Introduction

In this research paper, focus is on the flow and diffusion of three quantities (heat, species concentration, and nanoparticles) in flow, whereas the flow is maintained over a porous and stretching (shrinking) surface of variable thickness. The surface velocities are non-uniform and nonlinear; however, they can be converted into linear and uniform velocities for fixed values of certain parameters in these two quantities. The most relevant and published research work, which is strictly related to such types of flow, heat, and mass transfer problems were addressed here. Investigating these kinds of flow issues has produced notable results, and the researcher discovered fresh insights into these flow models. In this connection, the flow on the stretching surface was examined by Sakiadis and Chandra1,2 and his work was clarified through experiments in3. Whereas Crane4 has introduced variable stretching in the Sakiadis problem, and he found an analytical solution to this fluid flow problem. Moreover, Crane’s problem has been studied and extended by many other authors (see Refs5,6,7,8). Besides that, the stretching flow problems have been studied for multiple dynamical and kinematical conditions subjected to thermal and species mass effects (see Refs5,6,7,8). Banks9 has proposed a modified form of similarity solutions of the stretching flow problems. He also worked on the non-similar solution of such models. Eigen solutions of the stretching flow problems are studied, and its modified version is presented by Banks and Zaturska10. Moreover, Grubka and Bobba11 have used Kummar’s function for the solution of such a problem. Furthermore, more interesting and realistic features of these flows can be found in12,13,14,15. Truesdell and Rajapogal16 obtained unique results of the well-known Navier-Stokes equations in such dynamical situations. Chang17 and Lawrence18 found non-unique or different solutions to similar problems. On the other hand, rheological fluids are also studied on the surface of stretching and shrinking sheets in the presence of different dynamical and thermodynamic conditions (see Refs19,20). More precisely, nano-and micropolar fluids are studied in independent and joint models for the transport of thermal energy and nanoparticles. Extra comprehensive and modern investigations pertinent to the present study are enclosed in Refs21,22.

The diffusion of thermal energy and species mass in fluid flow has been extensively investigated, especially under various kinematical conditions such as stretching/shrinking and suction/injection of the surface. These studies are of significant interest due to their wide range of practical applications. For instance, such flow models are vital in the fabrication of polymer sheets, cooling of electronic devices, and extrusion processes. Furthermore, they are instrumental in the movement of heated materials in furnaces, the design of heat exchangers, and the production of metallic and non-metallic cables. Besides, the range of these sorts of uses comprises chemical processes that require precise thermo-solutal management as well as medicinal gadgets that provide regulated pharmaceutical administration. Most theoretical as well as experimental scientists have actively concentrated on reproducing and studying these events because of their commercial significance. In this context, Naramgari and colleagues23 inspected magneto-driven flow of a nano-liquid across a flexible surface that is leaking and stretching/shrinking while taking into account the physical effects of injection or suction. Furthermore, taking into account simultaneous thermal and solutal transfer, Anusha et al.24 carefully examined the impact of the velocity slip effect on the thermo-fluidic reactivity of Walters’ liquid B streaming over a permeable stretchy or contractible surface. Moreover, Jaafar et al.25 observed the combined impacts of suction and heat radiation when analyzing the MHD flow and thermal transport characteristics of a hybrid particle suspension flow across a nonlinear stretching/shrinking surface. Furthermore, ternary hybrid nanoparticle-driven flow over a stagnant section of a stretching/shrinking curved interface was mathematically scrutinized by Mahmood et al.26, taking into account the Lorentz force and the physical influence of suction. In another pertinent work, Alharbi27 integrated the contribution of suction phenomena and investigated the heat transfer characteristics of a ternary hybrid nanofluid using slip flow circumstances over a stretching/shrinking interface. Considering the impact of mass suction, Nazari et al.28 reexamined the effects of Rosseland radiative flux in mixed convective hybrid nanoliquid flow through an angled stretching/shrinking surface. Furthermore, Sreenivasulu et al.29 reported a computational analysis that used silver-gold hybrid particles to optimize targeted medication administration and thermodynamic performance in shrinking arteries.

