Nanofluid is an innovatively engineered heat transfer fluid that exhibits enhanced thermal conductivity at lower particle concentrations compared to conventional fluids. It is made by stably suspending and uniformly dispersing a few metallic or nonmetallic particles in ordinary heat transfer fluids. These particles are ultrafine and nano-scaled. Choi [1] developed the idea of nanofluid in 1995. Recent researchers have revealed that swapping traditional fluids with artificial nanofluids can be advantageous in enhancing the heat transfer performance in nuclear and engineering applications and cooling down automobile appliances [2] - [8] . Buongiorno [9] established a scientific model to explore heat transfer by convection in nanofluids. This model incorporates two crucial factors, namely Brownian and thermophoretic transmissions. Zangooee et al. [10] investigated the nanoparticle addition effects on the water flowing between two rotating plates using the Akbari-Ganji Method. They chose different nanoparticles which can enhance water’s thermodynamic and thermal properties. Seadawy et al. [11] clarified the analytical and mathematical dimensions of nanofluids having practical uses in mechanical devices. Rashidi et al. [12] analysed the entropy generation ratio in nanofluid drift concerning clutching disks.

Furthermore, the mutual effect of conductive fluids and electromagnetic fields represents magneto-hydrodynamics (MHD). Various purposes of nanofluids originated in industrial and engineering premises, primarily in the heat conversation devices design, accelerators, MHD generators, etc. [13] - [17] . Pavar et al. [18] discussed the convective MHD flow carried over a permeable exponentially prolonging surface, giving importance in clinical and medical research to generate 3D anatomical images from nuclear magnetic resonance signals. Li et al. [19] investigated the axisymmetric transient squeezing MHD flow of the Newtonian non-conducting fluid over a porous structure retaining the slip condition at the periphery.

A fascinating and unsettled tribological subject entails the viscoelasticity impact on thin-film flows. The preparation of mixing polymers with mineral lubricants has been renowned later the mid-1990s [20] - [22] . These mixtures drive the consequential oils to behave like non-Newtonian fluids by performing a viscosity dependency on the shear rate [23] - [25] . The classic Newtonian prototype covering the Navier-Stokes equations is unable to exhibit the extremely non-linear bondings between tangential stress and strain rate of non-Newtonian lubricants [26] - [30] . Harris [20] established the upper convected Maxwell Fluid (UCM) features to exhibit the lubricant behaviour of a non-Newtonian fluid. Upper convected Maxwell fluid is a regular rate-type viscoelastic material with fluid relaxation time features, namely the viscosity-modulus of elasticity ratio. It rejects the multifaceted properties of shear-related viscosity. It also establishes the effect of fluid elasticity on the physical appearance of its boundary layer [27] [31] . Due to the explosion of concrete applications in bioengineering and plastic manufacturing, paper fabrication, and food processing, investigators have paid attention to studying the boundary layer theory of non-Newtonian lubricant flow [32] - [36] .

Lie group transformation scheme is a developing field of mathematics with numerous purposes. Norwegian mathematician Sophus Lie developed a classic Lie group transformation scheme to discover invariant and similarity solutions [37] - [42] . The Lie group study offers an applicable method to concentrate on nonlinear equations. This method recommends a particular mathematical presentation of observant opinions of symmetry and deals with essential techniques for analysing non-linear differential systems. Investigators are now exploring the transformations of the Lie group for Newtonian and non-Newtonian fluid flows [40] [43] - [45] .

However, the Maxwell model enlightens the time-dependent stress relaxation of fluid accommodating substantial nanoparticles; the non-Newtonian Nanofluid with developed heat transfer has countless prospects to influence manufacturing and hemodynamic rationales. Even if the higher heat transfer procedure in non-Newtonian nanofluids is crucial, a study has been performed in this investigation. By perceiving the prior research, the principal attention is to analyse the time-dependent two-dimensional elastico-viscous nanofluid stream in two parallel convective stretched walls incorporating Maxwell’s rheological Nanofluid model using the Lie group scheme.

An unsteady elastico-viscous nanofluid flow is engaged through two parallel walls to demonstrate the physical configuration. The upper surface is stretched and porous, and the lower has convective boundary conditions. The Cartesian coordinate system is accommodated to explain the physical model. The $\bar{x}$-axis is occupied along the walls, and the $\bar{y}$-axis is upright to the walls, as displayed in Figure 1 .

