Numerical analysis of non-similar convection flows of a two-phase nanofluid past a semi-infinite vertical plate with thermal radiation

Abbreviations

NPs

Nanoparticles

NF

Nanofluid

BNF

Buongiorno nanofluid

Da–Fs

Darcy–Forchheimer

R

Rosseland thermal radiation

BL

Boundary layer

BLT

Boundary layer thickness

BLEs

Boundary layer equations

BCs

Boundary conditions

BVP

Boundary value problem

KBM

Keller box method

Nomenclature

b

Inertial drag coefficient

C

Concentration

c _p

Specific heat parameter (J/kg K)

C _f

Skin friction coefficient

C _w

Concentration at the wall surface

C _∞

Ambient concentration

D

Nanoparticle diffusivity

Da

Darcy number

D _B

Brownian diffusion coefficient (m²/s)

Dm

Molecular diffusivity (m²/s)

D _T

Thermophoretic diffusion coefficient (m²/s)

Fs

Forchheimer number

f

Non-dimensional steam function

g

Acceleration due to gravity (m/s²)

Gr

Grashof number

K

Permeability

k _f

Thermal conductivity (W/mK)

α _f

Thermal diffusivity (m²/s)

k*

The mean absorption coefficient

L

Characteristic length (m)

Nb

Brownian motion parameter

Nt

Thermophoresis parameter

Nr

Buoyancy ratio parameter

Nu

Heat transfer rate (local Nusselt number)

p

Pressure (Pa)

Pr

Prandtl number

Subscripts

w

Surface conditions

∞

Free stream conditions

q _r

Radiative heat flux

R

Radiation-conduction parameter

Sh

Mass transfer rate Sherwood number

Sc

Local Schmidt number

T

Temperature of the fluid (Kelvin)

T _w

Wall temperature (K)

T _∞

Ambient temperature (K)

x

Stream-wise coordinate (m)

y

Transverse coordinate (m)

V

Velocity field vector (m/s)

u, v

Non-dimensional velocity components along the x- and y-directions respectively (m/s)

Greek symbols

α _f

Thermal diffusivity (m²/s)

β

Volumetric volume expansion coefficient of the fluid (K)

σ _f

The electric conductivity of the fluid (S/m)

ρ _p

Fluid density of the nanoparticle

μ _f

Dynamic viscosity (kg m⁻¹ s⁻¹)

ν _f

Kinematic viscosity of base fluid (m²/s)

τ

The ratio of heat capacity of the nanoparticle by the heat capacity of the fluid

σ∗

Stefan-Boltzmann constant

θ

Dimensionless temperature

ϕ

Dimensionless concentration

ξ

Dimensionless tangential coordinate

η

Dimensionless radial coordinate

ψ

Dimensionless stream function

ρ _f

Density (kg/m³)

ρ c p p

Heat capacity for the nanoparticles (J/kgK)

1 Introduction

Nanofluids (NFs) have gained recognition for superior thermophysical properties, enhancing heat transfer in applications compared to traditional fluids like water, ethylene glycol, and lubricants. Introduced by Choi [1], NFs improve cooling efficiency via dispersed nanoparticles (NPs). Researchers have explored various NP types (metal, oxides, carbides) in base fluids through simulations and experiments [2]. NFs are now used in engineering, medical services, food hygiene, aerodynamic cooling, atomic reactors, and solar collectors. They enhance fluids like propylene glycol and mineral oil with low heat conductivity. Nanotechnology enables tailoring NFs by adding specific NPs to meet advanced thermal requirements. Kuznetsov and Nield [3] analyzed nanofluid flow over smooth layers using the BNF model [4]. Ramesh Reddy et al. [5] examined Hall and ion-slip effects in NF transport. Razzaq et al. [6] explored Maxwell fluid flow in a magnetized sheet with NPs. Cui [7] investigated 3D bioconvective boundary layers with NPs and microorganisms. H. Pandharikar et al. [8] explored the numerical and experimental approaches to evaluate nanoparticle-enhanced waste cooking oil biodiesel in a CI engine using a single-zone thermodynamic model. It revealed improved combustion efficiency, reduced emissions, and sustainable fuel performance with nanoparticle additives. Pandey et al. [9] studied the application of the inverse heat conduction algorithm to accurately estimate transient interfacial heat transfer coefficients between AZ31 alloy and H13 die during the casting process. A few more relevant works can be found in [10], [11], [12], [13].

Buongiorno’s nanofluid, engineered with enhanced heat transfer properties, improves efficiency in heat exchangers, auto-engines, and electronic cooling systems. Blending base fluids with nanoparticles (NPs) it enhances convection heat transfer and thermal conductivity, offering compact and efficient solutions for engineering applications. The Buongiorno nanofluid model incorporates thermal analysis and Brownian motion of fluid particles. Rasheed et al. [14] modelled two-dimensional magneto NFs with convective fluxes using the BVPh2 algorithm, considering heat absorption and generation. Bhavani [15] examined stable, ultrafast nano-coolants suitable for non-radiative designs. Humane et al. [16] studied magneto-micropolar NFs in porous, stretchable devices. Wang et al. [17] used a modified Buongiorno model to analyze Ag–H₂O nano-thin film heat transfer in permeable media. Rafique et al. [18] examined the combined effects of Brownian motion, thermophoretic diffusion, and the Soret and Dufour phenomena on micropolar nanofluid flow over an inclined sheet using Keller-box simulations to enhance the modeling of thermal and diffusive transport. Shahbaz et al. [19] employed an advanced intelligent computational framework to examine three-dimensional thermally laminated magnetohydrodynamic flow of non-Newtonian nanofluids over a stretched surface and demonstrated improved accuracy in predicting thermal and fluid dynamic characteristics.

