Chemically reactive Maxwell nanoliquid flow by a stretching surface in the frames of Newtonian heating, nonlinear convection and radiative flux: Nanopolymer flow processing simulation

1 Introduction

In recent years, non-Newtonian (rheological) fluid transport phenomena have received a great deal of interest from scientists and researchers due to diverse applications in industrial applications [1]. Many biotechnological and process mechanical engineering fluent media are non-Newtonian and common examples include gels, adhesives, polymers, foodstuffs, foams, slurries, emulsions, paints, blood, tribological suspensions, gypsum paste, glass melts, petroleum fluids, etc. Polymers in particular exhibit many complex rheological characteristics. For example, at a constant molecular weight, the amount of energy required to process a polymer is directly related to the shear rate dependence of the polymer viscosity. Some important manufacturing processes involving polymers are viscoelastic abrasive flow machining [2], micromixing [3], squeezing flows [4] and stretching and retraction dynamics of films/sheets [5]. Non-Newtonian materials have unique properties that cannot be explained by a single constitutive model. A number of constitutive equation models have therefore been constructed by researchers to simulate the behaviour of non-Newtonian liquids. These include the Oldroyd-B, Giesekus (Gsk), and Phan Thien-Tanner (PTT) models, which are viscoelastic models, the Bingham viscoplastic model, and the Ostwald–de Waele power-law model for pseudoplastic or dilatant fluids. In parallel with the rheometric experimental studies, considerable activity has been motivated in mathematical and computational modelling of non-Newtonian flows, often with heat and mass transfer. These studies feature highly nonlinear and coupled boundary value problems that generally require sophisticated analytical techniques, e.g. homotopy methods, variational iteration methods, Adomian decomposition, etc. Often, numerical methods are also used that may include finite element, finite difference and spectral methods. Many excellent studies of non-Newtonian flows from a stretching surface have been communicated since this regime is fundamental to polymeric flow processing systems. Huang et al. [6] utilized the Chebyshev finite difference method (ChFD) to simulate the significance of thermoviscosity (viscosity depending on the temperature) on the unsteady flow and heat transfer in a power-law liquid film from a stretching sheet. They showed that thermoviscosity strongly enhances the film thickness and the local heat-transfer rate (Nusselt number). Anwar Bég et al. [7] implemented an electrothermal network solver (PSPICE) to investigate the unsteady buoyancy-driven convection boundary-layer flow of a magnetic Walters-B viscoelastic polymer from a stretching surface in porous media, noting that velocity is increased with viscoelastic parameter while it is suppressed with the magnetic field. They also noted that an increasing thermal Grashof number inhibits the flow and an increasing suction at the sheet reduces the temperatures. An alternative rheological model is the upper-convected Maxwell viscoelastic model, which also has excellent accuracy for simulating certain polymer flows. This model is an example of a rate-type model in which the relaxation and retardation times are defined. It is particularly valuable for small molecular significance polymers since it allows for a reasonable estimation of stress relaxation and retardation. The viscosity of long-branched polymers is more shear rate-dependent than is the viscosity of linear polymers, and the long-chain branching affects the elasticity of the polymer melts, which is seen in the normal stress difference and the storage modulus [1]. Shahid et al. [8] investigated the non-Fourier heat and mass transfer in magnetized upper-convected Maxwell fluid flow from a permeable stretching sheet. They utilized the successive Taylor series linearization method (STSLM) with Gauss–Lobatto collocation and Chebyshev interpolating polynomials and noted a strong modification in the Nusselt number, Sherwood number and skin friction with a change in the viscoelastic parameter.