Nanofluids are recognized as more efficient heat transfer media that offer significant enhancements in convective transport capability and heat conductivity. They are created by distributing either metallic or nonmetallic nanomaterials into standard base liquids (carrier fluids). They are quite successful in a variety applications due to their improved heat conductivity, reliability, and adjustable qualities. In order to increase the efficiency of energy use, nanofluids are used in nuclear power plants, heating exchangers, photovoltaic collectors, and cooling mechanisms for microelectronics. Because of their special ability to interact with physiological surroundings, they are used in targeted medication delivery, hyperthermia therapy, laboratory, and bio-sensing in the healthcare area.

The idea of suspending tiny particles in fluids to improve thermal efficiency was first presented by Choi and Estman30, who established the groundbreaking paradigm of nanofluid research. Following on this, Buongiorno31 put forth a thorough model that included thermophoretic and Brownian diffusion as the two primary processes controlling the movement of nanoparticles. Later, Lone et al.32 used the physical effects of thermophoretic diffusion and Brownian dispersion to perform a theoritical analysis of magneto-driven bio-convective flow of a hybrid Casson non-Newtonian nano-liquid across a penetrable exponentially stretchable interface. The physical consequences of thermophoretic and Brownian diffusion, along with chemical reactions effects, on the convectional movement of a Williamson nanofluid across a permeable stretched interface was assessed by Nayak et al.33. The findings also revealed the roles that reactive diffusion and nanoparticle mobility play in influencing the thermal, solutal, and momentum distributions in the boundary layer region. Mahboobtosi et al.34 explored AI-driven modeling of bio-convective nanofluid flow in three-dimensional (3D) systems, predicting a robust predictor for tough fluid interactions. The influence of hybrid nanofluids on entropy and the temporal thermodynamic dynamics in a finned enclosure of a cylindrical obstruction was assessed by Chunyang et al.35 Chari et al.36 used analytical and computational tools to evaluate heat transport of GO/water nano-liquid flow driven by ohmic heating, chemical reaction, and magnetohydrodynamics (MHD), Hajizadeh et al.37 propose a new approach to investigate the effects of hybrid nanofluids on a non-Newtonian Maxwell model, with emphasis on improved heat regulation for biological engineering applications. Using mathematical simulations in Python, Shateri et al.38 tested the bio-convective efficiency of magnetized Casson/Maxwell Thermally augmented fluid mutual with Self-driven microbial particles. The amalgamated impacts of magnetic fields, changes in nanoparticles, and the relative distribution of microorganisms on flow resilience, heat transfer, and solutal transport channels were all integratedly evaluated in this study. The findings showed that the timing of the start of bio-convection and the ultimate transport efficiency are directly influenced by the interplay between magnetic forces and gyrotactic motility. These findings provide valuable insights into improved bio-nanofluid systems in the industry and the medical industry. Recently, Abbas et al. published new perspectives on the mechanism of thermal transport in a cubic thermo-stratified nanofluid under magneto-driven influences39 who explicitly included Newtonian heating and squeezing flow scenarios. The importance of melting heat transport in the stagnation point flow of Maxwell nanoliquid over a quadratically stratified Riga surface was shown by Khan et al.40 in a related work. Additionally, focusing on energy transport features, Khan et al.41 examined the convective dynamics in a nonlinearly stratified squeezed nanofluid flow, taking into account the impacts of Newtonian heating and solar thermal radiation.

Novelty

While recent advances in nanofluid conditions are significant, most studies still focus on linear kinematics, idealized geometries, and relatively simple boundary conditions that do not capture the coupled nonlinearities and surface complexity that characterize realistic transportation systems. The present research provides an innovative nonlinear computational model that combines the simultaneous effects of mass suction and injection, Brownian dispersion, thermophoretic forces, and unstable nonlinear kinematics for studying nanofludic transport across a rough, porous, and continuously drawn surface. This study employs a dependable MATLAB bvp4csolver to guarantee computing precision as well as stability while taking into account the combined influence of complex thermo-solutal transport qualities, in contrast to earlier research that focuses on both processes independently. The present investigation’s originality lies in.

  • The integration of surface roughness with nonlinear temporal dynamics.

  • The assessment of solution multiplicity and parameteric sensitivity across extended acceptable physical domain.

  • The determination of critical thresholds that control flow and thermodynamic transition.

  • Through the provision of a novel physical representation of heat-transporting nano-mixture flow in practical design contexts, the utilization of simulation and creation of models facilitates more efficient temperature monitoring, covering creation, and pharmaceutical delivery administration.