Non-similar transformations are mathematical tools used to analyze fluid flows where variables like velocity, temperature, or concentration do not maintain similarity across the flow field. These techniques introduce new variables to address non-uniform boundary conditions, geometries, or varying parameters, enabling the resolution of complex fluid dynamics problems. Non-similar flows arise when material properties depend on independent variables, preventing a unified dimensionless form. These transformations are vital for modelling real-world scenarios like varying freestream speeds, temperature differences, fluid injection, or surface mass transfer. Non-similar flows differ from similar ones along the streamwise direction. Techniques such as the LNS method, developed by Sparrow et al. [20], have been widely adopted to address non-similar boundary layer (BL) problems. Razzaq and Farooq [21] used non-similar forced convection to study Oldroyd-B fluid flow over an exponentially stretching substrate. Abdullah et al. [22] analyzed the effects of Brownian motion, thermophoresis, and mixed convection on nanofluid flow near a heated vertical surface. Anjum et al. [23] employed non-similar transformations and demonstrated that the inclusion of Ohmic heating and viscous dissipation notably improved thermal transfer performance and modified the flow behavior in the magneto-thermo-convective motion of nanofluids along a semi-infinite vertical surface.

Convection flows are fluid movements driven by density and temperature variations, where the process of convection, cooler, denser fluid sinks, while warmer than usual, less dense fluid ascends. These flows transfer heat within fluids and are critical in atmospheric circulation, ocean currents, and engineering systems like HVAC and cooking appliances. Natural convection, driven by density fluctuations and gravity, contrasts with forced convection, initiated by external sources like pumps or fans. Mixed convection combines both effects [24], 25]. In mechanical and petroleum-related engineering, convection fluxes in porous media are used in increased extraction of oil, geothermal power, and nuclear management of waste.

The Darcy number is a dimensionless measure representing fluid flow through porous media, named after Henry Darcy, who pioneered studies in porous media fluid dynamics. It is defined as the proportion of permeability to a characteristic length squared. The Forchheimer number, another dimensionless parameter, describes the significance of viscous forces relative to inertial forces in porous media and is used in analyzing flows through materials like packed beds and filters. Porous media, with voids or pores, can be natural (e.g., sandstone, human skin) or synthetic, tailored for industrial applications. Porosity is the proportion of the entire volume compared to pore volume. These media play vital roles in manufacturing, drug purification, water filtration, engineering construction, and packed bed reactors for heat and mass transfer. Darcy’s law links fluid flow to the Earth’s gravity and stress gradients in porous media. Larger pores increase porosity, enhancing flow but expanding viscous shear resistance. P. Forchheimer [26] modified Darcy’s law in 1901 by introducing a velocity-squared term to account for inertial forces, forming the Forchheimer number, further refined by Muskat [27] as the “Forchheimer term.” Ganesh et al. [28] noted increased temperature with viscous effects in magnetohydrodynamics. Makinde et al. [29] studied in a (Da-Fs) porous medium, fractional dynamics of entropy generation in unstable convection and conduction of a reactive nanofluid across a slippery, porous sheet, making porous media critical in engineering and manufacturing.

Thermal radiation involves heat energy release via electromagnetic waves due to temperature, primarily in the infrared range. Governed by Planck’s law, Stefan–Boltzmann law, and Wien’s displacement law, it significantly affects energy balance and heat transport in fluid systems, particularly at high temperatures. It interacts with conduction and convection in fields like atmospheric science, astrophysics, and thermal engineering. High-temperature engineering must account for thermal radiation’s impact on heat flow, as seen in industrial applications. Tunde et al. [30] used homotopy analysis for second-grade radiative MHD flows on surfaces. Jawad et al. [31] examined MHD hybrid nanofluid flow with heat radiation on expanding/contracting surfaces. Jile et al. [32] explored Maxwell nanofluid flows with Dufour, Soret, and thermal conductivity effects. Modifications in waste discharge concentration and the presence of quadratic thermal radiation significantly impacted the fluid flow behaviour and heat transfer of an oblique stagnation point Boger HNF over a cylindrical surface, as shown by Madhu et al. [33]. The Levenberg–Marquardt backpropagation neural network demonstrated by Raja et al. [34] effectively represented the effects of Hall currents and thermal radiation on the boundary layer flow past a stretching sheet, guaranteeing enhanced analytical precision and efficiency.

A semi-infinite vertical plate is an idealized model extending infinitely in height and one horizontal direction, simplifying the study of natural convection and boundary layer (BL) patterns by eliminating edge effects. This model aids in analyzing mass and heat transfer in buoyancy-driven flows relevant to heat exchangers, electronic cooling, building ventilation, pollutant dispersion, and oceanic thermal plumes. H. Pohlhausen [35] first used the momentum integral approach to study convection near a semi-infinite plate. S. Ostrach [36] introduced similarity solutions for free convection flows, solving coupled nonlinear ODEs using air as the fluid. B. Gebhart et al. [37] documented studies on convection with air or water, addressing the mass transfer effects of dissolved materials. B. Gebhart and L. Pera [38] developed the parameter (Nr) to relate flow density variations to chemical and heat diffusion. Amanullah et al. [39] explored non-Newtonian Nanofluid flow over a semi-infinite vertical surface with slip effects. Chamkha et al. [40] analyzed mass and heat transfer with magnetic conduction over a porous substrate. Gangadhar et al. [41] explored radiative second-grade nanofluid (NF) flows via a modified BNF model across a Riga plate with convective heating. Under various circumstances, many studies have been done on semi-infinite plates with myriad effects [42], [43], [44], [45].

