A Numerical Framework for Entropy Generation Using Second-Order Nanofluid Thin Film Flow over an Expanding Sheet: Error Estimation and Stability Analysis

Abstract

Thin film flow (TFF) problems received a lot of attention in recent times. Some frequent applications of TFF include polymer and metal extraction, elastic sheet drawing, food striating, heat exchanges, and device fluidization. Further improvement and enhancement of TFF need to be examined due to its practical applications. In the current analysis, viscoelastic nanofluid thin film flow through the vertical expanding sheet in the presence of a magnetic field with entropy function has been examined. The governing equations are transformed to first-order ODEs through similarity transformation and then solved numerically by using RK4 along with the shooting technique and ND Solve method. The impact of embedded parameters is discussed using graphs and tables. Physical quantities of interest are also discussed in detail. For the numerical solution, the error estimation and the residue error are calculated for the stability and confirmation of the mathematical model.

1. Introduction

Thin film flow (TFF) problems received a lot of attention in recent times. The application of TFF in numerous technical fields has a long history of loyalty and relevance. TFF problems are difficult to describe and classify since they are based on a variety of disciplines, ranging from flow measurement in respiratory systems to industrial lubricant challenges. The study of liquid TFF and its applications reveals an essential link between structural and fluid mechanics. Some frequent applications of TFF include polymer and metal extraction, elastic sheet drawing, food striating, exchanges, constant forming, and device fluidization. Further improvement and enhancement of TFF need to be examined due to these practical applications. Many scientists have tried a number of approaches using constructive geometries in the past for this goal. The expanding sheet is one such interesting geometry that has attracted a lot of attention and has become a research challenge [1,2].

Fluids with certain viscosity were first the focus of thin liquid film flow (TLFF). Viscosity-based classifications of such fluids have the area saturated. With time, the application was expanded to include viscous fluids. The effects of internal and exterior factors on viscoelastic micro-fluid are investigated and addressed. Sandeep et al. [3] explored heat transmission in viscoelastic micro-thin film fluid flow. Due to its time dependence and the sheet’s nature, the geometry of the problem is important in stretching sheet phenomena. Wang [4] looked into the movement of TLFF through a time-varying expanding sheet. Usha et al. [5] examined a similar geometry with a finite thin liquid. Liu et al. [6] examined the TLFF for improving energy transmission via an extending surface. The velocity of a TFF with energy production across an expanding surface was reported by Aziz et al. [7]. Tawade et al. [8] investigated TLFF for the transmission of heat in the vicinity of temperature radiations. To solve the modeled equations, they used the Newton–Raphson and the RK–Fehlberg methods. Anderssona et al. [9] provided a more concise overview of the heat transfer study on TLFF through an expanding surface. The expanding sheet problem has been studied extensively in the literature, and a quick summary of its uses revealed other technical improvements [10,11,12,13,14,15]. Apart from that, a number of fluids are studied using the same geometry. Using a slip velocity assumption, Haroon et al. [16] analyzed the MHD nanofluid in stagnation point over a stretching sheet with a chemical reaction. The magnetized Couette–Poiseuille flow with thermal radiation and variable viscosity between analogous plates was examined by Zeeshan et al. [17]. Zeeshan et al. [18] investigated the mathematical model for transfer in MHD Oldroyd-B fluid with a heat reservoir. Rasool et al. [19] provided a numerical investigation of EMHD nanofluid flows over a convectively heated Riga pattern positioned horizontally in a Darcy–Forchheimer porous medium. Nehad [20] studied the second-grade thermodynamic activity across a vertical surface. Raju et al. [21] investigated the ternary hybrid nanofluid movement with different shapes and variable densities over contracting porous walls. Tahir et al. [22,23] examined some novel changes to the TFF of nanofluids.

The huge properties of nanofluids contribute to their competence in many applications (heat transfer enhancement, cooling, etc.). Micro-fluids are employed in pharmaceutical procedures, engines, hybrid fuel cells, and micro-electrons, and many other applications from a practical point of view. Recently, its primary use is in the field of nanotechnologies. Electronic equipment and nanoboards are now required in many sectors. With the advancement of time, many boards and electrical accessories get heated, which reduces their efficiency. Nanofluids are employed as a coolant to minimize heat in this phenomenon [24]. According to a literature review, air is employed as a coolant in a variety of procedures. LED nanotechnology and projectors are employed to improve the performance of microchips [25,26,27]. Non-Newtonian fluid movement is abundant in science and is very dependent on the technology utilized. The peristaltic mechanism is one such mechanism that is important in both industrial and physiological processes. In this technique, sinusoidal waves propagate down the channel’s wall. Hose pumps, dialysis, and lungs heating are all examples of such wave applications. Further research into this issue leads to the examination of MHD flows. In medicine and bioengineering, the MHD study of peristaltic flows is essential. Srivastava et al. [28] investigated the variable viscosity for peristaltic flow. Abbasi et al. [29] studied the time-dependent viscosity in nanofluid flow.