The viscoelastic activities of fluid will be observed if elastic stress is applied, and the resultant strain will be contingent upon time, featuring the relaxation phase. The constitutive equation [27] of time-dependent stress relaxation is

$\tau =-pI+S$ (1)

When an extra stress tensor $S$ is applied, the elastico-viscous behaviour of the nanofluid, demonstrated by the Maxwell model [34] , is given by

$S+{\lambda }_S\left(\frac{\text{d}S}{\text{d}t}+V\cdot \nabla S-LS-S{L}^{\text{tr}}\right)={\mu }_{nf}\left(L+L^{\text{tr}}\right)$ (2)

Here $V$ means the nanofluid’s velocity, $L^{\text{tr}}=\nabla V$ is the velocity gradient, the ‘tr’ signifies a transpose, $L_{ij}=\frac{\partial u_i}{\partial x_j}$, ${\lambda }_S>0$ is the stress reduction time of non-Newtonian fluids, and ${\lambda }_S=0$ provides Newtonian fluids. Then the equations of flow are given by

$\frac{\partial \bar{u}}{\partial \bar{x}}+\frac{\partial \bar{v}}{\partial \bar{y}}=0$ (3)

$\begin{array}{l}\frac{\partial \bar{u}}{\partial \bar{t}}+\bar{u}\frac{\partial \bar{u}}{\partial \bar{x}}+\bar{v}\frac{\partial \bar{u}}{\partial \bar{y}}+{\lambda }_S\left(\frac{{\partial }^2\bar{u}}{\partial {\bar{t}}^2}+2\bar{u}\frac{{\partial }^2\bar{u}}{\partial \bar{t}\partial \bar{x}}+2\bar{v}\frac{{\partial }^2\bar{u}}{\partial \bar{t}\partial \bar{y}}+{\bar{u}}^2\frac{{\partial }^2\bar{u}}{\partial {\bar{x}}^2}+{\bar{v}}^2\frac{{\partial }^2\bar{u}}{\partial {\bar{y}}^2}+2\bar{u}\bar{v}\frac{{\partial }^2\bar{u}}{\partial \bar{x}\partial \bar{y}}\right)\\ =-\frac{1}{\rho }\frac{\partial \bar{p}}{\partial \bar{x}}+\nu \frac{{\partial }^2\bar{u}}{\partial {\bar{y}}^2}-\frac{\sigma B_0^2}{\rho }\left(\bar{u}+{\lambda }_S\left(\frac{\partial \bar{u}}{\partial \bar{t}}+\bar{v}\frac{\partial \bar{u}}{\partial \bar{y}}\right)\right)\end{array}$

(4)

Again, considering the Buongiorno model integrating thermophoresis and Brownian diffusion effects [9] [46] , the equations of energy and nanoparticle concentration are

$\begin{array}{l}\frac{\partial \bar{T}}{\partial \bar{t}}+\bar{u}\frac{\partial \bar{T}}{\partial \bar{x}}+\bar{v}\frac{\partial \bar{T}}{\partial \bar{y}}\\ =\frac{\kappa }{\rho c_p}\left(\frac{{\partial }^2\bar{T}}{\partial {\bar{y}}^2}\right)+\tau \left(D_B\left(\frac{\partial \bar{T}}{\partial \bar{y}}\frac{\partial \bar{C}}{\partial \bar{y}}\right)+\frac{D_T}{T_a}{\left(\frac{\partial \bar{T}}{\partial \bar{y}}\right)}^2\right)+\frac{\mu }{\rho c_p}\left({\left(\frac{\partial \bar{u}}{\partial \bar{y}}\right)}^2\right)\end{array}$ (5)

$\frac{\partial \bar{C}}{\partial \bar{t}}+\bar{u}\frac{\partial \bar{C}}{\partial \bar{x}}+\bar{v}\frac{\partial \bar{C}}{\partial \bar{y}}=D_B\frac{{\partial }^2\bar{C}}{\partial {\bar{y}}^2}+\frac{D_T}{T_a}\frac{{\partial }^2\bar{T}}{\partial {\bar{y}}^2}$ (6)

The resolutions of the model differ with the nature of the prescribed thermal boundary condition [32] . The corresponding boundary conditions are:

Case 1: Prescribed surface temperature (PST)

$\begin{array}{l}\bar{u}={\gamma }_h{u}_w,\bar{v}=v_h,\bar{T}=T_l,\bar{C}=C_l\text{at}\bar{y}=l\\ \bar{u}=0,\bar{v}=0,\bar{T}={\bar{T}}_0,\bar{C}={\bar{C}}_0\text{at}\bar{y}=0\end{array}\}$ (7)