In the present work, a mathematical model is developed for steady-state laminar boundary layer flow of a Buongiorno nanofluid past a semi-infinite vertical plate with the Darcy–Forchheimer (Da–Fs) model, with heat radiation impacts, unlike previous studies which focused on either the (Da–Fs) or Buongiorno’s model individually and have been confined to ordinary differential equations. The article fills a critical research gap by integrating the (Da–Fs) model with BNF theory and thermal radiation to study non-similar convection flows. The novelties of the current work are the simultaneous consideration of the combined effects of Darcy–Forchheimer drag and Buongiorno’s nanofluid model with radiation on non-similar convection flows past a semi-infinite vertical plate and ODEs which include fully two-dimensional NF flow. Moreover, velocity, temperature, concentration profiles, surface and contour graphs, and isothermal graphs are studied in detail. These have not been considered in previous studies. The dimensionless nonlinear multi-physical BVP with related wall and free stream BCs is solved using the robust 2nd order accurate implicit KBM. Validation using previous special instances that have been documented in the literature and authentication is also accomplished. Velocity, temperature, and concentration distributions are computed and visualized graphically for the influence of (Nb), (Nt), (Nr), (Da), (Fs), (R), (Pr), (Sc), (Gr). Additionally, graphs and tabular form for (C _f ), (Sh), (Nu), for parameters are also shown for R, Pr, Sc, Da, and Fs. The simulations are relevant and provide an in-depth parametric study, investigating the impact of various factors like permeability, porosity, nanoparticle volume fraction, and thermal conductivity on motion and heat transfer. The article’s findings can be used in the design and optimization of heating and cooling systems, and energy systems by providing accurate predictions of heat, mass transmission, and fluid flow patterns.

2 Mathematical analysis

A laminar, steady-state, incompressible convection flow of NF past a semi-infinite plate in an (x, y) coordinate system is studied, as illustrated in Figure 1a. Whereas, vertical plate is followed by the x-axis going higher, and the y-axis going normal to the plate. Buoyancy effects are often caused by gradients in a dispersed species’ thermal region, which drives the flow. The flow over the horizontal sheet described by Navier–Stokes becomes identical to boundary-layer equations as the characteristic value of natural convection, or the Grashof number (Gr), is increased indefinitely. Gravitational acceleration (g) exerts a downward force. Both NF and semi-infinite plates are first kept constant in terms of both temperature and concentration. Additionally, the temperature and concentration at the wall surface are higher than the constant free stream temperature and concentration T _w  > T _∞, and C _w  > C _∞, respectively.

Figure 1a:

Flow geometry of the problem.

The current flow assumption can be described as follows:

1. Darcy–Forchheimer flow has been used to examine the porous flow in the Buongiorno nanofluid model in the momentum equation.

2. The Rosseland approximation for thermal convection has been used in the energy condition.

To approximate the conductivity, a homogeneous and isotropic porous medium is considered. The second-order (Da–Fs) model’s pressure gradient is described in the following way:

where ∇ _p is the pressure, a = μ K and b = ρ K 1 are the constants, and U is the velocity. The governing equations for mass, momentum, energy, and nanoparticle species (concentration) for the BNF under the BL and Boussinesq approximations may be established using the models of Buongiorno [4], Ramesh Reddy et al. [5], and Anjum et al. [23], 46]. The vectorial forms of the conservation equations are:

The radiative flux is expressed via the Rosseland approximation [47], [48], [49], [50] as follows:

Furthermore, T ⁴ can be simplified using the Taylor series at about T _∞, and ignoring the higher order, we get [49]:

Thus, Eq. (6) becomes:

Using (8) and (9), we get:

The semi-infinite substrate surface (wall) and free stream (BL edge) are subject to the following boundary conditions, Anjum et al. [23], 46]:

Given the low concentration of nanoparticles and the use of an appropriate pressure option, in line with Kuznetsov and Nield [3], Eq. (3) can be written with the Oberbeck–Boussinesq approximation, yielding the linearized momentum equation, which is as follows:

The reduced BL conservation equations in an (x, y) coordinate system for the steady flow are as follows:

where:

In addition, the stream function ψ are defined as u = ∂ ψ ∂ y and v = − ∂ ψ ∂ x . Equation (12) is automatically satisfied subject to the velocity components expressed in terms of the stream function. Also, the following non-dimensional scaling variables are introduced (see Ramesh Reddy et al. [5] and Anjum et al. [23], 46]):

In addition, Eqs. (13)– (15) are thereby rendered into the following coupled nonlinear ODEs by virtue of the transformations in Eq. (17):

The transformed non-dimensional boundary conditions are:

In which

The above equations comprised distinct dimensionless influential parameters which are mathematically expressed in equation (22). Namely these parameters along with the other mathematical symbols or notations used in the paper are given in the nomenclature.

The shear stress components at the surface, known as the skin-friction coefficients (C _f ), the heat transfer rate known as the Nusselt number (Nu), and the mass transfer rate of NPs, known as the Sherwood number (Sh), are the physically key interesting engineering design parameters for the existing semi-infinite vertical surface. They are defined as follows:

where ΔT = T _w  − T _∞, and ΔC = C _w  − C _∞. Also, if ξ ∼ 0 then the BLEs (18)– (20) changed to the following similarity ODEs:

3 Methodology

The Keller box implicit finite difference scheme [51] has been applied to numerically solve the dimensionless boundary layer equations (18)– (20). This method remains one of the most influential numerical schemes for solving two-point BVPs. The Keller-box approach offers appealing extrapolation properties and second-order accuracy with flexible spacing. On a rectangular grid Figure 1b, a finite-difference technique is used (box) and converts the partial differential equations of BL into an algebraic set of equations. It attains remarkable accuracy, offers steady numerical meshing characteristics, and converges quickly. By utilizing fully implicit methods with customizable stepping, the Keller box approach improves accuracy on explicit or semi-implicit schemes. Another advantage of this KBM is that the non-similar solutions can be easily obtained via this scheme as compared to other numerical approaches. In line with the physics of parabolic systems, each discretization step is fully coupled.

