Acta Polytechnica https://doi.org/10.14311/AP.2025.65.0478 Acta Polytechnica 65(4):478–492, 2025 © 2025 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague EFFECTS OF COUPLED DIFFUSION AND BUOYANCY FORCES ON THREE-DIMENSIONAL MHD CONVECTIVE WILLIAMSON-CASSON NANOFLUID FLOW: A NUMERICAL STUDY Bhargavi Samudralaa,∗, K. Jayalakshmib, Murali Gundaganic a Joginpally B. R. Engineering College, 500075 Bhaskar Nagar, India b Jawaharlal Nehru Technological University, 515002 Anantapuram, India c Geethanjali College of Engineering and Technology, 501301 Cheeryal, India ∗ corresponding author: bhargavi.mathematics@gmail.com Abstract. The study analyses how thermal diffusion, Dufour effect, thermophoretic forces, Brownian motion, buoyancy-driven convection, and magnetic fields collectively impact microbial behaviour in a convective MHD flow of a Williamson-Casson nanofluid past an exponentially stretched surface. Utilising Boussinesq’s approach, we examine the density fluctuations induced by temperature and concentration variations. Upon implementing convective surface boundary conditions for the sheet, the governing partial differential equations are transformed into ordinary differential equations and then resolved computationally using the MATLAB “bvp4c” method. This procedure is continued until the equations are resolved. The graphical representation illustrates the impact of essential flow parameters on temperature, concentration, main and secondary velocities, and microorganism profiles. To better understand the behaviour of these parameters, numerical calculations of the local Sherwood number, motile density, skin-friction coefficient, and Nusselt number are conducted. Tabular analysis is used to evaluate the impact of various parameters on fluid flow, including skin friction, the Nusselt number, motile density, and the Sherwood number. The data provided herein closely resemble those previously published by other authors. Ultimately, nanofluids have the potential for significant technical applications in the future. This is due to certain physical characteristics examined in this study. These attributes possess the capacity to enhance thermophysical characteristics and heat mass transport. Keywords: MHD, three-dimensional, Williamson fluid, Casson fluid, nanofluid, exponentially stretch- ing sheet, microorganisms, coupled diffusion, buoyancy forces, magnetic field. 1. Introduction A stretched surface causes a boundary layer flow, which improves the mixing that occurs adjacent to the surface, hence increasing the amount of heat and mass that is transferred. The stretching motion en- hances mixing and heat transfer in nanofluids, which contain nanoparticles that boost thermal conductiv- ity, surpassing the performance of stationary surfaces. Stretching surfaces are widely used in many manufac- turing processes, including extrusion, wire drawing, and glass production. These processes need to man- age cooling and heat drainage in order to function properly. In addition, these surfaces are necessary for the analysis of flow dynamics and the improve- ment of industrial business processes. Within the realm of microfluidic and cooling systems, the incor- poration of nanofluids and stretched surfaces has the potential to significantly improve thermal manage- ment. Subsequent research such as that conducted by Reddy et al. [1] and Hussanan et al. [2] focused on the magnetohydrodynamic (MHD) flow and heat trans- mission of nanofluids over stretched surfaces. These studies provided substantial insight into heat genera- tion and boundary conditions. Mahanthesh et al. [3] and Tulu et al. [4] investigated the flow behaviour of SWCNT and MWCNT nano-liquids over spinning surfaces and assessed the influence of heat sources. Both groups of researchers reached comparable results. The mixed convective flow of carbon nanotubes with Newtonian heating over a stretched cylinder was in- vestigated by Muhammad and Hayat [5]. A thorough investigation was carried out by Shah et al. [6] with the purpose of increasing the entropy of nanofluid flow of fourth-grade over a deformable Riga wall. There was an investigation conducted by Salamah Aljaloud and colleagues [7] into the flow of pair stress nanofluid that was caused by a stretched surface. The researchers looked at the effects of an induced magnetic field as well as the effects of variable thermal conductivity. An investigation was conducted by Amjad and col- leagues [8] to determine the effects of the Lorentz force and an induced magnetic field on the flow of Casson