Nanoparticles have been discovered to have sizes of less than 100 nm in previous research [30,31]. Micofluids are mixtures of nanomaterial and are commonly used for heat transmission including ethylene glycol, oil, water, and glycol, among others. Nanoparticles may be made on a huge scale in labs and enterprises. Metals such as Al, Ag, Cu, Au, and metal oxides including CuO, Fe₃O₄, Al₂O₃, and TiO₂, nitrides such as SiN, AlN, and carbides (SiC) may all be used to make the nanoparticles. Because of their excellent thermal conductivity, nanoparticles made from such materials are utilized in extremely small volumes to increase heat transmission. The use of thermal systems for augmentation to improve heat transport is becoming more common. Malik et al. [32] used Buongiorno’s model to simulate nanofluid flow via a time-dependent stretching sheet. Using the Maxwell model, Nadeem et al. [33] addressed microsized fluid flow. Nanofluid flow was examined by Raju et al. [34] using non-Newtonian fluids on a stretched surface. The non-Newtonian film movement of nanomaterials through an angular extending surface was explored by Rokni et al. [35]. Using free natural convection and mass transmission, Nadeem et al. [36] investigated viscoelastic microsized MHD movement past a cone. Shehzad et al. [37] investigated the heat transmission of nanofluid flow across plates. On the stretched surface, Nadeem et al. reported numerical analyses of the flow of non-Newtonian nanoliquids [38]. Sheikholeslami et al. [39] investigated Jaffrey fluid nanomaterial magnetized flow with convective boundary restrictions. Mahmoodi et al. [40] used a kerosene-alumina with heat transfer for the cooling system. Shah et al. [41] investigated nanofluid flow for cooling and addressed the flow field’s heat sink. Shah et al. [42,43] investigated the influence of heat radiations and Hall current on spinning surfaces. Other interesting results presented in [44,45,46,47,48,49,50] will be useful for the reader. Sheikholeslami [51] used thin film nanofluid in the presence of a magnetic field with heat analysis. Sheikholeslami et al. [52] investigated the nanofluid in a porous channel. Aside from theoretical research, there is a wealth of experimental data on nanomaterial movement and its significance in heat enhancement analyses in the literature [53,54]. Akhgar et al. [55] used hybrid nanofluids with stability analysis and thermal conductivity. Keyvani et al. [56] used two convectional fluids (cerium oxide and ethylene glycol) to boost the volume friction and energy for a hybrid of copper oxide and titanium nanomaterial. This experiment was carried out in a laboratory at a temperature of (30–60) °C. They discovered a 41.5 percent enhancement in thermal at the highest limit of the temperature range. Ranjbarzadeh et al. [57] investigated an experimental process for the measurement of thermal conductivity using water-based graphene oxide. The thermal conductivity of the nanofluids was found to be 38.7% higher than that of the basic fluid. Refs. [58,59] reported similar experimental findings for increasing the thermal conductivity of microfluids.

The free existence of viscoelastic fluids and their use in industry attracted investigators to build models and future developments. The majority of organic substances were classified as non-Newtonian fluids. Molten plastics, food storage, lubricant oils, wall paints, and drilling mud are some of the broadly used significances of non-Newtonian fluids. According to the literature, several models have been presented and developed to categorize non-Newtonian liquids in terms of behavior. Walter’s B-fluid, Williamson fluid, Carreau fluid, Casson fluid, and others are widely used. The Newtonian generalized model [60] is another name for the Carreau fluid model. The Carreau fluid model’s importance in the field of water-based polymers, melts, and suspensions attracted researchers. Keeping the application of the Carreau model, scientists examined the Carreau fluid using various geometries. Here are some applications based on surveys connected to this concept. Using the Carreau fluid model, Hayat et al. [61] investigated thermo-solutal forced convective flow across two circular cylinders with magnetic field impact using Carreau fluid. The Carreau liquid flow across a perpendicular porous sheet with magnetic flux was studied by Alsarraf et al. [62] and Azari et al. [63]. The nature of the nanomaterial employed in the base fluid is solely dependent on the fluids model which is considered to be used. In the analysis of heat transfer, the shape of the nanomaterial employed is more relevant. The form of the nanoparticles utilized is also crucial for improving heat transmission and thermal and hydraulic capabilities. Munir et al. [64] investigated the Sisko fluid on a bidirectional stretchable sheet. Under high Reynolds numbers, they reported the data as percentages for both spherical and platelet-shaped nanomaterials. Olanrewaju et al. [65] also examined the free convection movement of radiative Sisko fluid over a flat plate.

Many researchers have looked at the development of entropy in thermal systems. Birkefeld and Weigand [66] used entropy generation to study incompressible movement through a flat surface. Makinde [67] used variable viscosity to study the second law for the magnetized flow of the boundary layer on a stretched sheet, as well as the heat analysis. Entropy production reduces when the Prandtl number and radiation parameter increase, according to Makinde. The convective flow of Darcy–Forchheimer CNT-based nanomaterials with heat transfer for entropy formation was studied by Hayat et al. [68]. They looked at heat transfer enhancement and entropy formation in both SWCNTs and MWCNTs. In another study of gravity-driven thin film flowi in the direction of a heated inclined sheet, Makinde [69] found that irreversibility of heat transfer is dominating at the liquid surface, whereas the converse is true at the plate surface. In nanofluids, a minor amount of work is performed on heat transport analysis using the second law of thermodynamics. Because nanomaterial flows are very uncommon, this is the case. Esmaeilpour and Abdollahzadeh [70] examined the use of entropy production to increase heat for nanofluid-free convection flows within an enclosure. Dawar et al. [71] proposed analytic consequences for the flow of CNT-nanofluid within rolling plates. The influence of MHD and entropy production was quantitatively considered in this study. Recently published work related to the present work is given in [72,73].