Figure 1b:

Keller box procedure, box cell, and boundary layer meshing.

The discrete algebra connected to the KBM is essentially independent of any other mimicking (physics-capturing) computation methods [52], [53], [54].

The four phases involved in the KBM are:

1. splitting the system of Nth order PDEs down to the N first-order ODEs,

2. finite difference discretization,

3. quasi-linearization of non-linear Keller algebraic equations,

4. block-tridiagonal elimination solution of the linearized Keller algebraic equations.

Step 1: Reduction of the Nth order partial differential equation system to N first-order equations.

Equations (18)– (21), together with new variables, are used to transform the boundary value problem into a multiple system of 1st order equations. Thus, adding the new variables results in a set of nine simultaneous first-order ODEs:

where primes denote differentiation with respect to η. In terms of the dependent variables, the BCs become:

Step 2: Finite difference discretization

In a Keller box (cell), a 2D computational grid is imposed in the ξ–η plane. The stepping process is defined by:

where k _n is the Δξ – spacing and h _j is the Δη – spacing. If g j n denotes the value of any variable at η j , ξ n , then the variables and derivatives of Eqs. (29)– (36) at η j − 1 / 2 , ξ n − 1 / 2 are replaced by:

The resulting finite-difference approximation of Equations (29)– (36) for the mid-point η j − 1 / 2 , ξ n , are:

Where:

The boundary conditions are:

Stage 3: Quasi-linearization of non-linear Keller algebraic equations

If we assume f j n − 1 , u j n − 1 , v j n − 1 , g j n − 1 , p j n − 1 , s j n − 1 , t j n − 1 to be known for 0 ≤ j ≤ J. This leads to a system of 7J + 7 equations for the solution of 7J + 7 unknowns f j n , u j n , v j n , g j n , p j n , s j n , t j n j = 0,1,2 , … , J . This non-linear system of algebraic equations is linearized using Newton’s method.

Stage 4: Block-tridiagonal elimination solution of linear Keller algebraic equations

The linearized system has a block-tridiagonal structure, and hence the block-elimination approach is used to solve this system. Block matrices make up the produced block-tridiagonal layout. The entire linearized system is expressed as a block matrix system, in which every element of the coefficient matrix is a matrix in and of itself. The effective Keller-box method is used to solve this system. The number of mesh points in both directions has a big impact on the numerical output. Following a few trials, a greater number of mesh points are chosen in the η-direction (radial coordinate), whereas a much smaller number of mesh points are used in the ξ-direction (tangential coordinate). η _max = 14 defines a sufficiently large value at which the specified BCs are accomplished. For this flow domain, ξ _max is set as 3. In the current computation, mesh independence is attained. The computational algorithm is run on a PC using MATLAB. As explained by Keller [51], the procedure exhibits outstanding stability, convergence, and consistency.

4 Results and discussion

The current portion of the work highlights the physical perspective of BNF’s flow past a semi-infinite vertical plate with the (Da-Fs) model. The Keller-box finite difference technique is applied to solve the requisite posited equations. A detailed graphical illustration of the solution is shown in Figures 1–13 utilizing MATLAB code, on velocity (f′), temperature (θ), and nanoparticle volume fraction concentration (ϕ), skin friction coefficient (C _f ), Nusselt number (Nu), Sherwood number (Sh) for the distinct dimensionless parameters. The numerical problem comprises two independent space variables (ξ, η), therefore, the non-similar results are obtained. The following default values like Pr = 0.71, Sc = 0.6, Nr = 0.1, Nb = 0.3, Nt = 0.3, R = Da = Fs = 0.5, Gr = 10, ξ = 1.0 are chosen for the simulation purposes.

Figure 1(c–e) depict the implications of thermal radiation (R), where (0 ≤ R ≤ 1.1) on velocity, temperature and concentration profiles. Mathematically, the radiative parameter is expressed as R = 16 σ ∗ T ∞ 3 3 k k ∗ . From the expression, it is clear that the radiative parameter effectively alters the nanofluid’s thermal conductivity, which significantly affects heat transmission. Thermal radiation ceases to exist at R = 0. Nevertheless, as R rises, thermal radiation becomes progressively more significant to the regime [56]. Although radiation only directly influences the temperature field while the coupling components in the momentum and nanoparticle concentration BLEs allow thermal radiation to have an indirect influence on these fields. Figure 1c shows that velocity increases due to increase values of the radiative parameter. Increasing R values correlate with the flow acceleration. Figure 1d illustrates that, as expected, a large increase in temperature occurs when the value of R increases. The regime is substantially electrified by radiative flux, resulting in a thicker thermal boundary layer. Furthermore, it reveals that greater R values result in improved heat transmission, which improves thermal profiles. Figure 1e demonstrates that the opposite effect is caused in nanoparticle concentration, with a decrease in concentrations accompanying higher values of R. When there is no radiative heat flux, nanoparticle concentration is clearly at its highest. Thinner species boundary layer thickness results in higher radiative effects. The physical relevance is that as thermal radiation increases, the velocity, temperature of the nanofluid increase because the radiative heat transfer raises the overall thermal energy in the regime. This additional heat energy reduces fluid viscosity, facilitating faster flow and stronger rotational motion. The increased temperature accelerates fluid motion, enhancing convective heat transfer. Simultaneously, the elevated thermal environment reduces the tendency for nanoparticles to cluster near the surface, leading to a decrease in nanoparticle concentration as they disperse more readily into the fluid.