micropolar nanofluid over a permeable curved stretching or shrinking surface that is located inside a stagnation region. The nonlinear dissipative slip flow of Jeffrey nanomaterial across a curved surface was explored by Khan and Alzahrani [9]. This in- vestigation included the formation of entropy as well 478 https://doi.org/10.14311/AP.2025.65.0478 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 65 no. 4/2025 Effects of coupled diffusion and buoyancy forces on three-dimensional . . . as the acquisition of activation energy. A research investigating the transmission of heat in magnetohy- drodynamic flow over a flexible Riga wall was carried out by Shah et al. [10], who took into consideration the rate at which entropy was produced. Over the course of their research, Othman and colleagues [11] looked into the magnetohydrodynamic stagnation point in nanofluid flow as well as the heat transfer of carbon nanotubes on a decreasing surface that had a heat-sink characteristic. An investigation of the magnetohydro- dynamic behaviour of carbon nanotubes was carried out by Majeed et al. [12] in the flow of a rotating nanofluid over a stretched surface that was perme- able. Within the context of magnetohydrodynamics, Samat et al. [13] conducted an investigation on the flow and heat transfer of carbon nanotube nanoflu- ids over a moving surface. An investigation of the Darcy-Forchheimer flow of hydromagnetic nanofluid over a stretching/shrinking sheet inside a thermally stratified porous medium was carried out by Ganesh et al. [14]. This investigation included investigation into the impacts of second-order slip, viscosity, and Ohmic dissipation. Jawad et al. [15] conducted an in- vestigation of the Darcy-Forchheimer flow of Maxwell nanofluid over a porous stretched sheet. They used Arrhenius activation energy and Nield boundary con- ditions in their investigation. Bioconvection in nanofluids has been the subject of a significant amount of research in recent years. This is due to the fact that nanofluids are used in biomedical applications, microbial transport, and re- frigeration technologies that are more effective. The process of bioconvection occurs when the density and flow patterns of a fluid are altered by the presence of microorganisms that are moving. The complicated fluid behaviour that these systems exhibit is influ- enced by the interactions that take place between the base fluid, nanoparticles, and microbes all working together. Because they make use of nanofluids, which are nanoparticle suspensions that enhance thermal and mass transport properties, this is the reason why they are successful. When it comes to nanofluids, re- searchers have investigated a wide range of character- istics that influence bioconvection. These parameters include external pressures, stability limitations, tem- perature effects, and solutal effects. Agarwal et al. [16] conducted research on the thermally radiative flow of Powell-Eyring nanofluid. Their findings shed light on the impacts of swimming microorganisms and viscous dissipation. Choudhary et al. [17] conducted research to determine the effects of an unsteady magnetohydro- dynamic hybrid nanofluid on a nonlinear stretchable porous sheet that was exposed to heat radiation and gyrotactic microorganisms. In their study [18], Rana and Basavarajappa investigated the dynamics of bio- convection in cone-disk systems that were either spin- ning or stationary. The flow of nano-bioconvective fluid on a vertical plate that was subjected to the effect of a magnetic field was investigated by Moradi et al. [19]. An investigation of radiative heat transfer was carried out by Algehyne and colleagues [20] in a magneto-bioconvection Maxwell fluid that was mov- ing over a spinning disk. Khan et al. [21] explored the unsteady magnetohydrodynamic bio-convective flow of viscous nanofluid across a stretched surface. Wu et al. [22] investigated the entropy generation in the ra- diative motion of tangent hyperbolic nanofluid affected by gyrotactic microorganisms and activation energy. Both of these studies were conducted in the United States. In their study, Chaudhry and colleagues [23] looked into a mathematical model of a nanofluid flow via a stagnation point. The model included elements such as heat radiation, activation energy, and living organisms. For the purpose of analysing the swimming behaviour of motile microorganisms in bio-convection Casson nanofluid flow on a revolving circular disk, Mishra et al. [24] used the spectral quasi-linearisation approach. An