The primary goal of this discussion is to look at the velocity of magnetized thin film flow with heat transfer analysis and entropy in a vertical stretchable surface. Algorithms for entropy creation and boundary-layer thermal expansion across a vertical stretched surface are developed for two-dimensional magnetized thin film nanofluids. These equations are obtained from the basic flow equations based on geometry and assumptions in the flow field. In the nanofluid model, many physical characteristics such as thermophoresis, concentration gradients, thermophoresis, and Brownian motion of the flow are deduced from configuration and predicated on fluid dynamic presumptions. The predicted foremost system in the pattern of PDEs (partial differential equations) is converted into first-order ODEs using similarity transformation. A numerical strategy is applied to solve the reduced system of ODEs. Due to its rapid convergence, NDSolve is integrated. The numerical and graphed convergence of the developed approach is explained. Physical parameters of interest such as Nusselt number, Sherwood number, and skin friction are numerically explored for their importance in boundary layer movement. Graphs for the entropy function explore the influence of Brinkman and Bejan numbers. Graphs show that the Reynolds number, magnetic parameter, and Prandlt number have an impact on the entropy function, Bejan number, and Brinkman number. For the numerical solution, the error estimation and the residue error are calculated for the stability and confirmation of the mathematical model.

2. Mathematical Modeling

Consider the movement of an unsteady second-grade nanofluid TFF through a vertically stretched surface. The magnetic field effect is also taken into account within the flow field since the fluid is presumed to be magnetized. A slit in the moving sheet causes it to begin its movement. Geometrically, the surface length is equal to the ox, and the oy is flat to the surface in the Cartesian coordinate system. There are two forces in opposing directions along the x-axis, yet the center of the flow remains stationary due to the stretching effects imparted to the surface of the flow. The geometry of the problem is presented in Figure 1.

The stretched sheet and x-axis are chosen so that they are contiguous, and the stress speed of the sheet is provided by [17]:

$$U_w\left(x,t\right)=\frac{\gamma x}{\left(1-\zeta t\right)}.$$

(1)

where $\zeta$ and $\gamma$ identify any constant numbers vertical to the x-axis while the temperature at the wall is and capacity of nanoparticles are expressed by [17,19]:

$$T_w\left(x,t\right)=T_r\left(\frac{\gamma x^2}{2v_f}\right){\left(1-\zeta t\right)}^{-1.5}+T_0,$$

(2)

$$C_w\left(x,t\right)=C_r\left(\frac{\gamma x^2}{2v_f}\right){\left(1-\zeta t\right)}^{-1.5}+C_0.$$

(3)

where $v_f$ explores the kinematic viscosity, $T_0$ and $C_0$ identify the temperature of the slit and volume friction of the nanoparticles, and $T_r$ and $C_r$ signify the ambient temperature and volume of the nanomaterial, correspondingly.

The magnetic flux is represented by the following expression [17]:

$$B\left(t\right)=B_0{\left(1-\zeta t\right)}^{-\frac{1}{2}},$$

(4)

where $B_0$ is the strength of the magnetic field.

The mathematical expression for the second-grade model is [23]:

$$\overrightarrow{T}=-p\overrightarrow{I}+\overrightarrow{S},$$

(5)

where

$$\overrightarrow{S}=\tau =\mu {\overrightarrow{A}}_{\left(1\right)}+{\alpha }_{\left(2\right)}{\overrightarrow{A}}_{\left(2\right)}+{\alpha }_{\left(2\right)}{\overrightarrow{A}}_{\left(2\right)}^2.$$

(6)

Here, $p\overrightarrow{I}$ is pressure, $\overrightarrow{S}$ is an extra-stress tensor, $\mu$ is the viscosity, ${\alpha }_{\left(1\right)}$ and ${\alpha }_{\left(2\right)}$ are the thermal stress moduli, and ${\overrightarrow{A}}_{\left(1\right)}$ and ${\overrightarrow{A}}_{\left(2\right)}$ are the stresses of the Rivlin–Ericksen tensors with the following mathematical formula:

$${\overrightarrow{A}}_{\left(1\right)}=\nabla \overrightarrow{V}+{\left(\nabla \overrightarrow{V}\right)}^T \mathrm{and} {\overrightarrow{A}}_{\left(2\right)}=\frac{D}{Dt}\left({\overrightarrow{A}}_{\left(1\right)}\right)+{\overrightarrow{A}}_{\left(1\right)}\nabla \overrightarrow{V}+{\left(\nabla \overrightarrow{V}\right)}^T{\overrightarrow{A}}_{\left(1\right)}.$$

3. The Governing Equations

The basic model equation is the continuity, momentum, and energy, which are expressed by the following relation [73]:

$$\nabla \cdot \overrightarrow{V},$$

(7)

$$\rho \frac{D}{Dt}\left(\overrightarrow{V}\right)=\nabla \cdot \overrightarrow{V}+\overrightarrow{J}\times \overrightarrow{B}+\overrightarrow{g},$$

(8)

$$\frac{\partial T}{\partial t}+\overrightarrow{V}\cdot \nabla T=\nabla \cdot \left[\frac{K\left(T\right)}{{\left(\rho c_p\right)}_f}\nabla T\right]+\tau \left[D_B\nabla C\cdot \nabla T+\left(\frac{DT}{T_0}\nabla T\cdot \nabla T\right)\right]-\frac{1}{\rho c_p}\frac{\partial q_r}{\partial y},$$

(9)

$$\frac{\partial C}{\partial t}+\overrightarrow{V}\cdot \nabla C=D_B{\nabla }^{2}C+\frac{D_T}{T_0}{\nabla }^{2}T.$$