An important development in the 21st-century engineering sciences has been the deployment of nanomaterials. Engineered at the nanoscale, such materials include graphene, matrix composites with embedded carbon nanotubes (CNTs) and nanopolymers. Nanopolymers are an example of nanofluids in which the base fluid is modified with the dispersal of nanosized particles. The nanoparticles may be metallic (e.g. copper, zinc, silver, etc.) or carbon-based (silicon, diamond, etc.), and the resulting nanofluid is termed unitary. Combinations of nanoparticles may also be used to produce hybrid nanofluids. Base fluids may be aqueous, oil, polymeric, etc. The inclusion of metallic nanoparticles, in particular, has been shown to achieve exceptional enhancement in thermal properties of the resulting solid–liquid composite material, i.e. colloidal suspension (nanofluid) [9]. Viscosity is also substantially modified in nanofluids as reviewed lucidly by Ilyas et al. [10]. Theoretical studies of nanofluid flows have also mobilized considerable interest and involve a combination of fluid dynamics, thermal convection and thermal conduction among other aspects. Subramanian et al. [11] have reviewed many developments in nanofluid simulation in a variety of technologies including chemical process engineering, materials fabrication, solar energy and lubrication systems. An important methodology for simulating nanoscale characteristics of nanofluids was introduced by Buongiorno [12], who developed a robust two-component model that includes both an energy balance and nanoparticle concentration balance, in addition to the momentum balance. The dominant transport mechanisms in this model are thermophoresis and Brownian dynamics. This model is accommodated comfortably in boundary layer theory and has been employed extensively in a wide spectrum of nanofluid dynamic transport problems. However, the Buongiorno model does not feature non-Newtonian characteristics, which have been confirmed via experimental studies by many researchers. For example, Hojjat et al. [13] have studied γ-Al₂O₃, TiO₂ and CuO–aqueous carboxymethyl cellulose (CMC) nanofluids and have observed strong pseudoplastic (shear thinning) behaviour and also viscoelastic behaviour at different volume fractions. Other investigations identifying viscoelastic behaviour of nanofluids and nanopolymers include the work by Ghanbarpour et al. [14] and Sica et al. [15] for metallic and graphene nanoparticles in polymers, respectively. Therefore, to simulate the viscoelastic behaviour of nanofluids (nanopolymers), a pragmatic approach is the combination of the Buongiorno model with a robust non-Newtonian model. Rana et al. [16] utilized an HP-finite element method, Buongiorno’s model and the Reiner–Rivlin second-grade non-Newtonian model to study the dissipative viscoelastic nanofluid transport from a stretching surface with energy dissipation effects. Vasu et al. [17] implemented HAM and modified differential quadrature schemes to examine the time-dependent magnetized viscoplastic Buongiorno nanoliquid boundary-layer flow from a stretched sheet doped with gyrotactic micro-organisms, as a model of bio–nanocoating dynamics. The Maxwell elastic–viscous model has however proved extremely popular in conjunction with the Buongiorno model for nanopolymeric materials processing analysis. Abbasi et al. [18] investigated the oblique stagnation bioconvection flow of a Maxwell nanofluid on a convective surface. They noted a strong depletion in the Nusselt number and Sherwood number with increasing Maxwell viscoelastic effects. Khan and Nadeem [19] used a bvp4c quadrature technique in MATLAB to compute the electrically conducting Maxwell nanofluid Hiemenz flow from a shrinking sheet (multiple solutions). Khan et al. [20] computed the Maxwell nanofluid stagnation point slip flow. Ahmed et al. [21] scrutinized the swirling flow induced by a rotating disk to a Maxwell nanoliquid with heat generation/absorption characteristics. Further studies of Maxwell nanofluid dynamics in materials processing systems concerning stretching surfaces include Ghasemian et al. [22], Bai et al. [23], Jagwal et al. [24], Ahmed et al. [25], Irfan et al. [26] and Acharya et al. [27]. All these analyses have confirmed the prominent effect of Maxwell viscoelasticity on transport characteristics and the inadequacy of a Newtonian model to capture the realistic behaviour of nanopolymer flows.