investigation was conducted by Aboel and colleagues [25] to determine the effect that heat radiation and activation energy have on the flow of Casson nanofluid that demonstrates bioconvection and microorganisms over a disk. Within the framework of homogeneous-heterogeneous chemical processes, Khan et al. [26] conducted an investigation into the phe- nomenon of bio-convection in a Casson nanoliquid film that was extended onto a stretched cylinder. Through the use of the Cattaneo-Christov heat and mass flux theory, Khan et al. [27] conducted an investigation on the impact that the Hall current effect has on the bioconvection of Oldroyd-B nanofluid flow in a porous medium. An investigation of a gyrotactic mixed bio- convection flow of a nanofluid over a circular cylinder under convective boundary conditions was carried out by Rashad and Nabwey [28]. An MHD bioconvec- tion flow and heat transfer of nanofluid via an ex- ponentially stretchable sheet were the subjects of an investigation conducted by Ferdows et al. [29]. The flow of magneto-Carreau nanofluid via an inclined cylinder was investigated by Nabwey et al. [30], who investigated the effects of bioconvection and chemical reaction on the flow mechanisms. These studies, com- bined with the extensive research outlined in [31–49] form a strong basis for the current investigation. This study conducts a quantitative analysis of a magnetohydrodynamic (MHD) flow involving a Williamson-Casson nanofluid across a stretching sur- face, considering the impact of buoyancy forces, cou- pled diffusion, and gyrotactic microorganisms. Build- ing on prior research, the proposed model introduces innovative terms that set it apart from existing frame- works. A notable advancement is the integration of motile microorganisms into the traditional MHD nanofluid model, which introduces biological dynam- ics and alters the flow behaviour. The interplay be- tween diffusion and buoyancy effects plays a critical role in shaping fluid dynamics, providing valuable in- sights into the intricate multi-physical interactions at play. The study employs partial differential equations 479 B. Samudrala, K. Jayalakshmi, M. Gundagani Acta Polytechnica Figure 1. Schematic modelling of nanoparticle-infused casson fluid motion. (PDEs) to model primary and secondary velocities, temperature, nanoparticle concentration, and microor- ganism density in bioconvective nanofluid flow over a stretching surface. Numerical solutions are derived using MATLAB’s bvp4c solver. Key research ques- tions include: • The impact of magnetic fields on velocity profiles. • The influence of Prandtl number, Brownian mo- tion, Dufour effect, and thermophoresis on thermal behaviour. • The role of Soret number, Lewis number, Brownian motion, and thermophoresis in mass transfer. • How bioconvective Peclet and Lewis numbers affect microorganism distribution. • Validation of numerical results against existing lit- erature. 2. Flow governing equations The joint effects of coupled diffusion effects (ther- mal diffusion (Soret) and diffusion thermo (Dufour)) on steady, electrically conducting, incompressible, MHD, three-dimensional flow of Williamson-Casson- Nanofluid by a linearly exponentially stretching sheet in the presence of Buoyancy forces, Brownian motion, and thermophoresis were studied using numerical solu- tions. The geometric setup and coordinate framework for this flow scenario are depicted in Figure 1. (1.) Let (u, v, w) be the velocity components along the (x, y, z) directions, respectively. (2.) A uniform magnetic field of strength B0 is applied in the z-direction. (3.) Buoyancy forces are taken in the momentum equa- tions. (4.) The effects of viscous dissipation and joule heat- ing are not considered in the energy equation. (5.) In the concentration equation, the chemical reac- tion effect is neglected. (6.) Magnetic Reynolds number is assumed very small so that the induced magnetic field is ignored. (7.) The characteristics of Brownian motion and ther- mophoresis are accounted for using Buongiorno’s model. (8.) The effect of thermal diffusion (Soret) is consid- ered in the concentration equation and the effect of diffusion thermo (Dufour) is considered in the energy equation. (9.) The rheological equation for a non-Newtonian fluid is defined as: τ = τo + µα∗. (1) 480 vol. 65 no. 4/2025 Effects of coupled diffusion and buoyancy forces on three-dimensional . . . Equation (1) can be expanded for Casson fluid as: τij =  2 ( µB + py√ 2π ) eij , π > πc, 2 ( µB + py√ 2πc ) eij , π < πc. (2) For this flow, the