Case 2: Prescribed convective boundary (PCB) case

$\begin{array}{l}\bar{u}={\gamma }_h{u}_w,\bar{v}=v_h,\bar{T}=T_l,\bar{C}=C_l\text{at}\bar{y}=l\\ \bar{u}=0,\bar{v}=0,-\kappa \frac{\partial \bar{T}}{\partial \bar{y}}=h_f\left(T_l-\bar{T}\right),-D_B\frac{\partial \bar{C}}{\partial \bar{y}}=h_f^{*}\left(C_l-\bar{C}\right)\text{at}\bar{y}=0\end{array}\}$ (8)

Here $T_l=T_0+\left(T_h-T_0\right)\frac{x}{{\left(1-t\right)}^2}$ and $C_l=C_0+\left(C_h-C_0\right)\frac{x}{{\left(1-t\right)}^2}$ are the temperature and the dilution at the upper plate, supposed to vary along the outward and in time, respectively.

The following dimensionless variables are used to make the conventional equations non-dimensional.

$t=\bar{t}\frac{u_w}{h},x=\frac{\bar{x}}{h},y=\frac{\bar{y}}{h},u=\frac{\bar{u}}{u_w},v=\frac{\bar{v}}{u_w},p=\frac{\bar{p}}{p_0},T=\frac{\bar{T}-T_0}{T_h-T_0},C=\frac{\bar{C}-C_0}{C_h-C_0}$ (9)

Then the dimensionless PDE model is specified by the following equations.

$\begin{array}{l}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+\frac{u_w{\lambda }_S}{h}\left(\frac{{\partial }^{2}u}{\partial t^2}+2u\frac{{\partial }^{2}u}{\partial t\partial x}+2v\frac{{\partial }^{2}u}{\partial t\partial y}+u^2\frac{{\partial }^{2}u}{\partial x^2}+v^2\frac{{\partial }^{2}u}{\partial y^2}+2uv\frac{{\partial }^{2}u}{\partial x\partial y}\right)\\ =-\frac{p_0}{u_w{}^2\rho }\frac{\partial \bar{p}}{\partial \bar{x}}+\frac{1}{\mathrm{Re}}\frac{{\partial }^2\bar{u}}{\partial {\bar{y}}^2}-\frac{\sigma B_0{}^{2}h}{\rho u_w}\left(u+\frac{u_w{\lambda }_S}{h}\left(\frac{\partial u}{\partial t}+v\frac{\partial u}{\partial y}\right)\right)\end{array}$

(10)

$\begin{array}{l}\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\\ =\frac{1}{PrRe}\left(\frac{{\partial }^{2}T}{\partial y^2}\right)+\frac{1}{Re}\left(N{b}_x\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}+N{t}_x{\left(\frac{\partial T}{\partial y}\right)}^2\right)+\frac{E{c}_x}{Re}{\left(\frac{\partial u}{\partial y}\right)}^2\end{array}$ (11)

$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=\frac{1}{ReLe}\frac{{\partial }^{2}C}{\partial y^2}+\frac{N{t}_x}{ReLeN{b}_x}\frac{{\partial }^{2}T}{\partial y^2}$ (12)

Let $\psi$ be the stream function. Set $u=\frac{\partial \psi }{\partial y}$ and $v=-\frac{\partial \psi }{\partial x}$ in the above equations to get

$\begin{array}{l}\frac{{\partial }^2\psi }{\partial t\partial y}+\frac{\partial \psi }{\partial y}\frac{{\partial }^2\psi }{\partial x\partial y}-\frac{\partial \psi }{\partial x}\frac{{\partial }^2\psi }{\partial y^2}+\frac{u_w{\lambda }_S}{h}(\frac{{\partial }^3\psi }{\partial t^2\partial y}+2\frac{\partial \psi }{\partial y}\frac{{\partial }^3\psi }{\partial t\partial x\partial y}\\ -2\frac{\partial \psi }{\partial x}\frac{{\partial }^3\psi }{\partial t\partial y^2}+{\left(\frac{\partial \psi }{\partial y}\right)}^2\frac{{\partial }^3\psi }{\partial x^2\partial y}+{\left(\frac{\partial \psi }{\partial x}\right)}^2\frac{{\partial }^3\psi }{\partial y^3}-2\frac{\partial \psi }{\partial y}\frac{\partial \psi }{\partial x}\frac{{\partial }^3\psi }{\partial x\partial y^2})\\ =-\frac{p_0}{u_w^2\rho }\frac{\partial p}{\partial x}+\frac{1}{Re}\frac{{\partial }^3\psi }{\partial y^3}-\frac{\sigma B^{2}h}{\rho u_w}\left(\frac{\partial \psi }{\partial y}+\frac{u_w{\lambda }_S}{h}\left(\frac{{\partial }^2\psi }{\partial t\partial y}-\frac{\partial \psi }{\partial x}\frac{{\partial }^2\psi }{\partial y^2}\right)\right)\end{array}$ (13)