Mathematical formulation of transport equations

Consider a two-dimensional (2D) incompressible nanofluid flowing steadily across a semi-infinite stretched sheet with a permeable rough surface. The sheet lies along the horizontal axis and exhibits surface roughness that influences the boundary layer behavior. It stretches nonlinearly with a velocity that fluctuates as a power of the distance from the origin, introducing nonlinear kinematic effects. The temperature and concentration at the sheet are also distributed nonlinearly, simulating thermal and solutal gradients (Fig. 1).

Fig. 1
Fig. 1
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Flow geometry.

The flow is further influenced by nanofluid characteristics such as Brownian diffusion and thermophoretic forces, which affect heat and mass transfer within the boundary layer. The formulation reveals how nonlinear and surface effects modulate the mass, momentum, thermal, and solutal transport within the nanofluid domain42,43

$$\:\frac{\partial\:u}{\partial\:x}+\frac{\partial\:v}{\partial\:y}=0$$
(1)
$$\:\frac{\partial\:u}{\partial\:t}+u\frac{\partial\:u}{\partial\:x}+v\frac{\partial\:u}{\partial\:y}=\nu\:\frac{{\partial\:}^{2}u}{\partial\:{y}^{2}}+{g}^{\text{*}}\left[{\beta\:}_{t}\left(T-{T}_{{\infty\:}}\right)+{\beta\:}_{c}\left(C-{C}_{{\infty\:}}\right)\right]$$
(2)
$$\:\frac{\partial\:T}{\partial\:t}+u\frac{\partial\:T}{\partial\:x}+v\frac{\partial\:T}{\partial\:y}=\alpha\:\frac{{\partial\:}^{2}T}{\partial\:{y}^{2}}+\tau\:\left[{D}_{B}\left(\frac{\partial\:T}{\partial\:y}\right)\left(\frac{\partial\:C}{\partial\:y}\right)+\left(\frac{{D}_{T}}{{T}_{{\infty\:}}}\right){\left(\frac{\partial\:T}{\partial\:y}\right)}^{2}\right]+\frac{\mu\:{c}_{p}}{\rho\:{c}_{p}}{\left(\frac{\partial\:u}{\partial\:y}\right)}^{2}+\frac{16{\sigma\:}^{\text{*}}{T}_{{\infty\:}}^{3}}{3{K}^{\text{*}}\rho\:{c}_{p}}{\left(\frac{\partial\:T}{\partial\:y}\right)}^{2}$$
(3)
$$\:\frac{\partial\:C}{\partial\:T}+u\frac{\partial\:C}{\partial\:x}+v\frac{\partial\:C}{\partial\:y}={D}_{B}\frac{{\partial\:}^{2}C}{\partial\:{y}^{2}}+\left(\frac{{D}_{T}}{{T}_{{\infty\:}}}\right){\left(\frac{\partial\:T}{\partial\:y}\right)}^{2}-{k}_{r}^{2}\left(C-{C}_{{\infty\:}}\right){\left(\frac{T}{{T}_{{\infty\:}}}\right)}^{n}{e}^{\frac{-Ea}{kT}}$$
(4)
$$\:u=U(x,t),v=V(x,t),T={T}_{w},C={C}_{w}\:at\:y=f(x,t)$$
(5)
$$\:u=0,T={T}_{{\infty\:}},C={C}_{{\infty\:}}\:as\:y\to\:{\infty\:}$$
(6)

where \(\:U\left(x,t\right)={U}_{0}{r(x,t)}^{m}\) and \(\:V\left(x,t\right)=\frac{{V}_{0}}{r(x,t)}\) and both \(\:V(x,t\)) and \(\:U(x,t)\) maybe positive and negative. \(\:f\left(x,t\right)=\delta\:r(x,t)\) and \(\:q\left(x,t\right)=\frac{{q}_{0}}{r(x,\text{t})}\) where \(\:r\left(x,t\right)=\gamma\:{({c}_{0}+{k}_{0}x+\omega\:t)}^{\frac{1}{\beta\:\text{*}}}\:\) with \(\:{\beta\:}^{*}=2+m\) and \(\:\gamma\:={{(\beta\:}^{*})}^{\frac{1}{\beta\:*}}\) \(\:{C}_{w}\), \(\:{T}_{{\infty\:}}\) and \(\:{C}_{{\infty\:}}\) constants. A set of new unseen similarity variables is considered42,43:

$$\:u=U\left(x,\:t\right)g\left(\eta\:,t\right),v=q\left(x,t\right)h\left(\eta\:,t\right)+V\left(x,t\right),\theta\:\left(\eta\:,t\right)=\:\frac{T-{T}_{{\infty\:}}\:}{{T}_{w}-{T}_{{\infty\:}}},\varphi\:\left(\eta\:,t\right)=\frac{C-{C}_{{\infty\:}}\:}{{C}_{w}-{C}_{{\infty\:}}},\:{t}^{\text{*}}=\omega\:t$$
(7)

here \(\:\eta\:=\frac{y-\:f(x,\:t)}{r(x,\:t)}\). The Eqs. (1–4) took the following form:

$$\:{H}^{{\prime\:}}+{\delta\:}_{3}\left(mg-\delta\:{g}^{{\prime\:}}-\eta\:g{\prime\:}\right)-\omega\:g{\prime\:}=0$$
(8)
$$\:{g}^{{\prime\:}{\prime\:}}{+\delta\:}_{2}{\delta\:}_{3}\left(\delta\:+\eta\:\right)g{g}^{{\prime\:}}-{(\delta\:}_{1}+{\delta\:}_{2}H){g}^{{\prime\:}}-m{\delta\:}_{2}{\delta\:}_{3}{g}^{2}-\omega\:g{\prime\:}+{\lambda\:}_{t}\text{}\theta\:+{\lambda\:}_{c}\text{}\varphi\:=0$$
(9)
$$\:\frac{1}{Pr}{\left(1+Rd\right)\theta\:}^{{\prime\:}{\prime\:}}+Nb{\theta\:}^{{\prime\:}}{\varphi\:}^{{\prime\:}}+{{Nt\theta\:}^{{\prime\:}}}^{2}\:{+\delta\:}_{2}{\delta\:}_{3}\left(\delta\:+\eta\:\right)g{\theta\:}^{{\prime\:}}-{(\delta\:}_{1}+{\delta\:}_{2}H){\theta\:}^{{\prime\:}}-\omega\:{\theta\:}^{{\prime\:}}+Ec(g{\prime\:}{\prime\:})2=0$$
(10)
$$\:\frac{1}{Sc}{\varphi\:}^{{\prime\:}{\prime\:}}+\frac{Nb}{Nt}{\theta\:}^{{\prime\:}{\prime\:}}+{{Nt\theta\:}^{{\prime\:}}}^{2}+Le{\delta\:}_{3}\left(\delta\:+\eta\:\right)g{\varphi\:}^{{\prime\:}}-\left(L{e}_{1}+LeH\right){\varphi\:}^{{\prime\:}}-\omega\:{\varphi\:}^{{\prime\:}}-\sigma\:(1+\delta\:\theta\:{)}^{n}\varphi\:{e}^{-\frac{E}{1+\delta\:\theta\:}}=0$$
(11)

The boundary conditions are simplified as:

$$\:g\left(0,t\right)=\theta\:\left(0,t\right)=\varphi\:\left(0,t\right)=1,H\left(0,t\right)=0,\:g\left({\infty\:},t\right)=\theta\:\left({\infty\:},t\right)=\varphi\:\left({\infty\:},t\right)=0$$
(12)

Note the dimensionless quantities are:

\(\:{\delta\:}_{1}=\frac{{V}_{0}}{\nu\:}\), \(\:{\lambda\:}_{t}=\frac{g{\beta\:}_{t}\left({T}_{w}-{T}_{\infty\:}\right)\nu\:}{{u}_{w}^{3}},{\lambda\:}_{c}=\frac{g{\beta\:}_{c}\left({C}_{w}-{C}_{\infty\:}\right)v}{{u}_{w}^{3}},Ec=\frac{{u}_{w}^{2}}{{c}_{p}\left({T}_{w}-{T}_{\infty\:}\right)}\) \(\:{\delta\:}_{2}=\frac{{q}_{0}}{\nu\:}\), \(\:Rd=\frac{16{\sigma\:}^{*}{T}_{\infty\:}^{3}}{3{K}^{*}\rho\:{c}_{p}}\) \(\:{\delta\:}_{3}=\frac{{k}_{0}{U}_{0}}{{q}_{0}}\), \(\:Nb=\frac{{\tau\:D}_{B}{\Delta\:}C}{\nu\:}\), \(\:Nt=\frac{{\tau\:D}_{T}{\Delta\:}T}{{T}_{\infty\:}\nu\:}\), \(\:\sigma\:=\frac{{k}_{r}^{2}}{{D}_{B}},E=\frac{{E}_{a}}{{K}_{T}{T}_{\infty\:}},\delta\:=\frac{{T}_{w}-{T}_{\infty\:}}{{T}_{\infty\:}}\) \(\:Pr=\frac{\nu\:}{\alpha\:}\), \(\:Le=\frac{{q}_{0}}{{D}_{B}}\), \(\:L{e}_{1}=\frac{{V}_{0}}{{D}_{B}}\), \(\:\omega\:\) signifies the time scaling parameter, \(\:{\Delta\:}T={T}_{w}-{T}_{\infty\:}\), and \(\:{\Delta\:}C={C}_{w}-{C}_{\infty\:}\), moreover, \(\:{C}_{f}\), \(\:Nu\) and \(\:Sh\) are:

$$\:{C}_{f}=\frac{{\tau\:}_{w}}{{\rho\:U}^{2}},\:Nu=\frac{{q}_{\omega\:}}{k{\Delta\:}T},\:Sh=\frac{{xq}_{m}}{{D}_{B}{\Delta\:}C},$$
(13)

Put the variables in Eq. (13) into the above equation, the following results are obtained:

$$\:\frac{{\left(x\phi\:\right(x,\:t\left)\right)}^{1-m}{\tau\:}_{w}}{\mu\:{U}_{0}}={g}^{{\prime\:}}\left(0\right),\:\phi\:(x,\:t)Nu=-{\theta\:}^{{\prime\:}}\left(0\right),\:\phi\:(x,\:t)Sh={\varphi\:}^{{\prime\:}}\left(0\right),$$
(14)

The function includes time dependence \(\:\phi\:=\frac{1}{x}{\left(\right(2\:+\:m\left)\right(\:{c}_{0}+{k}_{0}+t\left)\right)}^{1/(2+m)}\).

Numerical scheme (BVP4c method)

  • The coupled and nonlinear ODE system (Eqs. (8–11)) with boundary conditions (Eq. (12)) is solved using MATLAB’s bvp4c solver (Fig. 2).

  • The second-order differential equations are converted into a system of first-order ODEs by introducing substitution variables for each derivative.

  • A finite computational domain is selected by truncating the independent variable \(\:\eta\:\) at a sufficiently large value (typically \(\:\eta\:=10\)) to simulate the boundary at infinity.

  • An initial guess for the solution functions is provided over a uniformly spaced mesh grid within the domain.

  • The boundary conditions at \(\:\eta\:=0\) and \(\:\eta\:=\infty\:\) are implemented in a separate boundary function.

  • The solver uses collocation and finite difference methods with error control to obtain an accurate and stable numerical solution.

  • The output includes profiles for:

    • Velocity \(\:g\left(\eta\:\right)\).

    • Temperature \(\:\theta\:\left(\eta\:\right)\).

    • Concentration \(\:\varphi\:\left(\eta\:\right)\).

  • Results are validated by comparison with existing literature and extended to new parameter ranges to explore additional physical behaviors.

The strongly coupled and nonlinear ordinary differential Eqs. (8)–(11), subject to the boundary constraints (12), are numerically integrated using MATLAB’s bvp4c solver. This solver is particularly effective for two-point boundary value problems and handles systems with boundary conditions specified at different points (in this case, at \(\:\eta\:=0\) and \(\:\eta\:\to\:\infty\:\)). The steps involved:

Reduction to first-order system

Since bvp4c handles first-order ODE systems, we reduce the second-order equations to a set of first-order equations by introducing auxiliary variables. For instance:

Domain truncation

Since \(\:\eta\:\to\:\infty\:\) is not computationally feasible, a large finite value (typically \(\:{\eta\:}_{\infty\:}\approx\:8-10\)) is used, ensuring the field variables decay asymptotically.

Initial guess

A suitable initial guess for each variable is provided, usually based on expected physical profiles (e.g., exponential decay or linear behavior). Poor initial guesses may lead to divergence or selection of an unphysical solution branch.

Fig. 2
Fig. 2
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Flow chart of the bvp4c scheme.

Solver setup and execution

The first-order system and boundary conditions are coded in MATLAB functions:

  • One function defines the ODE system.