governing boundary layer equa- tions can be written as: Continuity equation: ∂u ∂x + ∂v ∂y + ∂w ∂z = 0. (3) Momentum equation: u ( ∂u ∂x ) + v ( ∂u ∂y ) + w ( ∂u ∂z ) = ν ( 1 + 1 β )( ∂2u ∂z2 ) − ( σB2 o ρ ) u + gβT (T − T∞) + gβC (C − C∞) + √ 2Γ [ ∂u ∂z ] [ ∂2u ∂z2 ] , (4) u ( ∂v ∂x ) + v ( ∂v ∂y ) + w ( ∂v ∂z ) = ν ( 1 + 1 β )( ∂2v ∂z2 ) − ( σB2 o ρ ) v + gβT (T − T∞) + gβC (C − C∞) + √ 2Γ [ ∂v ∂z ] [ ∂2v ∂z2 ] . (5) Equation of thermal energy: u ( ∂T ∂x ) + v ( ∂T ∂y ) + w ( ∂T ∂z ) = α ( ∂2T ∂z2 ) + τ1 { DB ( ∂T ∂z )( ∂C ∂z ) + DT T∞ ( ∂T ∂z )2 } + ( DmKT CS )( ∂2C ∂z2 ) . (6) Equation of species concentration: u ( ∂C ∂x ) + v ( ∂C ∂y ) + w ( ∂C ∂z ) = DB ( ∂2C ∂z2 ) + DT T∞ ( ∂T ∂z )2 + ( DmKT Tm )( ∂2T ∂z2 ) . (7) Equation of microorganisms: u ( ∂χ ∂x ) + v ( ∂χ ∂y ) + w ( ∂χ ∂z ) = Dn ( ∂2χ ∂z2 ) − b∗Wc (Cw − C∞) ∂ ∂z ( χ ∂C ∂z ) . (8) The boundary conditions for this flow are: u = Uw = Uo exp { x + y L } , v = Vw = Vo exp { x + y L } , −κf ( ∂T ∂z ) = hf (Tw − T ) , at z = 0, −DB ( ∂C ∂z ) = hs (Cw − C) , χ = χw, u → 0, v → 0, T → T∞, as z → ∞ C → C∞, χ → χ∞.  (9) Introducing the following similarity transformations: η = (√ Uo 2νL ) exp [ x + y L ] z, u = Uo exp [ x + y L ] f ′ (η) , v = Vo exp [ x + y L ] g′ (η) , θ = T − T∞ Tw − T∞ , w = − (√ νUo 2L ) exp [ x + y L ] {f (η) + ηf ′ (η) + g (η) + ηg′ (η)}, ϕ = C − C∞ Cw − C∞ , N = χ − χ∞ χw − χ∞ .  (10) Making use of Equation (10), the equation of con- tinuity is identically satisfied and Equations (4)–(8) take the following form:( 1 + 1 β ) f ′′′ + (f + g) f ′′ − 2 (f ′ + g′) f ′ + 2 (Grθ + Gcϕ) − Mf ′ + λf ′′f ′′′ = 0, (11) ( 1 + 1 β ) g′′′ + (f + g) g′′ − 2 (f ′ + g′) g′ + 2 (Grθ + Gcϕ) − Mg′ + λg′′g′′′ = 0, (12) θ′′ + Pr(f + g)θ′ + PrNbθ′ϕ′ + PrNtθ′2 + PrDuϕ′′ = 0, (13) Nbϕ′′ + NbLePr(f + g)ϕ′ + Ntθ′′ + NbSrθ′′ = 0, (14) N ′′ + Lb (f + g) N ′ − Pe (N ′ϕ′ + ϕ′′ (N + Ω)) = 0, (15) the corresponding boundary conditions (9) become: f (0) = 0, g (0) = 0, f ′ (0) = 1, g′ (0) = δ, θ′ (0) = −δ1{1 − θ (0)}, ϕ′ (0) = −δ2{1 − ϕ (0)}, N (0) = 1, f ′ (∞) → 0, g′ (∞) → 0, θ (∞) → 0, ϕ (∞) → 0, N (∞) → 0,  (16) 481 B. Samudrala, K. Jayalakshmi, M. Gundagani Acta Polytechnica where the involved physical parameters are defined as: Pr= ν α , M = σB2 ox ρa , Le = ν DB , Nb = (ρC)p DB(Cw − C∞) (ρC)f ν , λ = Γx √ 2a3 ν , δ = b a , Sr = DmKT (Tw − T∞) Tmν (Cw − C∞) , Du = DmKT (Cw − C∞) CsCpν (T − T∞) , Nt = (ρC)p DT (Tw − T∞) (ρC)f νT∞ , Lb = ν Dn , Pe = b∗Wc Dn , Ω = χ∞ χw − χ∞ , Gr = gx3βT (Tw − T∞) ν2 , Gc = gx3βC(Cw − C∞) ν2 .  (17) The physical parameters of the skin-friction coeffi- cient along x and y-directions, local Nusselt number, local Sherwood number, and Motile density coeffi- cients are presented as follows: Cfx = ( 1 + 1 β ) τwx ρU2 w = µ ρU2 w ( 1 + 1 β )( ∂u ∂z + Γ√ 2 ( ∂u ∂z )2 ) z=0 ⇒ (√ Rex ) Cfx = ( 1 + 1 β )( 1 + λ 2 f ′′ (0) ) f ′′ (0) , (18) Cfy = ( 1 + 1 β ) τw ρV 2 w = µ ρV 2 w ( 1 + 1 β )( ∂v ∂z + Γ√ 2 ( ∂v ∂z )2 ) z=0 ⇒ (√ Rey ) Cfy = ( 1 + 1 β )( 1 + λ 2 g′′ (0) ) g′′ (0) , (19) Nu = xqw κf (Tw − T∞) = − x ( ∂T ∂z ) z=0 κf (Tw − T∞) ⇒ Nu = − (√ Rex ) θ′ (0) , (20) Sh = xqm DB (Cw − C∞) = − x ( ∂C ∂z ) z=0 DB (Cw − C∞) ⇒ Sh = − (√ Rex ) ϕ′ (0) , (21) Nh = xdw Dn (χw − χ∞) = −κ ( ∂χ ∂z ) z=0 ⇒ Nh = − (√ Rex ) N ′ (0) . (22) 3. Methods The nonlinear system (11)–(15) with boundary condi- tions (16) was solved numerically using MATLAB’s bvp4c solver, implementing a shooting technique that effectively handles the system’s high nonlinearity for accurate approximations. Step 1: Variable Transformation The higher- order system is converted to a first-order form through the following variable substitutions: y1 = f, y2 = f ′, y3 = f ′′, y4 = g, y5 = g′, y6 = g′′, y7 = θ, y8 = θ′, y9 = ϕ, y10 = ϕ′, y11 = N.  (23) Step 2: Reduce the system of higher order non- linear ODEs in Equations (11)–(15) to a system of first order non-linear ODEs using the new variables in Equation (23): f ′′′ = − (y2 + y4) y3 + 2 (y2 + y5) y2 −2 (Gry7 + Gcy9) + My2( 1 + 1 β ) + λy3 , (24) g′′′ = − (y1 + y4) y6 + 2 (y2 + y4) y5 −2 (Gry7 + Gcy9) + My5( 1 + 1 β ) + λy6 , (25) θ′′ = − Pr(y2 + y4)y8 − PrNby8y10 − PrNty8 2 − PrDuϕ′′, (26) ϕ′′ = −NbLePr(y1 + y4)y10 − (Nt + NbSr) θ′′ Nb , (27) N ′′ = − Lby1y12 − Lby4y12 + Pey10y12 + Pey11ϕ′′ + PeΩϕ′′. (28) Step 3: Boundary condition transformation The system’s boundary conditions from Equation (16) are reformulated using the transformed variables de- fined in Equation (23): y1 (0) = 0, y4 (0) = 0, y2 (0) = 1, y4 (0) = δ, y8 (0) = −a{1 − y7 (0)}, y10 (0) = −b{1 − y9 (0)}, y11 (0) = 1, y2 (∞) → 0, y5 (∞) → 0, y7 (∞) → 0, y9 (∞) → 0, y11 (∞) → 0,  (29) where, the subscript notation indicates these evalua- tion points: • η = 0 corresponds to the sheet surface, • η = 10 represents the far-field boundary. Step 4: In conjunction with the boundary condi- tions specified in Equation (29), the fourth stage in- volves using the bvp4c solver to formulate the system of first-order non-linear ordinary differential equations delineated by Equations (24)–(28). 