further improved by the additional resistance of the fluid flow. The upshot of the porosity constraint ϵ vs. velocity field F ′ ξ is depicted in Figure 4. An escalation in ϵ is found to lead to a significant reduction in velocity field F ′ ξ. This is a value that represents the natural resistance to permeability of the porous matrix. As ϵ growths, the drag force within the solid matrix across fluid grows, which would reduce across fluid velocity. The effects of the fluid parameter α 0 across velocity distribution is depicted in Figure 5. The results show that the higher the value of the velocity profile, the greater the intensification α 0. This parameter is a measure of the non-Newtonian properties of the working fluid and is related to its elastic properties. An increment in α 0, the mixing ability of the fluid, is improved by promoting the ability of the fluid to transfer momentum.

Figure 6 demonstrates the effect of the Brownian motion parameter Nb across temperature profile S(ξ). It is evident that the temperature distribution increases with increasing values of Nb. Physically, Brownian motion describes the random movement of nanoparticles suspended within the base fluid. As Nb increases, the frequency of nanoparticle collisions becomes higher, enhancing the transfer of thermal energy from the nanoparticles to the surrounding fluid. The variation of the thermal radiation parameter Rd across temperature curve S(ξ) is presented in Figure 7. The data reveal a substantial increase in temperature as the value of Rd is amplified. The additional thermal energy generated by radiation across normal transfer processes is responsible for this phenomenon. Figure 8 demonstrates the Nt upon the thermal field. It is clear that an increase in Nt causes the temperature field to increase steadily. “Thermophoresis” is the process by which fine particles migrate along a temperature gradient, moving from hotter regions toward cooler zones. The effect of the thermal stratification parameter ϱ 1 across the temperature field is demonstrated in Figure 9. In contrast to the preceding parameters, an increase in ϱ 1​ results in a notable decrease in the temperature profile. Thermal stratification describes the variation of the ambient fluid temperature along the primary flow direction. A higher stratification parameter indicates a more pronounced difference in the ambient temperature, which effectively diminishes the thermal driving potential between the hot surface and the surrounding fluid.

Figures 10, 11 represent the parametric effects of the solutal stratification parameter ϱ 2 ​ and the chemical reaction index Kr across concentration function φ ξ . An incremental increase in ϱ 2 clearly shifts the concentration curve downward. This behavior is attributed to the solutal stratification effect: when the bulk concentration of the ambient fluid varies along the axis, the concentration difference between the surface and the fluid (mass-transfer gradient) is directly reduced. As far as the reactive effect is concerned, Figure 11 shows that the higher the value of variable y, the greater the amount of product formed. Kr ​ applies an extreme decay in φ ξ. Physically, this is driven by the destructive chemical process; a greater Kr ​escalates the strength of the reaction and hence faster consumption of diffusive species in the flow domain.

Figure 12 illustrates the effect of the parameter of the microorganism, referred to as ϱ 3. The average number of microorganisms per unit volume, as a function of the profile, is Λ ξ. The data points plotted reveal a definite pattern in which the upsurge in the number of trees correlates with an increase in the number of people. Value ϱ 3​ causes a significant decrease in the number of microorganisms throughout the domain. The changes of the ambient microorganism concentration of the bulk fluid are quantified in Figure 13, which looks at the effect of the Péclet number Pe of motile microorganism distribution Λ ξ. The results have a consistent trend which shows that increasing the Péclet number leads to a reduction of microorganism concentration in the region. The Péclet number is a dimensionless parameter which defines the relative importance of the advective transport relative to the diffusive transport of microorganisms.

The link between the viscoelastic parameter (α 0), the Forchheimer number (Fr), and the skin friction coefficient (Re 1 / 2 Cf) is shown in Figure 14. The magnitude of the skin friction coefficient significantly decreases with a boost in either parameter. The physical mechanisms connected to each parameter are responsible for this behavior. In particular, the Forchheimer number indicates the inertial resistance in the porous matrix; its increases make it harder for the fluid to flow, which lowers the shear stress applied to the wall. Simultaneously, the elastic properties of the Walter’s-B fluid are improved by an increase in the viscoelastic parameter.

Figure 15 shows (Nb) and (Rd) across the Nu. An increase in the Rd factor improves radiative heat transfer. Figure 16 indicates that the local Sherwood number (Re^(-1/2) Sh) is affected by variations in the thermophoresis parameter (Nt) and the Schmidt number (Sc). Figure 17 illustrates the relationship between the biological convection Lewis number (Lb), the Péclet ratio (Pe), and the motile microbial density number (R e − 1 / 2 Nh). According to these findings, Nh increases and Pe falls with increasing Lb. Microorganism diffusivity decreases as the bioconvection Lewis number increases. This makes them concentrate instead of dispersing away from the wall, which raises the density gradient close to the wall. On other hand, the Péclet number indicates the importance of advective movement in comparison to diffusive transport. Advection, which successfully moves microorganisms away from the border, becomes increasingly significant as Pe increases. Figures 18a,b portrays the contour plots of the given problem for different values of Forchheimer parameter.