(10)

Using all the above assumptions in Equations (7)–(10), we obtain:

$$\frac{\partial u}{\partial x}+\frac{\partial \nu }{\partial y}=0,$$

(11)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}-v\frac{{\partial }^{2}u}{\partial y^2}=\frac{{\alpha }_1}{\rho }\left(\frac{\partial }{\partial y}\left(u\frac{{\partial }^{2}u}{\partial y^2}\right)-\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial x\partial y}+v\frac{{\partial }^{2}u}{\partial y^2}\right)-\rho B_0^{2}u+g_r{\beta }_T\left(T-T_{\infty }\right)+g_r{\beta }_T\left(C-C_{\infty }\right),$$

(12)

$$\frac{\partial T}{\partial t}+\mu \frac{\partial T}{\partial x}+\upsilon \frac{\partial T}{\partial y}=\frac{1}{\rho c_p}\frac{\partial }{\partial y}\left[\frac{\partial u}{\partial y}\right]+\tau \left[D_B\left(\frac{\partial C}{\partial y}\frac{\partial T}{\partial y}\right)+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right]-\frac{16\sigma T_{\infty }^3}{3k}{\left(\frac{\partial T}{\partial y}\right)}^2$$

(13)

$$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+\upsilon \frac{\partial C}{\partial y}=D_B\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_T}{T_{\infty }}\left(\frac{{\partial }^{2}T}{\partial y^2}\right).$$

(14)

where $\rho$ denotes the fluid density and $\overrightarrow{V}$ is velocity, which can be expressed in component form as $\overrightarrow{V}=\left(u,v,0\right),T$, the Cauchy tensor of stress and $\overrightarrow{g}$ action force, respectively. $\overrightarrow{J}\times \overrightarrow{B}$ represents the famous Lorentz force, in which $\overrightarrow{J}$ is the current density, $\overrightarrow{B}$ is the magnetic field with magnetic field strength $B_0$, and $q_r$ is the radiator heat flux. Furthermore, $\overrightarrow{J}$ can be expressed as $\overrightarrow{J}=\sigma \left(\overrightarrow{E}+\overrightarrow{V}\times \overrightarrow{B}\right)$, also known as Ohm’s law, in which $\sigma$ and $\overrightarrow{E}$ describe the electrical conductivity and electric field, respectively, and assume that $\overrightarrow{E}=0,\frac{D}{Dt}$ represents the substantial derivative. Where ${\beta }_T$ is the thermal expansion coefficient, $C_{\infty }$ and $T_{\infty }$ denotes the concentration and temperature at a distance from the surface, and g is the gravitational acceleration.

The pertinent restrictions can be written as:

$$u=U_w, v=0, T=T_w, C=C_w \mathrm{at} y=0,$$

(15)

$$\frac{\partial u}{\partial x}=\frac{\partial T}{\partial x}=\frac{\partial C}{\partial x}=0, \upsilon =\frac{\mathit{dh}\mathit{(t)}}{dt} C\succ 0 \mathrm{at}y=h.$$

(16)

Using a Rosseland approximation for radiation, $q_r$ is introduced as:

$$q_r=\frac{-4{\sigma }^{\ast }{T}_{\infty }^3}{3k^{\ast }}\frac{\partial T^4}{\partial y}$$

where ${\sigma }^{\ast }$ is the Stefan–Boltzmann constant and $k^{\ast }$ is the mean absorption coefficient. Using the Taylor series expansion, $T^4$ can expand to about $T_{\infty }$ and ignoring the higher order, we obtain:

$$T^4\cong 4T_{\infty }^{4}T-3T_{\infty }^3$$

(17)

Introducing the following transformations [14]:

$$\begin{array}{l}\psi =x\sqrt{\frac{\upsilon \gamma }{1-\zeta t}}f\left(\eta \right),u=\frac{\partial \psi }{\partial y}=\gamma x\frac{f^{\prime }\left(\eta \right)}{1-\zeta t^{\prime }},\nu =\frac{\partial \psi }{\partial x}=-\sqrt{\frac{\gamma \upsilon }{\left(1-\zeta t\right)}}f\left(\eta \right)\\ \eta =\sqrt{\frac{\gamma }{\upsilon \left(1-\zeta t\right)}}y,h\left(t\right)={\left[\frac{\upsilon }{\gamma {\left(1-\zeta t\right)}^{-1}}\right]}^{\frac{1}{2}},\theta \left(\eta \right)=\frac{T-T_0}{T_w-T_0},\varphi \left(\eta \right)=\frac{C-C_0}{C_w-C_0}\end{array}$$

(18)

Here, the stream function is represented by $\psi$, the thickness of the fluid film is denoted by $h\left(t\right)$, and the kinematic viscosity is represented by $v=\frac{\mu }{\rho }$. The dimensionless film thickness is defined as:

$$\beta =\sqrt{\frac{\zeta }{\upsilon \left(1-\zeta t\right)}}h\left(t\right).$$

(19)

In other words, Equation (18) becomes:

$$\frac{dh}{dt}=-\frac{\zeta \beta }{2}\sqrt{\frac{\upsilon }{\zeta \left(1-\zeta t\right)}}.$$

(20)

With the help of the newly introduced variables, Equations (10)–(14) are reduced to the following equations, while the continuity equation is satisfied identically.