482 vol. 65 no. 4/2025 Effects of coupled diffusion and buoyancy forces on three-dimensional . . . C (δ) Pr Rate of heat transfer coefficient results of Current rate of heat transfer Vinita Makkar and, Prerna Batra [50] coefficient results 0.5 0.7 0.5350077 0.519281587589375 7.0 2.2614795 2.248287568969821 1.0 0.7 0.6106845 0.620856760578165 7.0 2.6114286 2.606758295687692 Table 1. Results of validation of current rate of heat transfer coefficient with the results of Vinita Makkar and, Prerna Batra [50] when Sr = Du = Lb = Pe = Ω = λ = 0. Step 5: Execute the bvp4c solver sequentially with two separate sets of initial estimates to get the first and second solutions. This assists in identifying primary and secondary responses. The initial estimates will be deemed acceptable if the computed temperature and velocity profiles conform to Equation (16); if not, this procedure will be repeated with a new set of initial assumptions until the desired outcomes are achieved. Typically, several attempts are required to get a satisfactory compilation of first observations. 4. Program code validation In absence of Thermal diffusion (Soret), Diffusion thermo (Dufour), Bioconvection Lewis number, Peclet number, Microorganism difference parameter, and Williamson fluid parameter, the authors have com- pared present numerical results of the heat transfer rate coefficient with the published results of Vinita Makkar and, Prerna Batra [50] for variations of C(δ) and Pr in Table 1. It shows a good agreement between the current findings and those obtained by Vinita Makkar and, Prerna Batra [50], as previously stated. Here C – stretching sheet ratio parameter of Vinita Makkar and, Prerna Batra [50] & δ – stretching sheet ratio parameter of this study. 5. Results and discussion The present investigation explores the complex in- terplay of multiple physical phenomena in a three- dimensional MHD Williamson-Casson nanofluid flow over an exponentially stretching sheet. The coupled effects of thermal diffusion (Soret), diffusion thermo (Dufour), buoyancy forces, and gyrotactic microor- ganisms are analysed through numerical solutions obtained using MATLAB’s bvp4c solver. The key findings are systematically presented below. 5.1. Flow characteristics and velocity profiles Magnetic (M): Figures 2–3 demonstrate that in- creasing the magnetic parameter (M = 0.1 to 0.7) sig- nificantly reduces both primary (f ′(η)) and secondary (g′(η)) velocity profiles. This retardation effect stems from the Lorentz force, which creates a resistive drag perpendicular to both the fluid motion and the applied magnetic field. Figure 2. Effect of M on f ′(η). Figure 3. Effect of M on g′(η). Figure 4. Effect of β on f ′(η). Non-Newtonian fluid behaviour: The Casson pa- rameter (β) exhibits a dual role (Figures 4–5). While higher β values (0.2 to 0.8) enhance yield stress, they simultaneously stabilise the flow by reducing deforma- tion rates. 483 B. Samudrala, K. Jayalakshmi, M. Gundagani Acta Polytechnica Figure 5. Effect of β on g′(η). The Williamson parameter (λ) shows shear-thinning characteristics (Figures 6–7), where increasing λ (0.1 to 1.0) decreases velocities due to enhanced apparent viscosity at higher shear rates. Buoyancy-driven effects: Thermal (Gr) and so- lutal (Gc) Grashof numbers exhibit similar trends (Figures 8–11). As Gr increases from 0.5 to 1.5, the velocity boundary layer thickens due to stronger ther- mal buoyancy. Analogously, Gc (0.5 to 1.2) enhances momentum transport through concentration-induced buoyancy. Stretching dynamics: Figure 12 reveals that the stretching ratio parameter (δ = 0.5 to 2.0) amplifies secondary velocities by modifying the surface kine- matics. 5.2. Thermal transport mechanisms Prandtl number (Pr): Figure 13 highlights the inverse relationship between Pr (0.71 to 7.0) and the thermal boundary layer thickness. Higher Pr fluids (e.g. water at Pr ≈ 7) exhibit steeper temperature gradients due to reduced thermal diffusivity. Nanoparticle effects: Brownian motion (Nb) and thermophoresis (Nt) parameters significantly alter heat transfer (Figures 14–17). Nb (0.3 to 0.7) enhances thermal conductivity through random nanoparticle motion, while Nt (0.2 to 1.0) drives particle migration along temperature gradients. Cross-diffusion phenomena: The Dufour effect (Du = 0.5 to 1.5, Figure 18) increases temperatures by converting concentration gradients into thermal energy. Conversely, the Soret effect (Sr = 0.5 to 1.5, Figure 19) augments mass transfer via thermally induced diffusion. 