$\begin{array}{l}\frac{\partial T}{\partial t}+\frac{\partial \psi }{\partial y}\frac{\partial T}{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial T}{\partial y}\\ =\frac{1}{RePr}\left(\frac{{\partial }^{2}T}{\partial y^2}\right)+\frac{1}{Re}\left(N{b}_x\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}+N{t}_x{\left(\frac{\partial T}{\partial y}\right)}^2\right)+\frac{E{c}_x}{Re}{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^2\end{array}$ (14)

$\frac{\partial C}{\partial \bar{t}}+\frac{\partial \psi }{\partial y}\frac{\partial C}{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial C}{\partial y}=\frac{1}{ReLe}\frac{{\partial }^{2}C}{\partial y^2}+\frac{N{t}_x}{ReLeN{b}_x}\frac{{\partial }^{2}T}{\partial y^2}$ (15)

Dimensionless boundary conditions for prescribed surface temperature (PST) are

$\begin{array}{l}\frac{\partial \psi }{\partial y}={\gamma }_h,\frac{\partial \psi }{\partial x}=-\frac{v_h}{u_w},T=\frac{x}{{\left(1-t\right)}^2},C=\frac{x}{{\left(1-t\right)}^2}\text{at}y=\frac{l}{h}\\ u=0,v=0,T=0,C=0\text{at}y=0\end{array}\}$ (16)

Here $Re=\frac{u_{w}h}{\nu }$ is the Reynolds number, $Pr=\frac{\nu \rho Cp}{\kappa }$ is the Prandtl number; $E{c}_x=\frac{u_w^2}{C_p\left(T_h-T_0\right)}$ is the Eckert number; $N{b}_x=\frac{\tau D_B\left(C_h-C_0\right)}{\nu }$ is the Brownian motion parameter; $N{t}_x=\frac{\tau D_T\left(T_h-T_0\right)}{\nu T_a}$ is the thermophoresis parameter; $Le=\frac{\nu }{D_B}$ is the Lewis number.

The prominence of similarity solutions in many areas of science and engineering is limitless. Looking for a universal symmetric approach that can be applied to specific mathematical models is obligatory. The Lie group transformation procedure is a sound technique for the theory of the continuous symmetry of numerical structures, which is immensely functional for various fields of modern mathematical physics. This analysis is anticipated to deliver a new methodology for investigating the continuous symmetries of model equations leading to heat transfer fluxes in elastico-viscous nanofluids. In the process, the Lie group theory trims down the number of independent parameters of the central PDEs reflected in the physical model. It also maintains the invariant structure of the model with the corresponding initial and boundary conditions. Because of the unsteady flow ( $0<t<1$) in a channel, the following two-parameter linear groups of transformations are to be considered for the Lie group analysis [47] :

$\begin{array}{l}\Gamma :\hat{t}=\left(1-t\right)e^{{\alpha }_1},\hat{x}=x{e}^{{\beta }_1},\hat{\psi }=\psi e^{{\alpha }_2}{e}^{{\beta }_2},\hat{y}=y{e}^{{\alpha }_3}{e}^{{\beta }_3},\hat{T}=T{e}^{{\alpha }_4}{e}^{{\beta }_4},\hat{C}=C{e}^{{\alpha }_5}{e}^{{\beta }_5},\\ \hat{p}=p{e}^{{\alpha }_6}{e}^{{\beta }_6},{\hat{\lambda }}_S={\lambda }_S{e}^{{\alpha }_7}{e}^{{\beta }_7},{\hat{B}}_0=B_0{e}^{{\alpha }_8}{e}^{{\beta }_8},E\hat{c}=E{c}_x{e}^{{\alpha }_9}{e}^{{\beta }_9},{\hat{\gamma }}_h={\gamma }_h{e}^{{\alpha }_{10}}{e}^{{\beta }_{10}},\\ {\hat{v}}_h=v_h{e}^{{\alpha }_{11}}{e}^{{\beta }_{11}},\hat{l}=l{e}^{{\alpha }_{12}}{e}^{{\beta }_{12}},\hat{N}b=N{b}_x{e}^{{\alpha }_{13}}{e}^{{\beta }_{13}},\hat{N}t=N{t}_x{e}^{{\alpha }_{14}}{e}^{{\beta }_{14}},\\ {\hat{h}}_f={\bar{h}}_f{e}^{{\alpha }_{15}}{e}^{{\beta }_{15}},{\hat{h}}^{*}{}_f={\bar{h}}^{*}{}_f{e}^{{\alpha }_{16}}{e}^{{\beta }_{16}}\end{array}$