Figure 1c:

Velocity profile for distinct values of R.

Figure 1d:

Temperature profile for distinct values of R.

Figure 1e:

Concentration profile for distinct values of R.

Figure 2(a–c) show the impact of Brownian motion (0.3 ≤ Nb ≤ 1.5) on velocity, temperature and concentration profiles. (Nb) defined as the ratio of the Brownian diffusion coefficient to the thermal diffusivity of the base fluid. This parameter is utilized to quantify the effects of Brownian motion on the heat and mass transmission characteristics of NFs. In Figure 2a Velocity profile is enhanced, although a stronger elevation is computed in the former with greater Nb values. Brownian motion also modifies the NF thermal conductivity and the propensity for heat transmission in the NF. As a result, the increased random motion of the NPs modifies the thermal pattern as well, and the momentum field experiences this influence through thermal buoyancy. As Nb increases, (θ) is positively affected across the BL regime, as seen in Figure 2b. There is also a significant change in the topology of (θ) profiles further from the substrate (wall) at very high Nb values. Raising the (θ) aggravates the motion of the NPs and ballistic collisions. Consequently, chaotic Brownian motion is increased even further. The increased heat conduction in the regime and the improved micro-convection surrounding the NPs are also influenced by the change in thermal conductivity with greater Brownian motion. This results in a thicker thermal BL due to a heating effect. Although the fluid’s molecules and NPs are always moving, there is a noticeable overall shift in (θ). Brownian motion, however, predominates in the random thermal motion of the NPs. As Nb is increased, there is a notable decrease in (ϕ) values, as observed in Figure 2c. Hence, the thickness of the nanoparticle concentration boundary layer is reduced. Our findings are consistent with those of Reddy et al. [5] and Gaffar et al. [44], who have demonstrated that higher viscosity operates to boost heat convection and flow while decreasing mass transfer rates. The physical significance of these observations lies in the complex interplay between Brownian motion and NF dynamics. As Nb increases, NPs experience enhanced random movement, facilitating more efficient thermal diffusion and increasing fluid velocity. This phenomenon is crucial in settings where efficient heat transport is necessary, like cooling structures and thermal management in engineering. However, the reduction in concentration gradient signifies a more homogeneous dispersion of NPs, which can affect processes reliant on localized concentrations, such as drug delivery or materials synthesis. The fluid’s Brownian motion is significantly impacted by increasing Nb levels because of the NPs’ random mobility.

Figure 2a:

Velocity profile for distinct values of Nb.

Figure 2b:

Temperature profile for distinct values of Nb.

Figure 2c:

Concentration profile for distinct values of Nb.

Figure 3(a–c) depict the influence of the thermophoresis parameter (0 ≤ Nt ≤ 0.8) on (f′), (θ), (ϕ) profiles. Driving NPs under a temperature gradient is known as thermophoresis. Nt quantifies the impact of thermophoretic force, which drives NPs from hot regions to cold regions, causing the fluid temperature to be modified within a fluid. The definition of the thermophoresis parameter is a ratio of the thermophoretic diffusion coefficient to the thermal diffusivity of the base fluid. From Figure 3a, an increase in Nt decelerates velocity (f′). The increasing thermophoresis parameter value enhances (θ) in Figure 3b and considerably boosts (ϕ) in Figure 3c. These trends are sustained at all distances transverse to the substrate. However, while asymptotic decays occur from the wall to the free stream for all (θ) profiles, the nanoparticle concentration profile is only a decay for Nt = 0, which has not been identified previously in the literature. As Nt increases, the topology is flipped from a convex to a concave one for Nt = 0.2. With subsequent elevation in the thermophoresis parameter, a peak in nanoparticle concentration emerges progressively further from the wall. Eventually, however, profiles for Nt > 0.2 do descend smoothly to the free stream. Overall, stronger thermophoresis elevates the thermal and nanoparticle species BLT, which inevitably influences the structure of the coating regime. The impact of a higher thermophoretic temperature gradient is significant on all transport characteristics, confirming the important role it plays in NF mechanics. Our findings support the patterns of Reddy et al. [7], 42] and Gaffar et al. [44]. The Physical significance is that this signifies that a stronger thermophoretic force leads to enhanced momentum, heat, and mass transfer, reflecting more pronounced thermal and solute gradients in the NF flow.

Figure 3a:

Velocity profile for distinct values of Nt.

Figure 3b:

Temperature profile for distinct values of Nt.

Figure 3c:

Concentration profile for distinct values of Nt.

Figure 4(a–c) elucidate the impact of the combined Buoyancy ratio parameter, (−0.7 ≤ Nr ≤ 0.1), on profiles (f′), (θ, (ϕ) through surface regime with transverse coordinate (η). Dual natural convection currents mobilized by temperature and nanoparticle species are present. The regime exhibits buoyancy from both thermal and nanoscale species, resulting in dual natural thermo-solute convection. If Nr < 0, the forces of thermal buoyancy and species buoyancy forces are opposed to each other. If Nr = 0, forced convection occurs, and buoyancy forces disappear. For Nr > 0, the buoyancy force helps the other. As seen in Figure 4a, Velocity (f′) is suppressed with positive Nr but is enhanced with negative Nr. In other words, assistive buoyancy damps the primary flow, whereas opposing buoyancy enhances it. This reduction weakens buoyancy-driven movement, tending to a decrease in fluid velocity. Consequently, with less buoyancy force driving fluid motion, the convective heat transfer process is hindered, resulting in depreciation of velocity profiles in the system.Figure 4(b and c) demonstrates that an increment of Nr strengthens the buoyancy-driven flows. This augmentation enhances fluid motion, facilitating more effective mixing and transport of heat and solute. Consequently, (θ) and (ϕ) profile appreciates. Notably, both (θ) and (ϕ) profiles show an upward trend, signifying that the heightened internal buoyancy forces improve heat, mass transfer. This results in higher thermal, solute gradients near the surface, causing increased (θ), (ϕ) levels within the boundary. Our outcomes align with the observed trends of Reddy et al. [7], 42] and Gaffar et al. [44]. Physically, this signifies that stronger buoyancy effects lead to more efficient energy and species transport in the fluid. This is essential for precisely estimating flow behaviour and maximising mass and heat transfer in a variety of technical applications.