$$f^{‴}+{\gamma }_1\left(f^{‴}{f}^{\prime }-{\left(f^{″}\right)}^2-f{f}^{iv}\right)+f{f}^{″}-{\left(f^{\prime }\right)}^2-St\left(f^{\prime }+\frac{\eta }{2}{f}^{″}\right)-Gr\theta +Gm\varphi +M{f}^{\prime }=0,$$

(21)

$$\left(1+Rd\right){\theta }^{″}+f{\theta }^{\prime }-2f^{\prime }\theta -\frac{St}{2}\left(3\theta +\eta {\theta }^{\prime }\right)+Nt{\left({\theta }^{\prime }\right)}^2+Nb{\theta }^{\prime }{\varphi }^{\prime }=0,$$

(22)

$${\varphi }^{″}+Sc\left[f{\varphi }^{\prime }-2f^{\prime }\varphi -\frac{St}{2}\left(3\varphi +\eta {\varphi }^{\prime }\right)\right]+\frac{Nt}{Nb}{\theta }^{″}=0.$$

(23)

The boundary conditions of the problem are:

$$f\left(0\right)=0,f^{\prime }\left(0\right)=1,\theta \left(0\right)=1,\varphi \left(0\right)=1,f\left(\beta \right)=\frac{S\beta }{2},f^{″}\left(\beta \right)=0,{\theta }^{\prime }\left(\beta \right)=0,{\varphi }^{\prime }\left(\beta \right)=0.$$

(24)

where $St=\frac{\gamma }{\epsilon }$ is the time-dependent factor, ${\gamma }_1=\frac{{\alpha }_1{\beta }^2}{\rho {\delta }^2}$ is the stretching parameter, $M=\frac{{\sigma }_f{B}_0^2}{b{\rho }_f}$ stands for the magnetic factor, $\mathrm{Pr}=\frac{\rho \upsilon c_p}{k}$ is the Prandtl number, $Nt=\frac{\tau D_w\left(T_w-T_{\infty }\right)}{\upsilon T_{\infty }}$ is the thermophoresis factor, $Nb=\frac{\tau D_B\left(C_w-C_{\infty }\right)}{\upsilon }$ is the Brownian motion, $Gm=\frac{g\beta L^3\left(C_w-C_{\infty }\right)}{\upsilon }$ is the mixing parameter, $Gr=\frac{g\beta L^3\left(T_w-T_{\infty }\right)}{\upsilon }$ represents the Grashof number, $Rd=\frac{4\sigma T_{\infty }^3}{3k{k}^{\ast }}$ signifies the radiation factor, and $Sc=\frac{\upsilon }{D_B}$ is the Schmidt number.

4. Physical Parameters of Interest

4.1. Skin Friction

The coefficient of skin friction in closed form is:

$$C_f=\frac{\left({\overrightarrow{S}}_{xy}\right)y=0}{\rho U_w^2/2},$$

(25)

where ${\overrightarrow{S}}_{xy}=\mu \frac{\partial u}{\partial y}+\rho a_1(2\frac{\partial u}{\partial x}\frac{\partial u}{\partial y}+\frac{{\partial }^{2}u}{\partial x\partial y})$

The dimensionless expression of the skin’s friction becomes:

$$C_f={\left({\mathrm{Re}}_e\right)}^{-\frac{1}{2}}\left(f^{″}\left(0\right)+3{\gamma }_1{f}^{″}\left(0\right)f^{\prime }\left(0\right)\right),$$

(26)

where $R_e=\frac{U_{w}x}{v}$ represents the local Reynolds number.

4.2. Nusselt Number

The close mathematical form of the Nusselt number is:

$$Nu=\frac{h{Q}_w}{k({\mathrm{T}}_0-\mathrm{T})}, \mathrm{where} Q_w=-k{(\frac{\partial T}{\partial y})}_y=0$$

$Q_w$ is the flux of heat. In the dimensionless form, the Nusselt number is given below physical parameters of interest, the skin friction, Nusselt number, and Sherwood number are expressed by the following equation:

$$Nu=-\mathrm{\Theta }\left(0\right).$$

(27)

4.3. Sherwood Number

In mathematical form, it can be expressed as $Sh=\frac{h{j}_w}{D_B(C_0-C_h)},$ where $j_w=-D_B(\frac{\partial C}{\partial y})$ is the mass flux. The dimensionless form of Sh is expressed as:

$$Sh=-\mathrm{\Phi }\left(0\right).$$

(28)

5. Mathematical Expression for Entropy Generation

The entropy generation (EG) for the viscoelastic second-grade fluid is [72,73]:

$$S^{‴}=\frac{K\left(T\right)}{T_0^2}\left[{\left(\frac{\partial T}{\partial y}\right)}^2+\frac{16\sigma T_{\infty }^3}{3k}{\left(\frac{\partial T}{\partial y}\right)}^2+\frac{\mu \left(T\right)}{T_0}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{Rd}{C_0}{\left(\frac{\partial C}{\partial y}\right)}^2+\frac{Rd}{T_0}\left(\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}+\frac{\partial C}{\partial x}\frac{\partial T}{\partial x}\right)+\frac{\sigma B_0^2{u}^2}{T_0}\right].$$

(29)