(17)

Here ${\alpha }_i,{\beta }_i\left(i=1,2,\cdots ,16\right)$ are the constants taken in such a way that the forms of the Equations (13)-(15) are invariant under the transformations connected given in the following relations

$\begin{array}{l}{\alpha }_2=-\frac{1}{2}{\alpha }_1,{\alpha }_3=\frac{1}{2}{\alpha }_1,{\alpha }_4=-2{\alpha }_1,{\alpha }_5=-2{\alpha }_1,{\alpha }_6=-2{\alpha }_1,{\alpha }_7={\alpha }_1,{\alpha }_8=-\frac{1}{2}{\alpha }_1,{\alpha }_9=0,\\ {\alpha }_{10}=-{\alpha }_1,{\alpha }_{11}=-\frac{1}{2}{\alpha }_1,{\alpha }_{12}=\frac{1}{2}{\alpha }_1,{\alpha }_{13}=2{\alpha }_1,{\alpha }_{14}=2{\alpha }_1,{\alpha }_{15}=-\frac{1}{2}{\alpha }_1,{\alpha }_{16}=-\frac{1}{2}{\alpha }_1\end{array}$

$\begin{array}{l}{\beta }_2={\beta }_1,{\beta }_3=0,{\beta }_4={\beta }_1,{\beta }_5={\beta }_1,{\beta }_6=2{\beta }_1,{\beta }_7=0,{\beta }_8=0,{\beta }_9=-{\beta }_1,\\ {\beta }_{10}={\beta }_1,{\beta }_{11}=0,{\beta }_{12}=0,{\beta }_{13}=-{\beta }_1,{\beta }_{14}=-{\beta }_1,{\beta }_{15}=0,{\beta }_{16}=0\end{array}$ (18)

With these relationships of ${\alpha }_i,{\beta }_i$ Equation (17) turns into

$\begin{array}{l}\Gamma :\hat{t}=\left(1-t\right)e^{{\alpha }_1},\hat{x}=x{e}^{{\beta }_1},\hat{\psi }=\psi e^{-\frac{1}{2}{\alpha }_1}{e}^{{\beta }_1},\hat{y}=y{e}^{\frac{1}{2}{\alpha }_1},\hat{T}=T{e}^{-2{\alpha }_1}{e}^{{\beta }_1},\hat{C}=C{e}^{-2{\alpha }_1}{e}^{{\beta }_1},\\ \hat{p}=e^{-2{\alpha }_1}{e}^{2{\beta }_1},\hat{\lambda }=\lambda e^{{\alpha }_1}{e}^0,{\hat{B}}_0=B_0{e}^{-\frac{1}{2}{\alpha }_1}{e}^0,E\hat{c}=E{c}_x{e}^0{e}^{-{\beta }_1},{\hat{\gamma }}_h={\gamma }_h{e}^{-{\alpha }_1}{e}^{{\beta }_1},\\ {\hat{v}}_h=v_h{e}^{-\frac{1}{2}{\alpha }_1}{e}^0,\hat{l}=l{e}^{\frac{1}{2}{\alpha }_1}{e}^0,\hat{N}b=Nb{e}^{2{\alpha }_1}{e}^{-{\beta }_1},\hat{N}t=Nt{e}^{2{\alpha }_1}{e}^{-{\beta }_1},{\hat{h}}_f={\bar{h}}_f{e}^{-\frac{1}{2}{\alpha }_1},\\ {\hat{h}}^{*}{}_f={\bar{h}}^{*}{}_f{e}^{-\frac{1}{2}{\alpha }_1}\end{array}$

(19)