Figure 4a:

Velocity profile for distinct values of Nr.

Figure 4b:

Temperature profile for distinct values of Nr.

Figure 4c:

Concentration profile for distinct values of Nr.

Figure 5(a–c) portray the impact of Darcy number (Da), where (0.2 ≤ Da ≤ 1.8), on profiles (f′), (θ), (ϕ). Particularly in the study of flow via porous media, Da is a dimensionless number that characterizes the relative permeability of the medium. Da is used to compare the effects of inertial forces and the resistance of the porous material to the movement of fluid. A high Da implies that inertial forces are more essential, whereas a low Da denotes that the flow is mostly controlled by the resistance of the porous medium. Figure 5a demonstrates that (f′) is amplified with a hike in Da, with the maximum effect occurring close to the surface. Greater permeability indicates a decrease in solid Fibers that obstruct axial flow; this lowers the Darcian resistance and accelerates the flow. Elevating pushes peak velocity even more Da. Figure 5b shows that boosting the Darcy parameter yields a significant suppression of (θ). Heat conduction is suppressed as a result of the loss of solid fibers in the porous matrix linked to increased penetration. This cools the regime by reducing heat diffusion inside it. As a result, the surface’s heat BLT is suppressed. Porous media with lower permeability reach far higher temperatures than those with higher permeability. Figure 5c eluciates that as the Da increases, this indicates increased resistance to pass through the porous substance. This resistance reduces the exchange of nanocrystals and leads to a low concentration of NPs in the vicinity. Consequently, the overall (ϕ) profile within the fluid decreases. Our outcomes are consistent with the patterns of [7], 17], 44]. Physically, fluid flow resistance falls as Da rises, showing increased permeability in the porous media. This lower resistance allows for an increased fluid velocity because the flow encounters less drag. However, the enhanced fluid velocity reduces leads to decay in both (θ) and (ϕ) profiles. Essentially, fluid moves more quickly through the medium, carrying less heat and fewer NPs along with it. Reduced heat transmission via thermal conduction in the system is facilitated by the steady reduction of solid fibres that have large Da values in a porous medium. This cools down the thermal BLT, which also diminishes and limits the transfer of thermal energy from the vertical surface into the system. Consequently, the presence of porous materials significantly affects the system’s speed and thermal dispersion, offering a solid foundation for regulating flow and temperature management. This parameter is crucial for analysing and modelling fluid flow patterns in applications involving porous media, such as filtration, groundwater flow, and enhanced oil recovery.

Figure 5a:

Velocity profile for distinct values of Da.

Figure 5b:

Temperature profile for distinct values of Da.

Figure 5c:

Concentration profile for distinct values of Da.

Figure 6(a–c) Influence of the Forchheimer number (0 ≤ Fs ≤ 2.3) on (f′), (θ), (ϕ) profiles across the farthest domain with transverse coordinates (η) is discussed. The dimensionless parameter (Fs) is used to describe how inertial forces affect porous media flow. It measures relative significance of viscous forces and inertial effects in a medium with pores. A greater Fs means that inertial implications, which affect the flow properties through the porous media, become of greater significance in comparison to viscosity forces. Figure 6a demonstrates how a greater inertial impedance that resists flow is present and causes a noticeable depreciation in velocity with an increase in (Fs). This impact is maximized close to the surface; a lesser deceleration in a flow with a higher (Fs) is calculated farther away. There is no discernible effect of (Fs) on flow in the free stream. Figure 6b reveals that a significant heating impact occurs in the regime as a result of the higher Fs. Temperature (θ) is always stronger for all transverse coordinate (η) values. Hence, the thickness of thermal BL is highest for the strong Forchheimer case (Fs = 1) and the smallest for the weak Forchheimer drag case (Fs = 0). Figure 6c illustrates that (ϕ) increases as (Fs) gradually rises because greater (Fs) signifies greater inertial effects in flow through a porous medium. These inertial effects enhance the mixing and dispersion of nanocrystals, leading to a greater (ϕ) of NPs near surface. Consequently, increased inertial forces facilitate better nanoparticle distribution and an elevated (ϕ) profile. Our outcomes are consistent with the patterns of [7], 16], 44]. Physical interpretation is that since the drag force and coefficient of inertia are connected, a surge in inertia causes the fluid’s drag force to grow, thus lowering its speed. The influence of the quadratic inertial drag is larger with closer proximity to the wall’s surface. Thus, velocity (f′) dampens while temperature (θ) and concentration (ϕ) profiles enhance with higher Fs numbers. (Fs) is used to optimize flow and performance in NF-based filtration systems and heat exchangers by accounting for inertial effects in porous media.

Figure 6a:

Velocity profile for distinct values of Fs.

Figure 6b:

Temperature profile for distinct values of Fs.

Figure 6c:

Concentration profile for distinct values of Fs.