In nanopolymeric manufacturing processes, chemical reactions arise. These contribute to the polymerization and compatibilization processes and enable the generation of different compositions of the fabricated product. Many excellent experimental studies have examined such reactions in multiple polymeric materials including polyethylene blends [28], viscoelastic coatings [29] and polycarbonate/polyamide bilayer films synthesized from stretching surfaces [30]. Chemical reactions strongly influence heat and mass transport and very sophisticated interplays arise between the thermal, momentum and mass diffusion processes. Chemical reactions can be divided generally into two categories: homogeneous chemical reactions, which occur uniformly across a phase, and heterogeneous chemical reactions, which occur inside or near the boundary of the segment. First-order, higher-order and binary reactions are examples of homogenous chemical reactions. The presence of chemical reactions in nanofluid manufacturing has therefore motivated engineers and mathematicians to develop models that simulate robust chemo-nanofluid mechanics transport [31]. Newtonian reactive nanofluid flows were initially studied. Shukla et al. [32] conducted the second law thermodynamic analysis of the transient stagnation point flow of Newtonian reactive nanofluids from a stretching sheet under orthogonal electrical and magnetic fields with a finite element method. They observed that a stronger homogenous first-order chemical reaction suppresses entropy generation, whereas both the increasing electrical and magnetic field strength elevate the entropy generation. Garvandha et al. [33] employed a finite difference scheme to simulate phase change in the Buongiorno nanoliquid boundary-layer reactive flow induced by inclined stretched cylinder under curvature and cross-diffusion aspects. They observed a dramatic depletion in the nanoparticle concentration with greater order of the chemical reaction and a reduction in the concentration boundary layer thickness. Chemically reacting non-Newtonian flows have also received attention more recently. Adesanya et al. [33] considered third-grade viscoelastic reactive heat transfer in a channel. Shamshuddin et al. [34] and Kamran and Wiwatanapataphee [35] used Eringen’s micropolar model to simulate reactive polymeric processing flows. Khan et al. [36] employed the Carreau–Yasuda rheological model with a chemical reaction and activation energy. Khan et al. [37] implemented the Casson viscoplastic model to study the reactive convection flow from a stretching vertical surface. Besides, reactive non-Newtonian nanoliquid flows have been investigated rigorously. Khan and Alzahrani [38] computed the Jeffrey viscoelastic nanofluid flow from a curved stretching surface with a binary chemical reaction, activation energy and viscous heating effects. The chemically reacting Maxwell viscoelastic nanofluid flow from a stretching cylinder was simulated using homotopy analysis by Irfan et al. [39], who observed that the mass transport rate (Sherwood number) for the constructive–destructive chemical reaction is opposite in nature to the response computed for the thermal Biot number.

In high-temperature polymeric materials processing systems, radiative heat transfer arises in addition to convection and conduction heat transfers [40]. Film stretching processes often utilize radiative flux to manipulate heat transfer characteristics [41]. The general equation of radiative transfer is formidable to solve. In coupled radiative–convective–conductive flows, a simpler approach is required. Algebraic flux models are often used, and the popular ones among these (in order of increasing complexity) are the Rosseland diffusion flux model, Schuster–Schwartzchild two-flux model and Hamaker 6 flux model [42]. Rosseland’s diffusion flux model is valid for optically thick fluids that can absorb or emit radiation at their boundaries and is quite accurate for most polymers [42]. However, it neglects scattering effects and is restricted to gray fluids. Many studies have featured this model in both non-Newtonian and Newtonian radiative polymeric flow simulations. Ramana Murthy et al. [43] studied the radiative heat flux effects on the entropy generation in a channel flow of two immiscible couple stress (polar) fluids. Janardhana Reddy et al. [44] used Rosseland’s flux model and the Eringen micropolar rheological model to simulate the time-dependent thermal coating flow of an electroconductive polymer on a cylinder. Uddin et al. [45] used the Rosseland diffusion model to compute radiative flux effects on the nanofluid convection boundary layer flow from extending/retracting sheet surfaces. The Maxwell nanofluid flow from a convectively heated surface with the Rosseland radiative flux was investigated by Hayat et al. [46]. They observed that the temperature and boundary layer thickness are both elevated with a stronger thermal radiative effect, whereas the Nusselt number (rate of heat-transfer rate to the wall) is depleted. Akinbo and Olajuwon [47] scrutinized the chemically reacting viscoelastic magnetic boundary layer flow from a stretching sheet with Newtonian heating and Rosseland radiative flux. They observed that the flow is augmented (skin friction is increased) with a higher radiative effect and the Nusselt number is suppressed. The analysis of the Buongiorno nanofluid model considering the viscous fluid radiative flow was addressed by Mahanthesh [48]. His study discloses that the nanoparticles suspension upsurges thermal conductivity and, therefore, the heat flux at the surface decays while the temperature is augmented. Rana et al. [49] modeled the buoyancy-driven nanofluid slip flow subjected to thermal radiation. Their results illustrate that the nanofluid velocity improves due to quadratic thermal convection. The two-dimensional buoyancy-driven nanofluid convected flow with thermal radiation was investigated by Rana et al. [50]. Their analysis witnessed that radiation and nanoparticle aggregation aspects improve the temperature distribution. Sabu et al. [51] evaluated the thermally radiative magnetohydrodynamic nanomaterial flow under quadratic convection. They noticed that the heat-flux sensitivity towards inclination angle upsurges at a 0.5094% rate for higher radiation factors. The chemically reactive three-dimensional nanofluid radiative flow featuring Joule heating was examined by Swain and Mahanthesh [52]. They observed that the thermal field is improved significantly due to nanoparticles aggregation.