In Equation (27), it is crystal clear that the entropy has six source generations, i.e., the first source on the right-hand side of Equation (27) is the local entropy generation (LEG) due to heat transfer across a finite temperature difference (E_H); the second source is the LEG due to thermal radiation (E_R); the third source is the LEG due to viscous dissipation (E_V); the fourth source is the LEG due to mixed product of thermal and concentration difference (E_RC); the fifth source is the LEG due to thermal, heat, and mass transfer across finite concentration difference (E_RTC); and the sixth term is the LEG due to the magnetic field (E_M). For the EG rate, it is reasonable to specify a dimensionless number N_G. To determine this number, the local volumetric EG rate $S^{‴}$ is divided by a characteristic EG rate $S^{‴}$. The typical EG rate for the specified boundary condition is [72,73]:

$$S_0^{‴}=\frac{K_0{\left(\nabla T\right)}^2}{L^2{T}_0^2}.$$

(30)

So, the EG number is:

$$N_G=\frac{S^{‴}}{S_0^{‴}}$$

(31)

The entropy generation number $N_G$ calculated using the following formulas for non-dimensional velocity, temperature, and concentration is:

$$N_G=\mathrm{Re}\left(1+\epsilon \theta +Rd\right){\left({\theta }^{\prime }\right)}^2+\frac{\mathrm{Re}Br}{\mathrm{\Omega }\left(1+\Lambda \right)}{\left(f^{″}\right)}^2+\frac{\mathrm{Re}Br}{\mathrm{\Omega }}M{\left(f^{\prime }\right)}^2+\mathrm{Re}\lambda {\left(\frac{\chi }{\mathrm{\Omega }}\right)}^2{\left({\varphi }^{\prime }\right)}^2+\mathrm{Re}\lambda \left(\frac{\chi }{\Omega }\right){\theta }^{\prime }{\varphi }^{\prime },$$

(32)

where $Br$ is the Brinkman number, $\lambda =\frac{Rd{C}_0}{k}$ the diffusion quantity, $M$ the magnetic parameter, $\mathrm{Re}=\frac{b{L}^2}{v}$ is the Reynolds number, $\Omega =\frac{\Delta T}{T_0}$, and $\chi =\frac{\Delta C}{C_0}$ signifies the dimensionless temperature and concentration change, separately.

An alternate irreversibility distribution parameter in the Bejan number was presented by Paoletti et al. [71]. This is applicable to all technical fields where entropy creation is a significant issue for engineers and industrialists.

$$Be=\frac{E_H+E_R}{E_V+E_M}.$$

(33)

6. Numerical Procedure and Stability Analysis

The set of Equations (7)–(12) are solved numerically due to high nonlinearity. First, the equations are altered to a system of first-order ODEs through similarity transformation and then solved numerically. MATLAB built-in function RK4 is used for this solution using the step size $\mathsf{\Delta }\eta =0.001$ with convergence 10⁻⁶, as shown in

Table 1. Twenty-fourth-order approximation for RK4 convergence, where ${\lambda }_1=Sc=0.4, Nt=Gr=b=St=$ 0.2, and $Pr=Nb=Gm$ = 0.2.

Order	${\mathit{f}}^{″}\left(0\right)$	${\Theta }^{\prime }\left(0\right)$	${\mathit{\varphi }}^{\prime }\left(0\right)$
1	−0.050000	−1.218600	0.064860
2	−0.215767	−0.218600	0.1967944
4	−0.226793	−0.217991	0.1946782
6	−0.229531	−0.217928	0.1966267
8	−0.229191	−0.217919	0.1196998
10	−0.229372	−0.217917	0.1972274
12	−0.229414	−0.217918	0.1972568
14	−0.229424	−0.217917	0.1972641
16	−0.229426	−0.217917	0.1972659
18	−0.229427	−0.217917	0.1972663
22	−0.229427	−0.217917	0.1972665
24	−0.229427	−0.217917	0.1972665

. For confirmation, the NDSolve approach is also applied and outstanding agreement is found, as depicted in Figure 2.

7. Error Analysis and Confirmation of the RK4 Method

RK4 results for most differential equations are often quite accurate. However, since its results are based on numerical sampling and error estimates, substantial inaccuracies are extremely unlikely. It is a good notion to perform some basic assessment of the solution when you need to be certain of its perfection. It is frequently convenient to examine outcomes by linking a solution that was generated with work precision greater than the usual machine precision (MP). Use RK4 to estimate the correct answer using the default work precision, and then repeat the process to determine the accurate results using working precision-22. Errors are thus usually relatively small; hence, it is advantageous to analyze them on a logarithmic scale. Real exponent [x] is an excellent choice for displaying differences that could be zero at some places since it effectively equals Log 10 (Abs [x]) without a singularity at zero. We approximated the error solutions for various physical parameters used in the model in the graphs below. These numbers demonstrate that our numerical answer is precise since the error is excessively small. The error analysis of the physical parameters $\beta ,$ $M,$ $G_r,$ $St,$ $G_m,$ and $Pr,$ respectively, is shown in Figure 3. It is evident in this figure that our mathematical computation is accurate. Similarly, the residue error is computed using the default work precision and working precision-22 on the basis of error analysis. Figure 4a–f show the relevant residue error for the physical parameters involved in the mathematical model and it is witnessed that our solution is correct, as the residue error is too small.

8. Table Discussion

Table 2. Influence of M, Gr, b, and St on skin friction and its stability.