  • Another defines the boundary conditions.

These are passed to bvp4c, which returns the numerical solution across the \(\:\eta\:\) domain.

Post-processing

The resulting profiles of \(\:g\left(\eta\:\right)\), \(\:\theta\:\left(\eta\:\right)\), \(\:\varphi\:\left(\eta\:\right)\), and \(\:H(\eta\:\)) are used to:

  • Plot velocity, temperature, and concentration profiles.

  • Compute physical quantities like skin friction, Nusselt number, and Sherwood number.

  • Analyze sensitivity of results to varying parameter values (e.g., \(\:\delta\:\), \(\:{\delta\:}_{1}\), \(\:{\delta\:}_{2}\), \(\:m\), \(\:Nb\), \(\:Nt\), \(\:Sc\), and \(\:Pr\)).

Why bvp4c?

  • Robust for stiff and nonlinear problems.

  • Adaptive mesh refinement improves accuracy where the solution changes rapidly.

  • Well-suited for multiple solution branches and parameter continuation techniques.

Results and discussion

The system of Eqs. (8–11), subject to the boundary conditions given in Eq. (12), was solved numerically using the bvp4c method. Initially, all the graphs and results from the study by Ali et al.43 were reproduced. However, this work extends their results by exploring new cases and solution behaviors not previously reported. These new results emerged from a deeper physical interpretation of the parameters and an adjustment of their ranges.

Numerical solutions were obtained for all field variables, including the skin friction coefficient and the rates of heat and mass transfer. Plots of \(\:g\), \(\:\theta\:\), and \(\:\varphi\:\) were generated for various parameter values. In some cases, \(\:\theta\:\) and \(\:\varphi\:\) were analyzed with respect to one varying parameter while keeping others fixed. Tables 1 and 2 presents the rate of heat transfer at the surface for different values of Prandtl number \(\:\varvec{P}\varvec{r}\) and stretching parameter \(\:\varvec{m}\), with other parameters held constant, respectively. Further, \(\:-\varphi\:^{\prime\:}\left(0\right)\) is evaluated is given in Table 3.

Table 1 \(\:-\theta\:^{\prime\:}\left(0\right)\) For \(\:\delta\:={\delta\:}_{1}=Nb=Nt=0\), \(\:{\delta\:}_{2}=2\) and various \(\:m\) and \(\:{\delta\:}_{3}\).
Full size table

The numerical values were compared with the results of Alam et al.48 and other authors. Notably, the new results did not match the previously published data, likely due to the existence of dual solutions. This explains the observed differences in temperature profiles compared to earlier studies.

Table 2 \(\:-\theta\:^{\prime\:}\left(0\right)\) Is evaluated for \(\:m={\updelta\:}={\delta\:}_{1}=0,{\delta\:}_{2}={\delta\:}_{3}=1\:and\:for\:Nb=0=Nt\).
Full size table
Table 3 \(\:-\varphi\:^{\prime\:}\left(0\right)\) Is evaluated for \(\:\delta\:=Le1={\delta\:}_{1}=0,m={\delta\:}_{2}=1,{\delta\:}_{3}=0.5\:and\:Pr=10\).
Full size table