A review of the literature has revealed that thus far no study has examined the combined effects of the Maxwell viscoelastic behaviour, Rosseland radiative flux, wall transpiration, Newtonian heating and chemical reaction in a nonlinear rheological nanofluid flow from a stretching permeable surface. This is the novelty and focus of the present work. Both wall suction and injection cases are considered and heat generation/absorption is additionally included. The convective, thermal and solutal boundary conditions are utilized to provide a more comprehensive evaluation of temperature and concentration distributions. Buongiorno’s nanofluid model featuring thermophoretic and Brownian motion effects is deployed. A first-order chemical reaction is considered. Dimensionless equations are used to create the flow model by utilizing the appropriate parameters. The computed models are presented through the convergent Liao homotopic analysis method (HAM) approach [53] with the help of Mathematica-12 symbolic software. Validation of HAM solutions with special cases from the literature is included. The impact of various thermophysical, nanoscale and rheological parameters (e.g. Deborash viscoelastic parameter, mixed convection parameter, chemical reaction parameter, etc.) on transport characteristics is visualized graphically and interpreted in detail. Convergence studies for HAM approximations are also included. The present study constitutes a novel contribution to the non-Newtonian reactive polymer processing simulation and has not been communicated thus far in the scientific literature.

2 Reactive viscoelastic nanofluid flow model

A steady, incompressible, non-Newtonian Maxwell nanofluid boundary layer flow towards a stretchable sheet surface is considered as a model for the nanopolymer flow processing. The sheet is oriented along the x-direction. The flow is limited to y ≥ 0 and is visualized in Figure 1. It is presumed that u w ( x ) = c x is the stretchable surface velocity and u e ( x ) = e x is the external flow velocity where c > 0 and e > 0 . The velocity field is defined by V = [ u ( x , y ) , v ( x , y ) , 0 ] . The nanoparticles in the nanofluid are assumed to be chemically reacting and simulated using a first-order homogenous model. The nanofluid is also assumed to be optically thick and grey. The Rosseland radiation diffusion flux model is deployed to simulate thermal radiation and assumes that the intensity is the black-body intensity at the fluid temperature. Convective boundary conditions are applied at the wall (sheet) and transpiration (lateral masss flux) is present, i.e. suction injection. Under these approximations, the boundary layer approximation and the usual Boussinesq approximation, the governing conservation boundary layer equations for mass, momentum, energy and nanoparticle concentration, under thermosolutal buoyancy forces, may be derived by extending the analysis of Irfan et al. [39] to evaluate the radiative transfer, Newtonian heating and wall transpiration, as follows

The prescribed boundary conditions at the sheet (wall) and in the free stream take the form

Here, v = μ ρ f is the kinematic viscosity, ρ f is the nanofluid (polymer) density, μ is the dynamic viscosity, λ ₁ is the relaxation time, g is the gravitational acceleration, ( Λ 1 , Λ 3 ) denote the linear (thermal, concentration) expansion coefficients, ( Λ 2 , Λ 4 ) denote the nonlinear (thermal, concentration) expansion coefficients, α = k ( ρ c ) f is the thermal diffusivity, τ = ( ρ c ) p ( ρ c ) f is the heat capacity ratio with ( ρ c ) f being the liquid heat capacity and ( ρ c ) p the nanoparticles’ effective heat capacity, σ ∗ is the Stefan–Boltzmann constant, ( D T , D B ) are (thermophoresis, Brownian) the diffusion coefficients, K 1 is the reaction rate, k ∗ is the coefficient of radiative mean absorption, ( T , C ) is for the liquid (temperature, concentration), u w ( x ) is the stretching velocity, u e ( x ) is the free stream velocity, k is the thermal conductivity, c is the stretching rate, ( h s , h c ) is the convective (heat, mass) transfer coefficient, ( T ∞ , C ∞ ) is for the ambient liquid (temperature, concentration), respectively, and ( u , v ) denote the components of velocity in the ( x , y ) directions, respectively.

Figure 1

Physical model illustrating the reactive radiative–convective Maxwell nanofluid stretched flow.