M	Gr	$\mathit{\beta }$	St	${\mathit{C}}_{\mathit{f}}$	Nehad et al. [20]
0.2	0.4	0.1	1.4	−0.2819	−0.2815
0.4				−0.3145	−0.3142
1.2				−0.3571	−0.3570
1.8	0.2			−0.2887	−0.2881
	0.4			−0.2819	−0.2812
	1.2			−0.2819	−0.2813
	1.7	0.2		−0.4545	−0.4548
		0.4		−0.8574	−0.8570
		1.2		−7.9971	−7.9973
		1.7	0.2	−2.9783	−2.9787
			0.4	−0.3328	−0.3329
			1.2	−0.2431	−0.2435

,

Table 3. Influence of Pr, M, Rd, and St on Nusselt number and its stability.

M	$\mathit{\beta }$	St	Pr	Nu	Nehad et al. [20]
0.2	0.2	1.4	1.5	0.4634	0.4636
0.4				0.4632	0.4637
1.2				0.4629	0.4623
1.8	0.2			0.4634	0.4630
	0.4			2.6256	2.6252
	1.2			2.3727	2.3725
	1.7	0.2		2.7359	2.7353
		0.4		0.3395	0.3390
		1.2		0.3915	0.3913
		1.7	1.2	0.4634	0.4638
			1.8	0.4354	0.4350
			4.0	0.4198	0.4197
			6.0	0.4123	0.4122

and

Table 4. Influence of Nb, Nt, Sc, Pr, and St, on Sherwood Number and its stability.

Nb	Nt	Sc	St	Pr	$\mathit{S}\mathit{h}$	Nehad et al. [20]
0.2	0.4	0.2	1.4	1.4	−2.4693	−2.4693
0.4					−0.3499	−0.3492
1.2					−0.1988	−0.1983
1.7	0.2				−0.1334	−0.1338
	0.4				−2.4693	−2.4690
	1.2				−3.8648	−3.8642
	1.7	0.2			−5.2566	−5.2566
		0.4			−4.9778	−4.9772
		1.2			−4.8564	−4.8566
		1.7	0.2		−2.9216	−2.9214
			0.4		−3.4169	−3.4160
			1.2		−3.9294	−3.9290
			1.7	1.4	−4.6374	−4.6372
				2.5	−6.7241	−6.7244
				4.0	−7.8817	−7.8812
				6.0	−8.4434	−8.4438

show the effects of various settings. M, Gr, b, and St all have an impact on skin friction, Nusselt number, and Sherwood number. The skin friction coefficient enhances rapidly when the unstable parameter St is increased, but the Nusselt number falls as the magnetic parameter M is increased. This is due to the Lorentz force which develops when the magnetic field is applied which resists the flow. Higher Grashof numbers Gr and thickness factor β values, on the other hand, reduce skin friction. It is clear from the mathematical expression of the Gr and β that the skin friction is inversely proportional to Gr and β. The Grashof number is generated in natural convection as a result of a variation in density. Similarly, viscous forces are functions of the Grashof number’s dependency. The Grashof number decreases as the viscosity coefficient is increased. As a consequence, the skin friction is reduced, as shown in Table 2. Table 3 shows the influence of $M, St, \beta ,$ and $Pr$ on $Nu$. The Nusselt number Nu decreases when M (the magnetic parameter) and St (the unsteadiness factor) increase in magnitude. The stability factor $St$ and thickness parameter β raise the Nusselt number $Nu$; on the other hand, the $Nu$ diminishes with rising Prandtl numbers. As a consequence, increasing $\beta$ raises the film’s viscosity, causing the velocity curve to fall further and the Nusselt number is enhanced. It is shown in Table 4 how Brownia motion $Nb$, temperature $Nt$, Schmidt number $Sc$, Prandlt $Pr$, and unsteadines parameter $St$ influence Sherwood numbers. The Sherwood number rises as the thermophoretic parameter rises in magnitude. The Schmidt number and the Sherwood number have an inverse relationship. Stability parameter St and Prandtl number $Pr$ for the Sherwood number exhibit the same trend of reduction. The Sherwood number decreases exponentially as the Prandtl number increases. Additionally, the present work is validated with published work reported by Nehad et al. [20] and excellent agreement is found, as shown in Table 2, Table 3 and Table 4.

9. Results and Discussion

The aim of this research is to understand the thin film movement characteristics of nanoliquids. The parameters in our model equations are clearly shown in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25. Figure 5 depicts the thickness of the liquid layer $\beta$ in the direction of melting flow. Since the dimensionless thin film thickness $\beta$ is directly connected to the melting thickness $h\left(t\right)$ and is a function of viscosity, the flow velocity curve decreases as the film thickness increases. As a consequence, increasing $\beta$ raises the film’s viscosity, causing the velocity curve to fall further. This occurs because of the indirect link between $\beta$ and the profile of flow, i.e., the greater amount of $\beta$ reduces the viscosity of the fluid, lowering the velocity field. Figure 6 depicts the effect of the unsteadiness parameter $St$ on the velocity field for various embedded parameters. In Figure 5, a direct relationship between fluctuations in the unsteadiness factor and velocity profile is observed. The influence of the stretching parameter is responsible for these variances. The unsteadiness factor $St$ is a function of the liquid film thickness, which changes directly with the stretching factor, causing the velocity profile to grow. The fluid’s velocity increases as it is enhanced. Investigation reveals that the solution exists for $St\in$ [0, 2] and is highly dependent on the factor $St$.