The values of the skin-friction coefficient computed from the above are summarized in Table 2. For Cf, using the various values of the inertia coefficient Fr, the porosity parameter ε is called the fluid parameter α 0. The data indicate that when the number of Fr from 0.1 to 0.7 raises Cf ​ from 1.321212 to 1.427697, an increase in inertial drag due to porous structure is responsible for this upward trend, which directly increases the wall shear stress. Similarly, an increase in the parameter of porosity ε is accompanied by an overall enhancement in Cf ​ The porous skeleton provides a greater resistance to the working fluid passing through it, resulting in a greater frictional interaction at the boundary. Furthermore, as the values increase, it is clear that there is a significant increase in the skin-friction coefficient. As this fluid parameter increases from 0.1 to 1.0, Cf jumps from 0.165271 to 0.897528. This behavior indicates that the momentum exchange in the flow is enhanced by the non-Newtonian rheology increasing the velocity gradient in the near-wall region, and hence the surface drag. The response of the Nusselt number is shown in Table 3. Nu to variations in the parameter of thermal radiation Rd, the heat source parameter δ, the coefficient of the Brownian motion Nb, thermophoresis coefficient Nt, the thermal stratification parameter ϱ 1. A notable increase in Nu is observed with an escalation in Rd. The enhancement is due to the extra radiative flux which contributes to the overall thermal energy transport away from the heated boundary. An increase of the Brownian motion parameter Nb also enhances the performance of heat transfer, since the enhanced chaotic motion of nanoparticles leads to improved thermal diffusion. On other hand, higher values of the thermophoresis parameter Nt have a decreasing effect on Nu. Table 4 shows the behavior of the Sherwood number Sh, Schmdt number Sc, Brownian motion Nb, thermophoresis coefficient Nt, stratifying parameter of solutal stratification ς c, and concentration stratification ϱ 2. The results in the table show a gradual increase in Sh with rising Sc. Table 5 shows the effect of the Lewis number of bioconvection Lb, Péclet number Pe, concentration difference of microorganisms ω, the stratification of microorganisms ϱ 3, and the number of motile microorganisms across number Nh. The results show that Nh escalations with Lb intensifications. The thermal to microorganism diffusivity is represented in the term Lb, which increases the dispersion of motile species, thus augmenting the density gradient close to the wall. Pe also increases, which suggests that Nh will continue to increase. Table 6 reveals that the comparison analysis agrees well with Rehman et al. (2025a).

TABLE 2. Computation of Cf for various values of Fr , ϵ , α 0.

TABLE 3. Computation of Nu for various values of Rd , Nb , Nt , ς e , ϱ 1.

TABLE 4. Computation of Sh for various values of Sc , Nb , Nt , ς c , ϱ 2.

TABLE 5. Computation of Nh for various values of Lb , Pe , ω , ϱ 3.

TABLE 6. Comparison of − F ′′ 0 with existing literature for varying ϵ taking all other parameters equal to zero.

4 Conclusion

This research studies the coupled thermal and mass transport phenomenon of a nanofluid consisting of motile microorganisms in a Darcy–Forchheimer porous material for a stretching solid surface. The theoretical framework combines the effect of thermal radiation, thermophoretic effect, Brownian diffusion, chemical reaction, triple stratification (thermal, solutal, and microbial), and the Cattaneo–Christov double-diffusion model. Once non-dimensionalized, the governing equations were solved analytically with the homotopy analysis method (HAM), and the convergence of the solution was carefully monitored using ℏ-curves and tabulated residual errors. This study has produced the following results:

• The Forchheimer inertia term and the porosity parameter cause very significant reduction in velocity profile.
• Additional factors to the augmentation of temperature distribution and thermal boundary layer thickness are the Brownian motion, radiation, and thermophoretic effects.
• However, thermal stratification reduces the thermal driving force (the difference between wall and ambient temperatures) and thus slows the temperature field.
• Both solutal stratification and chemical reaction parameters decrease the concentrations gradients of nanoparticles, which means that the mass diffusion from lower concentration difference and the consumption of species due to weak chemical reaction.
• The motile microorganism density is decreased due to microbial stratification and higher Péclet numbers, which further highlights the controlling role played by advection and environmental gradients on bioconvection patterns.
• The increase in wall shear stress (represented by skin-friction coefficient) indicated by the increase in Forchheimer number, porosity, and Walter’s-B parameter suggests that the momentum exchange is strengthened near the wall.
• Radiation and Brownian motion enhance heat transfer performance (local Nusselt number), whereas thermophoresis reduces it. Opposing effects of molecular diffusion and nanoparticle drift in mass transport are shown in the Sherwood number, which increases when Schmidt numbers increase but decreases when the thermophoresis is enhanced.
• The bioconvection Lewis number and concentration difference parameter are observed to increase the density of motile microorganisms, thus highlighting the importance of microorganism diffusivity in the transport of bioconvective flux.

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Footnotes

Edited by: Saranya Shekar, United Arab Emirates University, United Arab Emirates

Reviewed by: Sharif Ullah, University of Malakand, Pakistan

Israr Khan, Kohat University of Science and Technology, Pakistan