From the absolute invariants, the similarity parameters are designated as

$\begin{array}{l}\eta =\frac{y}{\sqrt{1-t}},\psi =\frac{x}{\sqrt{1-t}}f\left(\eta \right),p=\frac{x^2}{{\left(1-t\right)}^2}{p}_{\eta },{\lambda }_S=\left(1-t\right){\lambda }_0,B_0=\frac{B}{\sqrt{1-t}},\\ E{c}_x=x^{-1}Ec,T=\frac{x}{{\left(1-t\right)}^2}\theta \left(\eta \right),C=\frac{x}{{\left(1-t\right)}^2}F\left(\eta \right),{\gamma }_h=\frac{x}{1-t}\gamma ,v_h=\frac{v_w}{\sqrt{1-t}},\\ l=\sqrt{1-t}h,N{b}_x=\frac{{\left(1-t\right)}^2}{x}Nb,N{t}_x=\frac{{\left(1-t\right)}^2}{x}Nt,{\bar{h}}_f=h_f{\left(1-t\right)}^{-\frac{1}{2}},{\bar{h}}_f^{*}=h_f^{*}{\left(1-t\right)}^{-\frac{1}{2}}\end{array}$

(20)

Using the above similarity transformations in Equations (13)-(15), we obtain the following similarity equations,

$\begin{array}{l}\left(\frac{1}{Re}-\frac{{\beta }_S}{4}\left({\eta }^2-4\eta f+4f^2\right)\right)f^{iv}-\frac{1}{2}\left(\eta f^{‴}+3f^{″}+2f^{\prime }{f}^{″}-2f{f}^{‴}\right)\\ -\frac{{\beta }_S}{4}\left(9\eta f^{‴}+4\eta {f^{″}}^2-16f{f}^{‴}+8f^{\prime }{f}^{″}-8f{f^{″}}^2-8{f^{\prime }}^2{f}^{″}+15f^{″}\right)\\ -M\left(f^{″}+\frac{{\beta }_S}{2}\left(\eta f^{‴}+3f^{″}-2f{f}^{‴}-2f^{\prime }{f}^{″}\right)\right)=0\end{array}$ (21)

${\theta }^{″}-PrRe\left(\frac{\eta }{2}{\theta }^{\prime }+2\theta +f^{\prime }\theta -f{\theta }^{\prime }\right)+Pr\left(Nb{\theta }^{\prime }{F}^{\prime }+Nt{{\theta }^{\prime }}^2\right)+PrEc{f^{″}}^2=0$ (22)

$F^{″}-ReLe\left(\frac{\eta }{2}{F}^{\prime }+2F+f^{\prime }F-f{F}^{\prime }\right)+\frac{Nt}{Nb}{\theta }^{″}=0$ (23)

And the transformed PST boundary conditions are

$\begin{array}{l}{f}^{\prime }=\gamma ,f=f_w,\theta =1,F=1\text{at}\eta =1\\ f^{\prime }=0,f=0,\theta =0,F=0\text{at}\eta =0\end{array}\}$ (24)

Following a similar procedure, the PCB boundary conditions take the form

$\begin{array}{l}{f}^{\prime }=\gamma ,f=f_w,\theta =1,F=1\text{at}\eta =1\\ f^{\prime }=0,f=0,{\theta }^{\prime }=-Bi\left(1-\theta \right),F^{\prime }=-B{i}^{*}\left(1-F\right)\text{at}\eta =0\end{array}\}$ (25)

Here ${\beta }_S=\frac{u_w{\lambda }_S}{h}$ is the Deborah number; $M=\frac{B_0^2\sigma h}{\rho u_w}$ is the Magnetic field parameter; $\gamma$ is the stretching parameter; $f_w=\frac{v_w}{u_w}$ is the suction parameter; $Bi=\frac{h_{f}h}{\kappa }$ and $B{i}^{*}=\frac{h_f^{*}h}{\kappa }$ are the Biot numbers for convective heat and mass transfer, respectively.

Finally, the skin friction coefficient estimates the friction force functional to the surface, and the Nusselt number characterises the heat flux from a heated surface to a fluid. The skin friction coefficient (C_f), the local Nusselt number (Nu), and mass transfer (Sh) are defined as

$C_f\propto f^{″},Nu\propto {\theta }^{\prime },Sh\propto F^{\prime }$ (26)

The fourth-order collocation approach based on cubic polynomials is used by the bvp4c package, which offers a fair trade-off between computing cost and accuracy. It is appropriate for a variety of engineering challenges due to its ability to efficiently solve nonlinear differential equations and nonlinear boundary conditions. One of its important characteristics is automated mesh adaptation, which maintains accuracy without needless processing by fine-tuning the solution grid where necessary. bvp4c, which is integrated into MATLAB, features an easy-to-use interface and supports multipoint boundary conditions, vectorised formulations, and parameter continuation. bvp4c is dependable and quick for issues with a moderate number of differential equations.