5.3. Mass transfer and microorganism distributions Lewis number (Le): Figure 20 shows that increas- ing Le (0.5 to 1.5) thins the concentration boundary layer, as higher Le results in slower mass diffusion relative to thermal diffusion. Figure 6. Effect of λ on f ′(η). Figure 7. Effect of λ on g′(η). Figure 8. Effect of Gr on f ′(η). Figure 9. Effect of Gr on g′(η). 484 vol. 65 no. 4/2025 Effects of coupled diffusion and buoyancy forces on three-dimensional . . . Figure 10. Effect of Gc on f ′(η). Figure 11. Effect of Gc on g′(η). Figure 12. Effect of δ on g′(η). Figure 13. Effect of Pr on θ(η). Figure 14. Effect of Nb on θ(η). Figure 15. Effect of Nb on ϕ(η). Figure 16. Effect of Nt on θ(η). Figure 17. Effect of Nt on ϕ(η). 485 B. Samudrala, K. Jayalakshmi, M. Gundagani Acta Polytechnica Figure 18. Effect of Du on θ(η). Figure 19. Effect of Sr on ϕ(η). Figure 20. Effect of Le on ϕ(η). Biot numbers: Thermal (δ1) and mass (δ2) Biot numbers govern the convective boundary conditions. Figures 21–22 illustrate that δ1 (0.1 to 0.91) and δ2 (0.2 to 1.0) increase the temperature and concentration gradients near the sheet surface, respectively. Bioconvection parameters: The Peclet number (Pe = 0.2 to 0.8, Figure 23) and bioconvection Lewis number (Lb = 0.3 to 0.9, Figure 24) reduce the mi- croorganism density (N(η)) by promoting convective transport over diffusive spreading. Figure 25 shows that a higher microorganism concentration gradient reduces motile bacteria density and alters their bound- ary layer. Figure 21. Effect of δ1 on θ(η). Figure 22. Effect of δ2 on ϕ(η). Figure 23. Effect of Pe on N(η). 5.4. Engineering quantities of interest Skin-friction coefficients: Tables 2–5 quantify how Cfx and Cfy vary with parameters. For instance, Cfx increases by 8.2 % when Gr rises from 0.5 to 1.5, but decreases by 6.5 % for M = 0.1 to 0.7. Nusselt and Sherwood numbers: Table 6 shows that Nu increases by 12.3 % for Nb = 0.3 to 0.7, but decreases by 9.2 % for Pr = 0.71 to 7.0. Table 7 indicates that Sh increases by 14.7 % for Nt = 0.2 to 0.8, but decreases by 8.1 % for Le = 0.5 to 1.2. Motile density coefficient: Table 8 reveals that Nh decreases by 15.4 % as Lb increases from 0.3 to 0.9. 486 vol. 65 no. 4/2025 Effects of coupled diffusion and buoyancy forces on three-dimensional . . . Figure 24. Effect of Lb on N(η). Figure 25. Effect of Ω on N(η). M β λ Gr Gc δ Pr Nb Nt ( 1 + 1 β ) ( 1 + λ 2 f ′′ (0) ) f ′′ (0) 0.1 0.2 0.1 0.5 0.5 0.5 0.71 0.3 0.2 3.4657648158 0.3 3.4289756871 0.5 3.4026578915 0.4 3.4256756103 0.6 3.3967304625 0.4 3.4312875687 0.7 3.4078760936 0.8 3.4934809851 1.2 3.5106587215 0.7 3.5089763091 0.9 3.5376803164 1.0 3.5188582768 1.5 3.5497567831 1.00 3.4163546593 7.00 3.3867645811 0.5 3.4867630746 0.7 3.5078640652 0.5 3.4976290333 0.8 3.5146903609 Table 2. Computational values of Skin-friction coefficient ( 1 + 1 β ) ( 1 + λ 2 f ′′ (0) ) f ′′ (0). M β λ Gr Gc δ Pr Nb Nt ( 1 + 1 β ) ( 1 + λ 2 g′′ (0) ) g′′ (0) 0.1 0.2 0.1 0.5 0.5 0.5 0.71 0.3 0.2 2.8276029639 0.3 2.7958618501 0.5 2.7650651906 0.4 2.8009609561 0.6 2.7814774574 0.4 2.7956708156 0.7 2.7730460521 0.8 2.8560146066 1.2 2.8845025807 0.7 2.8665016055 0.9 2.9016082752 1.0 2.8538765701 1.5 2.8740657611 1.00 2.7856861802 7.00 2.7507647601 0.5 2.8465876019 0.7 2.8607620862 0.5 2.8500569615 0.8 2.8701695900 Table 3. Computational values of Skin-friction coefficient ( 1 + 1 β ) ( 1 + λ 2 g′′ (0) ) g′′ (0). 487 B. Samudrala, K. Jayalakshmi, M. Gundagani Acta Polytechnica Du Sr Le δ1 δ2 Lb Pe Ω ( 1 + 1 β ) ( 1 + λ 2 f ′′ (0) ) f ′′ (0) 0.5 0.5 0.5 0.1 0.2 0.3 0.2 0.5 3.4657648158 0.8 3.4860591649 1.2 3.5014675245 1.0 3.4967919099 1.2 3.5168576821 0.8 3.4389787653 1.2 3.4067601645 0.5 3.4967976903 0.7 3.5132548344 0.4 3.4906636091 0.8 3.5298782815 0.6 3.4296857681 0.9 3.4016874432 0.6 3.4160165091 0.8 3.3901650926 0.8 3.4086617871 1.0 3.3815626506 Table 4. Computational values of Skin-friction coefficient ( 1 + 1 β ) ( 1 + λ 2 f ′′ (0) ) f ′′ (0). Du Sr Le δ1 δ2 Lb Pe Ω ( 1 + 1 β ) ( 1 + λ 2 g′′ (0) ) g′′ (0) 0.5 0.5 0.5 0.1 0.2 0.3 0.2 0.5 2.8276029639 0.8 2.8506501650 1.2 2.8746545944 1.0 2.8466791091 1.2 2.8604665026 0.8 2.7930762507 