The nanofluid velocity field increases noticeably as the magnetic constraint (M) increases, as seen in Fig. 3, indicating that the fluid motion is accelerated by the applied magnetic field. The main cause of this improvement is the Lorentz force, which works in the direction of flow and is produced inside the electrically conducting nanofluid. Upon aligning the magnetic field with the direction of flow, the Lorentz force efficiently eliminates resistive drag, accelerates the fluid layers, and evens up the trajectory of vector speed. In the same way, Fig. 4 illustrates how the nanofluid velocity rises as the surface curvature constraint grows. This pattern implies that greater curvature improves stretching and speeds up the mobility of liquid particles along the edge. By reducing resistivity in the boundary layer and adjusting the flow domain, the curved design allows the fluid to flow more swiftly and effectively across the boundary layer, generating higher degrees of physical variation. As seen in Fig. 5, the velocity distribution rises with an increase in the mass flux parameter, indicating that enhanced surface mass transfer fortifies fluid motion. According to physical point of view, this phenomenon implies that greater suction or injection at the boundary has a noteworthy impact on the momentum border layer. In the event of fluid injection, the circulation field is accelerated, which increases the velocity along the wall, decreases toward-wall blockage, and increases the fluid molecules. Additionally, In Fig. 6, the velocity contour curve varies with the values of the other vector flow parameters. It is evident from the plot that the velocity profile rises from 0.0 to 1.0; the velocity profile rises as the longitudinal flow feature rises from 0.0. This pattern implies that the higher main flow velocity is produced by stronger transverse (cross-flow) properties. An additional volume is being pushed or injected into the boundary layer from an orientation transverse to the main flow as the transverse flow factor rises. The velocity field rises as a result of the increased streamwise velocity caused by this additional momentum. The upward tilt in the flow curves indicates that larger transverse flow values of parameters allow the fluid to sustain higher rates over farther distances by reducing velocities degrading at the level of the surface. As the Eckert number (Ec) rises, the thermal distribution exhibits an increasing trend, as seen in Fig. 7, signifying that viscous heating (resistive heating) is the responsible source for the improved thermal power output. From a physical viewpoint, this phenomenon specifies that viscous dissipation within the fluid intensifies as the Eckert number increases, indicating a larger ratio of kinetic energy to enthalpy. The enhanced method for transforming mechanical force into internal heat raises the fluid thermal energy, which extends the thermal barrier layer and boosts heat generation in the flow region. Figure 8, illustrates how naoliquid temperature increases as radiation variable rises, highlights the role of thermal radiation in augmenting heat movement within the fluid. In terms of physics, this indicates that as Rosseland thermal radiation becomes more momentous, supplementary energy is sent to the stream in the form of radiant heat. The resultant enhancement of the total heating energy of the boundary layer leads to a thicker thermal boundary layer and a higher nanofluid thermal energy profile. It is clear from Fig. 9 that when the Prandtl constraint upsurges, the thermal energy profile declines. As thermophoresis constraint values upsurge, the thermal distribution of nanofluids rises, as shown in Fig. 10. In physical terms, this phenomenon can be described by the shift of nanoparticles from warmer to cooler places when subjected to a gradient of temperatures. As the thermophoresis effect increases, larger particles migrate away from the heated region, hampering the progress of local heat exchange and causing thermal energy to build up in the fluid. Consequently, the thermal energy profile typically rises and the thermal boundary layer deepens. It is clear from Fig. 11 that the thermal profile grows as the Schmidt number does. Physically speaking, momentum diffusion is more common than mass diffusion when the Schmidt number is large, which reduces the fluid’s solutal diffusivity. This reduced mass diffusion slows the migration of species further from the surface, perhaps lessening the effects of convective cooling. Because more thermal energy is retained within the boundary layer, the thermal power of the nanofluid increases. As the Lewis number climbs, the nanofluid thermal energy profile clearly rises (Fig. 12). The Lewis number provides a physical description of the proportionate effectiveness of thermal energy transmission in a liquid situation as opposed to mass distribution. Heat diffusion outperforms mass diffusion as the Lewis number (Le) rises, meaning that heat spreads more effectively than species accumulation. This enhanced thermal diffusion leads to a higher temperature within the boundary layer, resulting in an elevated temperature profile. Figure 13 noticeably displays that the thermal profile upsurges with growing values of the Brownian diffusion constraint. Physically, this indicates that intensified random motion of solid-nanoparticles improves the thermal energy exchange between the nanoparticles and the carrier liquids. As Brownian diffusion becomes stronger, it contributes to increased energy transfer at the microscopic level, thereby raising the overall temperature within the thermal boundary layer. Figure 14 presents the 3D velocity profile for varying values of the time scaling parameter time scaling constraint. It is evident that the velocity cuts with growing time scaling parameter. Physically, this suggests that as the unsteadiness in the flow upsurges, the inertial effects become more dominant relative to the viscous forces. The amplified unsteady effects generate resistance to fluid motion, thereby diminishing velocity and compressing the momentum boundary layer region. The 3D surface plot for thermal profile for different time scaling constraint values is displayed in Fig. 15. It has been noted that when the time scaling parameter grows, so does the temperature distribution. The flow’s unsteadiness is physically represented by the time scaling parameter; larger values indicate more time-dependent effects. More thermal energy can build up inside the momentum border layer (BL) region as a result of transient heating’s increasing influence with increasing time scaling constraints. As a result, the fluid thermal field increases and the temperature boundary layer (TBL) thickens overall. In Fig. 16 visibly shows that the specicies concentration field declines with growing values of the time scaling parameter. Physically, the time scaling parameter represents the unsteady characteristics of the flow. As time scaling parameter increases, the unsteadiness becomes more pronounced, which enhances temporal diffusion effects and causes the solute or nanoparticle concentration near the surface to dissipate more rapidly. This leads to a thinning of the concentration boundary layer (BL) and a noticeable decline in concentration throughout the domain.