For boundary value issues requiring convective or specified temperature boundary conditions in heat transfer, bvp4c and its Collocation Method are perfect. It can be used to represent issues with elastic stability, vibration, and beam deflection. It is also employed to address flow in porous media and laminar boundary layer issues. Modelling thermal profiles in engine parts, brake systems, or even battery thermal management is one of the main applications for this technique.

Equations (21)-(23) combined with the boundary conditions Equation (24) and Equation (25) individually are solved numerically using the collocation method developed by a MATLAB package, known as bvp4c. The analysis is made for various parameters like Deborah number or Maxwell parameter ( ${\beta }_S$), stretching parameter ( $\gamma$), suction parameter ( $f_w$), magnetic field parameter ( $M$), Prandtl number ( $Pr$), Eckert number ( $Ec$), thermophoresis parameter ( $Nt$), Brownian motion parameter ( $Nb$), and Lewis number ( $Le$). The mesh size is $\eta =0.01$, and the tolerance factor is set 10⁻⁶. Based on the contemporary model, [0, 1] is considered the channel problem domain.

The renovation of the model equations trims down the mathematical work extensively. Graphical representations of consequences are very constructive in discussing the physical features offered by the solutions.

In nanofluid systems, the Brownian diffusion of particles is an essential factor in studying the nanoparticle effects on the flow field, temperature and nanoparticle volume fraction (NVF) distributions since the particle size approaches the nanometer scale. Brownian diffusion is accompanied by the arbitrary motion of particles suspended in a conventional fluid after collision with the molecules of the base fluid, which is an imperative mechanism to improve heat transmission. Figure 2 and Figure 3 illustrate the impact of Brownian motion parameter on thermal and mass boundary layers, respectively, for both cases: PST (solid line) and PCB (dashed line). From Figure 2 , it is noticed that the temperature distribution enhances as Nb improves in both cases. The higher Brownian motion parameter yields enhanced diffusion; that’s why the collision of particles increases. So, the temperature is raised after collation and produces an augmented thermal boundary layer; an analogous conclusion was expressed by Khan and Pop [48] in their work.

Figure 3 demonstrates that the NVF initially increases for higher Nb near the lower wall. But apart from the lower wall after $\eta =0.22$, those profiles reverse the trend and begin to fall due to growing Nb for the PST state. Again, as the Brownian motion parameter is higher, the NVF reduces throughout the flow regime for the PCB boundary condition; a parallel discussion is ended by Aziz and Khan [49] .

Thermophoresis diffusion occurs due to a temperature gradient force, for which the nanoparticles in the mixture reveal dissimilar reactions. The local temperature in the thermal boundary layer improves laterally with the liquid flow regime since the thermophoresis parameter Nt increases, as revealed in Figure 4 , for both PST (continuous line) and PCB (discontinuous line) boundary conditions. Eldabe and Abou-zeid [50] discovered a similar result in 2017. Figure 5 indicates that the NVF profile raises for higher thermophoresis parameter Nt throughout the flow system for PCB boundary conditions (dashed line). But the distribution of NVF decreases in the domain $\eta \in \left[0,0.23\right]$ and increases in the remaining regime for higher Nt for PST boundary conditions (solid line).

The Eckert (Ec) number states the affiliation of kinetic energy and the enthalpy variance of the boundary layer in a fluid flow. A considerable Eckert number produces an escalation in the kinetic energy of nanofluid particles. Accordingly, the fluid heats up, and the temperature rises. Figure 6 illustrates that the Eckert number develops the heat transportation in the liquid, which is a vital tool for the advancement of energy transference in nanofluids, in accordance with Atif et al. [51] . Figure 7 indicates the NVF profile dropping for higher Eckert numbers over the PST boundary condition flow system (solid line). But the outline of the nanoparticles volume fraction escalates in the domain $\eta \in \left[0.341,0.718\right]$ and declines in the remaining regime for higher Ec for PCB boundary conditions (dashed line).

The Deborah number ( ${\beta }_S$) is a dimensionless variable that treats the fluid relaxation time to its characteristic time scale. Here ${\beta }_S=0$ gives the incompressible Newtonian fluid feature. Liquids with a small Deborah number have a liquid-like behaviour, while a large Deborah number communicates that solids-like substances can better conduct and retain heat. Therefore, it is witnessed that regularly increasing the Deborah number boosts the fluid viscosity, which increases the confrontation in momentum. Consequently, the hydrodynamic boundary layer thickness reduces because of the Maxwell fluid, as revealed in Figure 8 , leading to a good connection with Bilal et al. [52] . No deviation is created for different boundary conditions. An exciting fact, i.e., a crosswise flow occurs here between $\eta =0.22$ and $\eta =0.71$.