Introducing the following similarity variables [55]:

Equation (1) is fulfilled automatically. Equations (2)–(6) reduce to the following self-similar momentum, thermal and concentration boundary layer equations with associated boundary conditions:

Here (′) designates differentiation with respect to the transformed dimensionless y-coordinate, i.e. η , β is the Deborah (viscoelastic) number, δ is the mixed convection variable, Gr x is the local thermal Grashof number (thermal buoyancy parameter), N is the ratio of concentration/thermal buoyancy, Gr x ⁎ is the local solutal Grashof number (solutal buoyancy parameter), ( β t , β c ) are nonlinear (thermal, concentration) convection variable quantities, A is the ratio of free stream velocity/stretching velocity, S denotes the wall lateral mass flux parameter:for suction ( S > 0 ) or for injection ( S < 0 ) , N t is a thermophoresis variable, Pr is the Prandtl number, ( γ 1 , γ 2 ) denotes the conjugate (heat, mass) transfer, R is a radiation parameter, N b is a Brownian motion parameter: ( γ > 0 ) for generative chemical reaction and ( γ < 0 ) for destructive chemical reaction and Sc is the Schmidt number. These variables are defined as follows:

The heat transfer rate and mass (concentration) transfer rate may be expressed using local Nusselt and Sherwood numbers, which take the following definitions:

In non-dimensional form, using the transformations defined in equation (7) the desired expressions are as follows:

Here, Re _x denotes the local Reynolds number based on the coordinate x.

3 Homotopy analysis method (HAM) solutions and convergence

To solve the nonlinear transformed boundary value problem, defined by equations (8)–(11), the powerful semianalytical power series expansion technique known as HAM is implemented [48]. This method achieves exceptional accuracy and can successfully accommodate nonlinear fluid mechanics simulations. It has been used to study Maxwell viscoelastic flows [54], Carreau reactive flows [55], thermosolutal transport in porous media [56], biological propulsion of Oldroyd-B liquids [57] and magneto-tribological micropolar flows [58]. For the present problem, the initial guesses ( f 0 ( η ) , θ 0 ( η ) , ϕ 0 ( η ) ) and auxiliary linear operators ( L f , L θ , L ϕ ) are defined as follows:

with

Here, C i ( i = 1 − 7 ) denote the arbitrary constants.

HAM is utilized to identify solutions in terms of convergence. This method involves an auxiliary parameter h , which facilitates us to modify the convergence area of f ″ ( 0 ) , θ ′ ( 0 ) , ϕ ′ ( 0 ) The supplementary parameter h demonstrates a critical role. Figures 2–4 show the “h curves”. It is observed that the approved values of h f , h θ and h ϕ in Figures 2–4 are − 1.4 ≤ h f ≤ − 0.2 , − 1.3 ≤ h θ ≤ − 0.1 and − 1.3 ≤ h ϕ ≤ − 0.1 . The convergences of velocity gradient f ″ ( 0 ) , temperature gradient θ ′ ( 0 ) and concentration gradient ϕ ′ ( 0 ) are presented in Table 1. It is evident that the 35th order of approximations is appropriate for f ″ ( 0 ) and θ ′ ( 0 ) , whereas the 25th order of approximation is necessary for ϕ ′ ( 0 ) .

Figure 2

H curve stimulus for f ″ ( 0 ) .

Figure 3

H curve stimulus for θ ′ ( 0 ) .

Figure 4

H curve stimulus for ϕ ′ ( 0 ) .

4 Authentication of HAM solutions

To validate the correctness of HAM computations, a comparison with several earlier studies from the literature is conducted. The comparison results of the skin friction coefficient in a limiting case agree very closely with the literature as displayed in Tables 2 and 3. Confidence in the present HAM solutions is therefore justifiably very high.

5 Analysis of results

Extensive HAM calculations have been conducted and are shown in Figures 5–19.

Figure 5

f ′ ( η ) versus A .

Figure 6

f ′ ( η ) versus β .

Figure 7

f ′ ( η ) versus δ .

Figure 8

f ′ ( η ) versus S .

Figure 9

f ′ ( η ) versus N .

Figure 10

θ ( η ) versus R .

Figure 11

θ ( η ) versus N t .

Figure 12

θ ( η ) versus N b .

Figure 13

θ ( η ) versus Pr .

Figure 14

θ ( η ) versus γ 1 .

Figure 15

ϕ ( η ) versus Sc .

Figure 16

ϕ ( η ) versus γ .

Figure 17

ϕ ( η ) versus γ 2 .

Figure 18

ϕ ( η ) versus N b .

Figure 19

ϕ ( η ) versus N t .