Fig. 3
Fig. 3
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Variation in g’ with magnetic field strength.

Fig. 4
Fig. 4
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Variation in g’ with curvature parameter.

Fig. 5
Fig. 5
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Variation in g’ with mass flux parameter.

Fig. 6
Fig. 6
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Variation in g’ with transverse flow parameter.

Fig. 7
Fig. 7
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Variation in θ’ with Eckert number.

Fig. 8
Fig. 8
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Variation in θ’ with Radiation parameter.

Fig. 9
Fig. 9
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Variation in θ’ with Prandtl number.

Fig. 10
Fig. 10
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Variation in θ’ with thermophoresis parameter.

Fig. 11
Fig. 11
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Variation in \(\:\varphi\:\left(\eta\:\right)\)’ with Schmidth number.

Fig. 12
Fig. 12
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Variation in \(\:\varphi\:\left(\eta\:\right)\)’ with Lewis number.

Fig. 13
Fig. 13
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Variation in \(\:\varphi\:\left(\eta\:\right)\)’ with Brownian motion.

Fig. 14
Fig. 14
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3D velocity profile illustrating the effect of the time scaling parameter.

Fig. 15
Fig. 15
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3D temperature profile illustrating the effect of the time scaling parameter.

Fig. 16
Fig. 16
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3D concentration profile illustrating the effect of the time scaling parameter.

Final remarks and future scope

The current study used computational simulations to systematically examine the nonlinear flow mechanics effects on nanofluid flow on a rough, leaky surface. It focused on the ways in which the time-scaling parameter (ω) regulates unsteady flow behaviors. The research considered the combination of physical impacts of mass flux, curvature, transverse flow, viscous resistive heating, thermophoretic, Brownian dispersion, the Rosseland radiative thermal energy, magnetization (MHD), and temporally unsteadiness. The key conclusions of the study are summarized as follows:

The thermal energy profile declines with rising Prandtl and Lewis numbers, indicating a drop in thermal diffusivity, but it rises with increasing Eckert, radiation, thermophoresis, and Brownian dispersion characteristics. Due to limited mass diffusion, the nanofluid solutal concentration declines with increasing Schmidt and Lewis numbers but increases with Brownian diffusion. Through increased temporal diffusion, a rise in the time-scaling parameter (ω) further reduces the concentration.

Future directions

This research can be expanded in a number of encouraging ways. In-depth stability and bifurcation analysis may help identify physically attainable solution branches and assess flow stability. Investigating complex rheological models such as Casson, Maxwell, Williamson, Burgers hybrid/ternary/tetra/penta nanofluids may reveal the impact of non-Newtonian features on heat and mass transport. Three dimensions and rotating porous structures might enhance the existing model’s mathematical realism. Using artificial intelligence methods surrogate modeling can improve forecasts and sensitivity analysis over large parameter ranges. Finally, the connections between theoretical modeling and practical applications might be strengthened by using benchmark datasets or doing experimental validation. Combining these strategies could greatly improve the prediction performance and technical applicability of the proposed model.

Advantages, and limitations

By adding several slip conditions, bio-convection phenomena, and nonlinear roughness of the surfaces, the proposed model extends conventional nanofluid practices and agreements a more realistic depiction of transport mechanisms. Similarity scaling conversions are used to efficiently reduce the controlling PDEs to ODEs, and the bvp4c solver ensures reliable and accurate computational results while revealing a wide variety of branches and bifurcation aspects that were often missed in earlier evaluations. Notwithstanding these benefits, the computational cost of bvp4c and the presumption of stable, laminar, two-dimensional flow may restrict its applicability to more complex or dynamic regimes. Moreover, the assumption of constant thermal physical factors and the absence of empirical validation limit the model’s predictability. Further studies should incorporate data-driven optimal schemes, shifting material features, and unstable fluid characteristics to increase authenticity and practical utility.