Figures 9-11 represent the consequences of suction (f_w) on momentum, thermal, and NVF distributions, respectively. It is particularly from these figures that momentum and thermal distributions are increasing due to higher suction for both the PST and PCB boundary surfaces. Physically, the imposed suction brings a force to the liquid in the waterway, improving the fluid’s movement and energy. In addition, the NVF profile declines with the more incredible suction.

From Figure 12 , it is discovered that the wall stretching factor enhances the fluid velocity to increase the momentum distribution with the stretching effect ( $\gamma$) growth nearby the upper wall, where the momentum distribution is reduced within the flow regime $\eta \in \left[0,0.732\right]$. As a result, across-flow occurs.

The magnetic field parameter (M) effects on momentum are demonstrated in Figure 13 for both instances. It is developed from the magnetic field strength imposed on an electrically conductive fluid, which generates a drag force called the Lorentz force against the flow direction along the surface to slow down the velocity. It is perceived that a rise in the magnetic field parameter values monotonically

reduces the momentum distribution. That is why the imposed magnetic field is responsible for smoothing the fluid flow velocity. Finally, also crosswise flow exists for magnetic field parameter variation.

Additionally, the effect of the stress relaxation parameter on the skin fraction coefficient ( $f^{″}$), the local Nusselt number ( $-{\theta }^{\prime }$) and the Sherwood number ( $-F^{\prime }$) are arranged in Table 1 for PST and Table 2 for PCB, considering $Re=1$, $M=1$, $Pr=21$, $Ec=0.1$, $Le=5$, $Nb=0.16$, $Nt=0.1$, $\gamma =0.1$, $f_w=0.1$, $Bi=0.1$ and $B{i}^{*}=0.3$.

Heat Transfer Rate for Brownian Motion Parameter

Figure 14 is prepared to portray the consequence of the Brownian motion parameter (Nb) on the Nusselt number (Nu) at the lower wall $\eta =0$ for the Deborah number ( ${\beta }_S$) for PST and PCB. It is noticeable from these figures that the heat transmission rate decays for the progressive values of the Brownian motion parameter (Nb) as the Deborah number increases. Figure 15 is arranged to define the control of the Brownian motion parameter on the Nusselt number at the upper surface $\eta =1$ over the Deborah number for PST and PCB. It is visible from these figures that the energy transfer rate (Nu) is enhanced for the more significant Brownian motion parameter as the Deborah number develops.

Table 1 is tabulated to investigate the mathematical computations of friction drag coefficient $f^{″}$, heat transfer ( $-{\theta }^{″}$), and mass transfer ( $-F^{\prime }$) using dissimilar values of Deborah number ( ${\beta }_S$) for PST conditions at the lower ( $\eta =0$) and upper ( $\eta =1$) plates, respectively. The local skin-friction factor increases with the lessening of the boundary layer. At the lower plate ( $\eta =0$), the friction drag and mass transfer rate escalate because of the greater Deborah number, while the heat transmission rate decreases. However, an opposite approach is seen at the upper plate ( $\eta =1$).

Table 2 describes the numerical estimations of friction drag coefficient, heat and mass transmission rates using the Deborah number for prescribed convective boundary (PCB) conditions at the lower ( $\eta =0$) and upper ( $\eta =1$) plates, respectively.

In automobile engineering, accurate thermal analysis is critical for ensuring the performance, safety, and longevity of various components. Two important thermal boundary conditions often applied in simulations and experimental setups are PST and PCB conditions. These conditions represent controlled thermal environments that allow engineers to analyse heat transfer behaviour under specific scenarios. PST condition is particularly useful when studying systems like engine blocks, exhaust manifolds, or battery packs, where surface temperature data can be measured or estimated reliably. PCB condition, on the other hand, is appropriate for simulating and designing effective thermal management systems that require more realistic interactions between solid components and cooling medium, such as air or coolant, which are reflected in these boundary conditions.

The following is a summary of the important implications derived from the report of the current model.

In conclusion, this research asserts that the model demonstrates velocity control phenomena and enhances heat transfer in nanofluids, presenting a significant possibility to enhance the cooling efficiency of mechanical systems with reduced friction.