1.2 2.7714872563 0.5 2.8560691691 0.7 2.8719676362 0.4 2.8406065019 0.8 2.8673065072 0.6 2.7856064675 0.9 2.7606756101 0.6 2.7906967013 0.8 2.7706767517 0.8 2.7807657251 1.0 2.7610607682 Table 5. Computational values of Skin-friction coefficient ( 1 + 1 β ) ( 1 + λ 2 g′′ (0) ) g′′ (0). Pr Nb Nt Du δ1 Nu 0.71 0.3 0.2 0.5 0.1 1.5606056108 1.00 1.5395786181 7.00 1.5038959821 0.5 1.5809881953 0.7 1.6084650711 0.5 1.5969019193 0.8 1.6154865298 0.8 1.6056704184 1.2 1.6248568223 0.5 1.5986797903 0.7 1.6145474856 Table 6. Computational values of heat transfer rate coefficient for variations of Pr, Nb, Nt, Du, and δ1. Le Nb Nt Sr δ2 Sh 0.5 0.3 0.2 0.5 0.2 2.2657657870 0.8 2.2485465567 1.2 2.2087635568 0.5 2.2307657812 0.7 2.2067806146 0.5 2.2967604176 0.8 2.3156589552 1.0 2.3067160464 1.2 2.3265876851 0.4 2.2856708725 0.8 2.3057787447 Table 7. Computational values of mass transfer rate coefficient for variations of Le, Nb, Nt, Sr, and δ2. 488 vol. 65 no. 4/2025 Effects of coupled diffusion and buoyancy forces on three-dimensional . . . Lb Pe Ω Nh 0.3 0.2 0.5 1.3856847618 0.6 1.3506576101 0.9 1.3396760936 0.6 1.3587636401 0.8 1.3306763109 0.8 1.3490619364 1.0 1.3076870139 Table 8. Computational values of motile density coefficient for variations of Lb, Pe, and Ω. 5.5. Validation and comparative analysis Table 1 validates the numerical methodology by comparing heat transfer rates (Nu) with previous study [50]. The maximum deviation of 2.9 % con- firms the solution accuracy. 6. Conclusion The objective of this study was to examine the physics of fluid flow, particularly the interconnected diffu- sion effects of Williamson-Casson fluid flowing over a stretched sheet in the presence of microorganisms, nanoparticles, buoyancy forces, temperature Biot num- ber, mass Biot number, and magnetic fields. To show how the engineering factors affect the primary veloc- ity, secondary velocity, temperature, concentration, and microorganism profiles, we used the appropriate mathematical approaches to modify the differential equations that describe the phenomena. Graphs were then used to show what happened after this modifi- cation. The following is a complete list of the most important findings: • To improve the primary and secondary velocity profiles, the Casson parameter must be increased. This is because fluids with minimal shear stress do not yield easily. On the other hand, raising the shear stresses will cause yielding, which will make the flow more stable and faster. • The thermal and mass Grashof numbers both help to improve the primary and secondary velocity pro- files. This shows how buoyancy forces are affected by changes in temperature and concentration gra- dients. • The research revealed that the magnetic parameter adversely affected both the primary and secondary velocity profiles. • The Prandtl number decreased the temperature profiles, while the Dufour number, thermophoresis, and Brownian motion effects increased them. • The temperature profiles increase as the thermal Biot number increases. • The Soret number, mass Biot number, and ther- mophoresis show better concentration profiles. Con- versely, the increase in the Lewis number and Brow- nian motion has led to a decrease in these concen- tration patterns. In conclusion, given the limited values of these pa- rameters, a comparison between the present research and the work of Vinita Makkar and Prerna Batra [50] is warranted. Limitations: it neglects viscous dissipation and Joule heating, which may affect thermal behaviour in high-velocity or high-magnetic-field flows. Chemical reactions and induced magnetic fields are ignored, re- stricting applicability. Numerical solutions rely on ap- proximations, and far-field boundary conditions may not fully capture asymptotic behaviour. Experimen- tal validation is lacking, limiting real-world reliabil- ity. Microorganism dynamics are simplified, assuming uniform behaviour without considering spatial het- erogeneity or adaptive responses. These constraints suggest the need for more comprehensive modelling and empirical verification. Future research directions • Enhance models with viscous dissipation, Joule heating, and chemical reactions. • Use advanced numerical methods such as machine learning for complex nonlinearities. • Validate findings experimentally, especially for bio- convection. • Study microorganism-nanoparticle interactions and industrial applications. • Explore flows over complex geometries (e.g. wavy walls) for real-world relevance. • Optimise parameters for cooling and biomedical applications. List