Figure 7(a–c) indicate the impact of Prandtl number (Pr), where (0.3 ≤ Pr ≤ 1) on profiles (f′), (θ), (ϕ). Pr is a non-dimensional number; it characterizes momentum diffusion’s (viscous effects) relative significance to thermal diffusion (heat conduction) in a fluid. Its literal meaning is the kinematic viscosity to thermal diffusivity ratio. Figure 7(a–c) demonstrate that when the Prandtl number Pr increases, the velocity (f′) profile appreciates, but temperature (θ) profiles decrease because a higher Pr implies that viscous forces dominate over thermal diffusion, slowing the flow and reducing heat transfer efficiency. A higher Pr means less thermal diffusion, which leads to slower heat transfer within the BL. Conversely, the nanoparticle (ϕ) increases because the reduction in thermal diffusion limits the influence of thermal gradients, allowing more NPs to accumulate near the surface. The Physical significance is that this indicates that in high-Prandtl-number fluids, heat conduction is less efficient, while (ϕ) remains more stable, emphasizing the dominance of viscous and concentration effects over thermal effects in such flows. Realistic application of Pr involved improving the thermal efficiency of heating, ventilation, and air conditioning (HVAC) systems by using NFs with suitable Prandtl numbers for better heat transfer rates. In the unimpeded stream, every profile is shown to converge seamlessly.

Figure 7a:

Velocity profile for distinct values of Pr.

Figure 7b:

Temperature profile for distinct values of Pr.

Figure 7c:

Concentration profile for distinct values of Pr.

Figure 8(a–c) Features impact of Schmidt number (Sc), where (0.71 ≤ Sc ≤ 5) on profiles (f′), (θ), (ϕ) through surface regime with transverse coordinate (η). Schmidt number (Sc) dimensionless number that characterizes the relative significance of mass diffusion (molecular diffusion) over momentum diffusion (viscous effects) in a fluid. Figure 8(a–c) elucidates that as Schmidt number Sc increases velocity profile rises because a higher Sc implies that mass diffusivity is much lower than momentum diffusivity, resulting in less resistance to momentum transfer, thus enhancing fluid flow. Meanwhile, the temperature (θ) and concentration (ϕ) decrease because higher Sc values reduce mass diffusivity, limiting both heat and nanoparticle diffusion away from the surface. This leads to a thinner BL for (θ) and (ϕ) profiles. The Physical viewpoint is a higher Sc indicates that mass diffusion is restricted, allowing for faster fluid flow (higher velocity) but reduced efficiency in both heat transfer and nanoparticle dispersion, which affects the overall convection and transport processes in the fluid. The physical application of Sc is to optimize mass transfer processes in applications such as chemical reactors and separation technologies by selecting NFs with appropriate momentum and mass diffusion properties.

Figure 8a:

Velocity profile for distinct values of Sc.

Figure 8b:

Temperature profile for distinct values of Sc.

Figure 8c:

Concentration profile for distinct values of Sc.

Figure 9(a–c) demonstrate the impact of Grashof number (0.71 ≤ Gr ≤ 10) on profiles (f′), (θ), (ϕ) through surface regime with transverse coordinate (η). (Gr) a non-dimensional number which quantifies the proportionate importance of buoyant and viscous forces in a fluid flow. It helps in understanding and optimizing heat transfer processes driven by buoyancy effects. Figure 9(a–c) As Gr increases, (f′) increases due to stronger buoyancy forces driving the flow, enhancing fluid motion. Temperature (θ) and Concentration (ϕ) profiles decrease because the intensified convection transfers heat and NPs more rapidly away from the surface, thinning the BLs. The physical significance is that a higher Gr indicates dominant buoyancy effects, leading to faster fluid flow and more efficient heat and mass transfer, but reduced BLT for (θ), (ϕ) fields. It helps predict how buoyancy-driven flow affects heat and mass transfer, allowing for improved efficiency in these thermal management systems.

Figure 9a:

Velocity profile for distinct values of Gr.

Figure 9b:

Temperature profile for distinct values of Gr.

Figure 9c:

Concentration profile for distinct values of Gr.

Notably, the preceding graphs show that all profiles in the free stream converge smoothly, indicating the use of an appropriate large infinity boundary condition.

Figures 10(a–c) illustrate the impacts of the Darcy parameter (Da) on higher magnitudes of skin friction (C _f ), Sherwood number (Sh), and Nusselt number (Nu), are computed with an increase in streamwise coordinate (0≤ ξ ≤ 3). A substantial boost of Da decreases flow resistance within the porous medium, leading to higher fluid velocities. This reduction in resistance enhances shear stress at the surface, increasing C _f , momentum BLT is enhanced. Additionally, with improved flow velocities, convective mass and heat transfer are more efficient, resulting in higher Sh, Nu. The higher velocity promotes better mixing and dispersion of NPs, which improves Nu, Sh near the surface produces significant heating, and nanoparticle volume, and escalates thermal concentration BLT. Physical relevance is, higher Da enables more effective convective transport, leading to improved thermal and mass transfer performance. In real-life applications, increasing the Da enhances the efficiency of mass and heat transmission in systems such as advanced cooling systems, chemical reactors, and filtration units. Improved (C _f ), Sh and Nu indicate better performance in heat dissipation and particle transport, optimizing processes like electronic cooling and water treatment.”

Figure 10a:

Variation in skin friction for distinct values of Da.

Figure 10b:

Variation in Sherwood number for distinct values of Da.

Figure 10c:

Variation in Nusselt number for distinct values of Da.

Figure 11(a–c) demonstrate the impacts of (Fs), on (C _f , Sh, Nu), along surface regime at higher streamwise coordinate values (0≤ ξ ≤ 3). A substantial rise in Fs signifies a rise in inertial effects relative to viscous forces in the porous medium. This shift causes increased flow resistance and turbulence, leading to reduced fluid velocities. Consequently, C _f decreases due to less effective shear stress at the surface. Additionally, enhanced inertial effects disrupt the BLs, reducing the efficiency of both mass and heat transmission. As a result, Nu, Sh declined due to decreased convective effectiveness, reflecting poorer thermal and particle transport performance. Moreover, this elucidates that appreciation of Fs values causes a remarkable decline in the heat and mass transport to the substrate. Thermal and concentration BLT is condensed and has a large heating influence because of a net heat, mass transfer into BL.