Figures 5–9 depict the velocity evolution f ′ ( η ) with the variations in A (ratio of the external velocity/stretching rate velocity, i.e. e c ), β (Deborah number), δ (mixed convection parameter), S (suction/injection parameter) and N (ratio of the concentration/thermal buoyancy force parameter). Figure 5 indicates that with increment in the velocity ratio, A, f ′ ( η ) experiences a strong enhancement, i.e. the flow is accelerated. When ( A > 1 ) , the stream velocity (e) exceeds the linear velocity of the sheet stretching (c). This induces a momentum boost via the external free stream that manifests in strong acceleration at all values of transverse coordinates, η. The momentum (hydrodynamic) boundary layer thickness is therefore reduced. However, for ( A < 1 ) , the stretching velocity exceeds the external free stream velocity and the opposite effect is computed, i.e. the velocity f ′ ( η ) is suppressed and the boundary layer thickness increases. When ( A = 1 ) , both external and stretching velocities are equal and this case naturally falls between the other two cases of A > 1 and A < 1. Clearly, greater stretching of the sheet is inhibitive to the momentum development, whereas a greater external velocity is assistive. A reflective symmetry is computed about the line A = 0 for A < 1 and A < 1. Figure 6. exhibits the influence of β on f ′ ( η ) and a substantial elevation in velocity is witnessed with greater values of the Deborah number. This parameter arises in the augmented shear terms in the momentum boundary layer equation (8), viz + β (2ff′f″ − f ² f‴). β = λ ₁ c and larger values of this parameter correspond to greater viscous effects and lower elasticity in the nanofluid polymer. This also implies a greater relaxation time of the nanofluid that assists momentum diffusion and accelerates the flow. The momentum boundary-layer thickness is consequently diminished with higher Deborah number values. The inclusion of a viscoelastic rheological model is therefore important since it captures acceleration effects, which would be neglected with Newtonian modelling in which the velocity is under-predicted. Figure 7 implies that with greater values of mixed convection variables δ , there is also an upsurge of velocity f ′ ( η ) . Clearly, the thermal buoyancy force is intensified with greater values of δ = ( Gr x ) ( Re x 2 ) . This energizes the nanopolymer boundary-layer regime and the flow is accelerated leading to a reduction in the momentum boundary-layer thickness. Figure 8 illustrates that with the increment in wall suction ( S > 0 ) , there is depletion in velocity magnitudes. Flow deceleration is induced since the nanopolymer boundary layer adheres more strongly to the wall and the momentum is reduced. The hydrodynamic boundary layer thickness is thereby enhanced in the regime. However, velocities are always positive, i.e. no flow reversal or separation is induced in the boundary layer. Conversely with increment in the injection factor ( S < 0 ) , there is a boost in the velocity owing to the introduction of greater momentum in the regime via blowing through the porous (permeable) stretching sheet. This accelerates the nanopolymer boundary layer flow and leads to a thinner momentum boundary layer thickness. With greater values of the species/thermal buoyancy ratio, N, a significant elevation in velocity occurs throughout the regime (see Figure 9). In all cases, N ≫ 1 implies that the nanoparticles’ species buoyancy force greatly exceeds the thermal buoyancy force. This generates a reduction in the momentum boundary layer thickness. In all the plots, asymptotically smooth profiles are achieved in the free stream, confirming that a sufficiently high value is prescribed in HAM computations for the infinity boundary condition.