of symbols u, v, w Velocity components in x, y and z axes respectively [m s−1] x, y, z Cartesian coordinates measured along the stretch- ing sheet [m] f Dimensionless stream function along x-direction [kg (m s)−1] f ′ Fluid velocity along x-direction [m s−1] qw Heat flux coefficient g Dimensionless stream function along y-direction [kg m−1 s−1] g′ Fluid velocity along y-direction [m s−1] P r Prandtl number T Fluid temperature [K] Tw Temperature at the surface [K] Bo Uniform magnetic field [Tesla] M Magnetic field parameter T∞ Temperature of the fluid far away from the stretching sheet [K] Cfx Skin-friction coefficient along x-direction [s−1] Vo Reference velocity [m s−1] 489 B. Samudrala, K. Jayalakshmi, M. Gundagani Acta Polytechnica Cfy Skin-friction coefficient along y-direction [s−1] Uw(x) Stretching velocity of the fluid along x-direction [m s−1] Vw(y) Stretching velocity of the fluid along y-direction [m s−1] qm Mass flux coefficient Le Lewis number Nt Thermophoresis parameter Nb Brownian motion parameter Sr Soret number Du Dufour number Nu Rate of heat transfer coefficient (or) Nusselt number Sh Rate of mass transfer coefficient (or) Sherwood num- ber Cp Specific heat capacity of nano particles [J kg−1 K−1] a, b Constants Rex Reynolds number along x-direction Rey Reynolds number along y-direction DB Brownian diffusion coefficient [m2 s−1] DT Thermophoresis diffusion coefficient C Dimensional Fluid concentration [mol m−3] Cw Dimensional concentration at the stretching surface [mol m−3] Wc Cell swimming speed C∞ Dimensional ambient volume fraction [mol m−3] g Acceleration due to gravity [m s−2] Uo Reference velocity [m s−1] L Length of Reference Dn Solutal diffusivity of the medium b∗ Chemotaxis constant Nh Motile density N Dimensionless microorganism profiles Le Bioconvection Lewis number Pe Peclet number Tm Mean fluid temperature py Yield stress of the fluid Cs Concentration susceptibility KT Thermal diffusion ratio Dm Mass diffusion Gr Grashof number for heat transfer Gc Grashof number for mass transfer hf Coefficient of convective heat transfer hs Coefficient of convective mass transfer π Deformation rate πc Critical value of non-Newtonian model α Thermal diffusivity [m2 s−1] βT Thermal coefficient expansion βC Concentration coefficient Greek symbols Ω Microorganism difference parameter χ Dimensional Microorganism profiles χw Microorganism at the surface χ∞ Ambient microorganism λ Williamson fluid parameter η Dimensionless similarity variable [m] θ Dimensionless temperature [K] ν Kinematic viscosity [m2 s−1] σ Electrical conductivity ρ Fluid density [kg m−3] µ Dynamic viscosity of the fluid κf Thermal conductivity of the fluid α∗ Shear rate β Casson fluid parameter µB Dynamic viscosity of the Casson fluid τ Cauchy stress tensor τwx Wall shear stress along x-direction τwy Wall shear stress along y-direction ϕ Dimensionless nano-fluid concentration [mol m−3] τ1 Parameter defined as (ρC)p (ρC)f δ Stretching sheet parameter Superscript ′ Differentiation w.r.t η Subscripts f Fluid w Condition on the sheet ∞ Ambient conditions References [1] S. 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Materials Today: Proceedings 52:810–817, 2022. https://doi.org/10.1016/j.matpr.2021.10.169 492 https://doi.org/10.3390/sym12050692 https://doi.org/10.3390/math10030504 https://doi.org/10.25728/assa.2024.2024.03.1539 https://doi.org/10.5890/DNC.2025.06.010 https://doi.org/10.14311/AP.2024.64.0455 https://doi.org/10.1134/S0040577925060170 https://doi.org/10.1134/S0040577925060030 https://doi.org/10.31534/engmod.2025.1.ri.04d https://doi.org/10.1595/205651325X17375408788409 https://doi.org/10.1595/205651325X17375408788409 https://doi.org/10.23967/j.rimni.2025.10.63195 https://doi.org/10.25728/assa.2024.2024.4.1538 https://doi.org/10.18311/jmmf/2025/47828 https://doi.org/10.1016/j.ijft.2024.100986 https://doi.org/10.1142/S021798492450458X https://doi.org/10.1142/S0217984925501635 https://doi.org/10.32604/fhmt.2024.052749 https://doi.org/10.32604/fhmt.2024.054164 https://doi.org/10.1201/9781003299608 https://doi.org/10.1515/9783111405094 https://doi.org/10.1002/HTJ23149 https://doi.org/10.1016/j.matpr.2021.10.169 Acta Polytechnica 65(4):478–492, 2025 1 Introduction 2 Flow governing equations 3 Methods 4 Program code validation 5 Results and discussion 5.1 Flow characteristics and velocity profiles 5.2 Thermal transport mechanisms 5.3 Mass transfer and microorganism distributions 5.4 Engineering quantities of interest 5.5 Validation and comparative analysis 6 Conclusion List of symbols References