Figure 11a:

Variation in skin friction for distinct values of Fs.

Figure 11b:

Variation in Sherwood number for distinct values of Fs.

Figure 11c:

Variation in Nusselt number for distinct values of Fs.

Figure 12(a–e) illustrates how engineering interest quantities (C _f , Sh, Nu) respond to a variety of associated components. In Figure 12a. Substantial increase (C _f ) is noted with an improvement in the Forchheimer number (Fs) and Darcy number (Da) and readings. A surface plot of C _f for different values of (Nb) and (Nt) is shown in Figure 12b. It portrays that skin friction appreciates significantly as (Nb) and (Nt) parameters expand. The surface plot in Figure 12c shows that Nu improves as the Forchheimer number (Fs) and Darcy number (Da) grow. Moreover, in Figure 12d, the Nusselt number Nu improves as the Buoyancy number (Nr) and Thermal radiation (R) elevate. The implication of (Sc) and thermophoresis (Nt) on Sh is illustrated in Figure 12e. The mass profile thickening and the (Sc) pattern are further amplified as the value of (Nt) grows, which also contributes to the enhancement of the dimensionless Sherwood number Sh (Figure 13(a and b)).

Figure 12a:

Surface contour plot variation for distinct values of Da, Fs against C _f .

Figure 12b:

Surface contour plot variation for distinct values of Nb, Nt against C _f .

Figure 12c:

Surface contour plot variation for distinct values of Da, Fs against Nu.

Figure 12d:

Surface contour plot variation for distinct values of Nr, R against Nu.

Figure 12e:

Surface contour plot variation for distinct values of Sc, Nt against Sh.

Figure 13a:

Isotherms for different Darcy values when Pr = 0.71.

Figure 13b:

Isotherms for different Forchheimer values when Pr = 0.71.

Table 4 shows (C _f , sh, Nu) for various values of R, Pr, Sc, Da, and Fs along with a variation in the stream-wise coordinate value ξ, (1 ≤ ξ ≤ 3). Here Nr = 0.1, Nb = 0.3, Nt = 0.3, Ec = 0.5, Gr = 10. It is noted that as radiation parameter (R) appreciates (C _f ), (Sh) damps conversely (Nu) hikes. Moreover, the Prandtl number Pr appreciates the fluid’s thermal diffusivity decreases, leading to a thicker thermal BL, reducing (Nu). Elevation in Pr also means higher momentum diffusivity, which depreciates (C _f ). Conversely, mass diffusivity is less affected, resulting in a thinner concentration BL and an increased Sh, indicating enhanced mass transfer. As (Sc) boosts, it leads to a thinner concentration BL. This results in higher mass transfer rates, thus steadily rising (Sh). The reduced mass diffusivity also means higher momentum diffusivity, which increases velocity gradient at the wall, thereby raising (C _f ). Furthermore, the enhanced flow mixing due to higher (Sc) contributes to surge heat transfer, resulting in a higher (Nu). As (Da) appreciates, the permeability of porous medium hikes, reducing resistance to fluid flow. This leads to higher fluid velocities near the wall, which elevates (C _f ). The enhanced fluid motion improves both heat and mass transmission, resulting in higher (Nu), (Sh) due to thinner thermal and concentration BL. As (Fs) rises significantly, the inertial resistance in porous media becomes more significant. This increased resistance reduces (f′), leading to a decrease in (C _f ). Additionally, the lower (f′) diminishes both heat and mass transmission, resulting in reduced (Nu), (Sh) due to thicker thermal and concentration BL.

5 Main findings

In this study, we conducted numerical simulations to investigate the steady state laminar, incompressible, thermal radiative flow past a semi-infinite vertical surface embedded with Darcy–Forchheimer porous medium. Additionally, we considered the influence of the two-phase nanofluid model along with thermal convection, nanoparticle heat, and mass transfer in this context. After utilizing the pertinent variables, the required obtained equations are additional exercised by the well-known Keller-box scheme. The non-similar solutions are obtained in the form of graphs and tables for the several distinct controlling parameters. Consequences divulge the subsequent practical implications for enhancing the design and optimization of cooling systems, electronic thermal management, and energy systems in circumstances where accurate control of temperatures and effective heat transmission are essential. The significant findings from our study can be summarized as follows.

1. The velocity and temperature profiles increased with rising values of the Brownian motion parameter while the concentration profile decreased.

2. With increasing values of the thermophoresis parameter, the velocity profile shrinkages whereas the concentration and temperature profiles boosted up.

3. The temperature distribution and concentration profiles raised due to the larger impacts of the buoyancy ratio parameter but the velocity profile shrank.

4. The heat transfer rate upsurges due to the superior impacts of the radiation parameter while the mass transfer rate and shear stress abruptly decline.

5. Owing to the rising values of the Darcy number, the gradients (shear stress, heat and mass transfer rates) escalated.

6. The shear stress, heat and mass transfer rates decayed owing to the larger influences of the Forchheimer parameter.

6 Future work

Future research studies can be extended by incorporating non-Newtonian fluid characteristics, variable thermophysical properties, and time-dependent boundary conditions to interpret better the real-world scenarios. Future work may also involve coupling the numerical results with experimental validation to enhance reliability. Moreover, exploring three-dimensional and transient flow configurations, along with optimization through machine learning algorithms, could further enrich the understanding of nanofluid transport in advanced thermal systems.