Figures 10–14 illustrate the influence of the radiation parameter, R , on the thermophoresis variable, N t on the Brownian motion variable, N b on the Prandtl number, Pr and the conjugate heat transfer γ 1 , respectively, on temperature profiles, θ ( η ) . Figure 10 shows clearly that when R increases, θ ( η ) is significantly enhanced. A stronger radiative flux energizes the stretching nanopolymer regime and augments thermal diffusion. Moreover, it manifests a growth in the thermal boundary-layer thickness. Physically, the presence of stronger thermal radiation (higher R values) generates greater heat (thermal energy). Therefore, neglecting the radiative transfer would lead to an under-prediction in the temperature computed in the nanopolymer flow processing. Figure 11 indicates that with greater values of the nanoscale thermophoresis parameter, N t , there is an upsurge in the temperature, θ ( η ) . Greater mobility of nanoparticles towards colder zones in the boundary layer regime is induced under a temperature gradient with a stronger thermophoretic body force. This exacerbates thermodiffusion in the regime and heats the boundary layer, resulting in a greater thermal boundary-layer thickness. Maximum temperatures for any case examined always however arise at the wall, i.e. the sheet and all profiles decay monotonically to the free stream. Figure 12 demonstrates that with increment in the Brownian motion parameter N b , the temperature θ ( η ) is also elevated substantially. The intensification in ballistic collisions with greater Brownian motion effect leads to dissipation of the kinetic energy as thermal energy. This heats the regime and also enhances the thermal boundary layer thickness. Larger Nb values in the Buongiorno model imply smaller diameter spherical nanoparticles. There is an upsurge in the volume fraction, which leads to an exacerbation in chaotic nanoparticle collisions. The temperature of the nanopolymer can therefore be successfully manipulated with a change in the dimensions of the nanoparticles, as noted lucidly in Das et al. [9]. Nanopolymer constitution therefore exerts a significant role in the thermal field. With the increment in the Prandtl number, Pr , the temperature θ ( η ) is depleted. The Prandtl number expresses the relative rate of momentum diffusion to thermal diffusion in the boundary layer. It also expresses the ratio of the momentum boundary layer thickness/thermal boundary layer thickness. For Pr < 1, the energy diffusion rate exceeds the momentum (viscous) diffusion rate. An increment in the Prandtl number also implies a decrease in the thermal conductivity of the nanopolymer. This suppresses thermal convection in the boundary layer and decreases the temperature, i.e. induces a cooling effect. The effective Prandtl number of the nanopolymer can be manipulated by the selection of appropriate nanoparticles to modify thermal characteristics. Figure 14 reveals that with an upsurge in the conjugate thermal parameter γ 1 , there is an enhancement in the temperature, θ ( η ) . Additionally, the gradient of the temperature plot is strongly modified at the wall and it is reduced with greater values of the conjugate parameter. This effect is simulated via the modified convective boundary condition θ = −γ ₁(1 + θ(η)) featuring in equation (11), which quantifies the strength of Newtonian heating. Clearly, neglecting this effect prevents the correct near-wall thermal distribution from being computed accurately. The rate of change of temperature with the transverse coordinate cannot be simulated properly at the sheet surface and in proximity to it without a Newtonian heating condition. Stronger Newtonian heating produces a greater thermal boundary layer thickness than the conventional thermal boundary conditions at the wall. These trends have also been observed in a number of other investigations including Kamran et al. [35] and Khan et al. [37]. Again, it is noteworthy that the correct asymptotic profiles in the free stream are computed in all temperature distributions confirming that an adequately large infinity boundary condition has been deployed in the HAM analysis.

Figures 15–19 show the impact of the Schmidt number Sc, chemical reaction parameter γ , conjugate mass transfer parameter γ 2 , Brownian motion parameter N b , and thermophoresis parameter N t on the dimensionless nanoparticle concentration, ϕ ( η ) . Figure 15 implies that with an increase in Sc , there is a distinct depletion in the concentration profile ϕ ( η ) . This is accompanied with a decrease in the species boundary layer thickness. Physically, Sc is inversely proportional to the molecular (mass) diffusivity of the nanoparticles in the polymer base fluid. As Sc increases, the molecular diffusion rate decreases and this inhibits the transport of nanoparticles in the boundary layer regime. The concentration magnitudes are therefore reduced. Manipulation of the distribution of nanoparticles in the resulting nanopolymer can be achieved therefore by judicious selection of nanoparticles with a particular diffusivity. Nanopolymers can therefore be effectively engineered using carefully selected nanoparticles [9]. Figure 16 shows that with a greater destructive chemical reaction factor γ > 0 , the concentration ϕ ( η ) is depleted since a greater quantity of the original nanoparticles is converted to another species. The opposite trend is induced with a stronger generative (constructive) chemical reaction γ < 0 for which concentrations are increased and species boundary layer thickness is also elevated. The case of non-reactive flow (γ = 0) falls between these two types of chemical reactions. A homogenous chemical reaction is simulated via the first-order term, +Sc(−γϕ) in the species boundary layer equation (10). This term becomes assistive (positive) for γ < 0 and inhibitive (negative) for γ > 0 . Clearly, when the first-order homogenous chemical reaction is neglected in the model, this either under-predicts the concentration or overpredicts the concentration magnitudes, depending on whether destructive or constructive chemical reactions are being considered in nanopolymer processing. Figure 17 shows that as conjugate solutal (mass transfer) parameter γ ₂ is increased, there is a boost in the magnitudes of the concentration of nanoparticles ϕ ( η ) . This parameter arises in the augmented wall concentration boundary condition (11), i.e. ϕ = −γ ₂(1 + ϕ(η)). Higher values of γ 2 strongly increase the dimensionless concentration ϕ ( η ) and simultaneously reduce the concentration gradient at and near the wall. An upsurge in the species, i.e. the solutal boundary layer thickness is therefore produced with the stronger conjugate mass transfer effect. Figures 18 and 19 show that the elevation in thermophoresis N t and Brownian motion N b parameters, respectively, elevate and reduce ϕ ( η ) . The intensified migration of nanoparticles under a thermophoretic body force results in a more homogenous distribution of nanoparticles and a higher concentration value. However, greater chaotic motion via Brownian motion produces more ballistic collisions that inhibit nanoparticle migration and reduce their concentration and also the species boundary layer thickness in the viscoelastic nanopolymer.

Table 4 shows the local Nusselt number Nu x Re x − 1 2 values for selected parameters Pr , R , N b , N t , A , β , δ , N and γ 1 when β t = β c = 0.1 , S = 0.4 , γ = 0.3 , Sc = 1.2 , γ 2 = 0.2 . Here A < 1, i.e. the stretching velocity exceeds the external velocity. The local Nusselt number magnitudes clearly increase, i.e. the heat transfer rate to the wall improves with elevation in Pr , R , A , β and δ , whereas they are reduced with increment in γ 1 , N b , N t . Further, it is noticed that the parameter N does not significantly modify the heat transfer rate to the wall (porous sheet). Table 5 exhibits the local Sherwood number values Sh x Re x − 1 2 for various selected parameters. It is evident that the mass transfer rate to the wall is enhanced with greater values of Sc , γ and γ 2 , whereas it is depleted with increment in N t , N b .

6 Conclusions

Inspired by the recent developments in high-temperature nanopolymeric materials processing, a theoretical study has been presented to investigate the combined effects of the chemical reaction and radiative heat flux in a nonlinear mixed thermosolutal convection flow of a viscoelastic nanoliquid from a stretchable surface theoretically. Newtonian heating (modified thermal convective boundary conditions) has also been included. The upper-convected Maxwell (UCM) model has been deployed to represent non-Newtonian characteristics. The model also includes the influence of thermal radiation, which is simulated via an algebraic flux model. Buongiorno’s two-component nanofluid model has been implemented for thermophorestic and Brownian motion effects. The transformed, self-similar nonlinear boundary layer flow model with appropriate wall and free stream conditions has been solved with a convergent homotopic analysis method (HAM) approach with the help of Mathematica-12 symbolic software. Validation of HAM solutions with special cases from the literature has been included. A detailed parametric analysis highlighting the influence of all thermophysical, rheological and nanoscale factors on transport attributes has been elaborated. The principal findings of the present simulations can be summarized as follows:

1. Temperatures are strongly enhanced with the Brownian motion and thermophoresis parameter.

2. Velocity is boosted with increment in the Deborah viscoelastic number, and the mixed convection parameter and hydrodynamic boundary layer thickness are reduced.

3. Stronger generative chemical reaction enhances concentration magnitudes, whereas an increment in the destructive chemical reaction reduces them and also depletes the concentration boundary layer thickness.

4. Temperature is elevated with the conjugate thermal parameter and thermal boundary layer thickness is increased.

5. Nanoparticle concentration is boosted with greater values of the conjugate solutal parameter with an accompanying increase in the solutal boundary-layer thickness.

6. Greater radiative flux strongly upsurges the temperature and allied thermal boundary-layer thickness.

7. Stronger suction at the wall depletes velocity and increases the momentum boundary-layer thickness. However, an opposite result is induced for greater injection (blowing).

8. Higher external velocity relative to the stretching velocity induces strong flow acceleration, whereas the opposite case induces strong flow deceleration.

9. Increasing the Schmidt number and Brownian motion parameter diminishes the concentration values, whereas they elevate the local Sherwood number magnitudes, i.e. enhance the nanoparticle mass transfer rate to the wall.

10. Local Nusselt number magnitudes are boosted with elevation in the thermal buoyancy factor and Prandtl number, while they are suppressed with a greater conjugate thermal parameter, thermophoresis and Brownian motion parameter values.

Excellent stability, convergence and accuracy are achieved with HAM in nanopolymeric viscoelastic thermal flow processing simulation. The present study has neglected unsteady flow effects and also has been restricted to a single non-Newtonian model. Future investigations may consider time-dependent flow and other rheological models, e.g. Eringen’s micropolar model and the Oldroyd-B viscoelastic model. HAM appears to be an excellent tool for simulating such flows.