Figure 7 depicts the flow’s mixed convective impact $Gm$. Figure 7 depicts the characteristics of buoyant forces, which are beneficial to the state variable velocity. The concentration difference, length, and kinematic viscosity of the nanofluid all have an impact on the mixing parameter. The velocity and viscosity of the nanomaterial have an inverse relationship. Physically, as the mixing parameter rises, the liquid film concentration enhances immediately while the viscosity lowers, causing the velocity profile to rise. It is observed that the velocity profile is proportional to $Gm$. Figure 8 shows the order 2-fluid velocity distribution parameter ${\gamma }_1$. This characteristic is physically inversely linked to density, maintaining the thickness constant. As a result, rising the amount of the factor ${\gamma }_1$ causes the density of the fluid to drop, resulting in an increment in the velocity distribution. In other words, it makes the fluid less dense, which causes the fluid velocity to increase. The relationship between the Grashof parameter $Gr$ and the velocity distribution is seen in Figure 9. The properties of buoyant forces are shown here, which provide a desirable behavior for the velocity profile. The Grashof number $Gr$ is the ratio of buoyant force to viscous force in physical terms. As the buoyant forces increase, the viscous forces decrease, resulting in quicker motion. In summary, rising values of $Gr$ causes the velocity profile to rapidly grow.

The fluctuation of the magnetic factor $M$ across the velocity profile is depicted in Figure 10. To raise the strength of the magnetic field that bends the surface of the plate, the magnetic parameter must be applied horizontally, which means that an enhancement in the magnetic parameter would significantly increase the magnetic field strength. The velocity profile decreases as a result of the bending, but the magnitude remains the same. In brief, as the magnetic factor $M$ is enhanced, the velocity profile decreases.

Figure 11 illustrates the influence of thermal radiation on $Rd$ and $\theta \left(\eta \right)$. The thermal radiation parameter $Rd$ and the temperature profile have an inverse relationship, as seen in the graph. A quick decline in the temperature profile can be seen with larger values of $Rd$, and vice versa. The influence of the thermophoresis parameter $Nt$ on the temperature field is seen in Figure 12. The constraints of thermophoresis aid raise the surface temperature. As a consequence, the kinetic energy is developed due to the nanomaterial; the irregularity in motion (Brownian motion) produces a temperature rise, resulting in a thermophoretic force. Due to the force’s strength, the fluid begins to flow in the opposite direction of the stretched surface. As a consequence, higher values of $Nt$ produce an increment in temperature, which causes an enhancement in surface temperature.

Figure 13 shows the effect of thin film thickness $\beta$ on temperature for various values of the embedded parameter. Because the thickness parameter is a function of the kinematic viscosity and fluid thickness, increasing it raises the viscosity, which causes the temperature profile to fall further. As a result, the temperature profile declines with greater values of $St$. The temperature field and concentration profiles under the Brownian motion parameter $Nb$ are shown in Figure 14 and Figure 15, respectively. A collision is formed between particles owing to the irregular motion of the particles. The graphic indicates that when the Brownian motion parameter $Nb$ is increased, the heat of the fluid increases and, therefore, the free surface nanoparticle volume friction decreases. Because of the increasing values of Brownian motion, the boundary layer thicknesses declines, resulting in a drop in the concentration profile.

Figure 15 depicts the behavior of the concentration profile $\varphi \left(\eta \right)$ when the unsteadiness parameter $St$ is varied. The unsteadiness parameter $St$ and the concentration field $\varphi \left(\eta \right)$ seem to have a direct relationship. Larger values of the unsteadiness factor $St$ raise the temperature field that enhance the fluid’s kinetic energy, causing a rise in the liquid film’s concentration. The influence of the thermophoresis factor $Nt$ on the concentration field is seen in Figure 16. The graphic shows how an increase causes the concentration profile to rise. This is due to larger levels of upsurges in the kinetic energy of nanofluid atoms, which causes the concentration to rise.

Figure 17 depicts the influence of thin film thickness $\beta$ on concentration profiles $\varphi \left(\eta \right)$ for various embedded parameters. The thickness factor is inversely connected to the kinematic viscosity; as kinematic viscosity is inversely proportional to the fluid density, raising the thickness parameter $\beta$ causes the concentration profile to drop. As a result, it is clear that as the number of $\beta$ decreases, the concentration profile decreases. A similar impact was seen for $\beta$ in both the velocity and temperature distributions. Figure 18 depicts the inverse information described in the temperature profile under various factors. The graphic indicates that when the Schmidt number $Sc$ increases, the concentration profile drops, lowering the boundary layer thickness.

Figure 20 and Figure 21 depict the differences between Brinkman numbers $Br$, magnetic factors $M$, and Bejan numbers $Be$. The fluctuations in the Bejan number under different values of the magnetic parameter $M$ are shown in Figure 20. The graph depicts that the Bejan number enhances as the value of $M$ increases. Figure 21 shows similar changes in the Bejan number for numerous values of the Brinkman number. For the lesser amount of the magnetic factor in the domain of 0.85 < $\eta$ $\le$ 1.0, small changes are observed while large variances are examined in the domain of 0.0 < $\eta$ $\le$ 0.2. Figure 22 shows how the entropy number changes when the Prandtl number $\mathrm{Pr}$ changes. The temperature field increases as the Prandtl number $\mathrm{Pr}$ grows and the entropy function boots, as depicted in Figure 22. Within the range of 0.0 $\le$ $\eta$ $\le$ 1.0, the fluctuations in the entropy function increase. The impact of the Reynolds number $Re$ on $N_G\left(\eta \right)$ is seen in Figure 23. The entropy regime grows when $Re$ is increased. As a result, the Reynolds number and the entropy generation immediately vary. The figure shows that the fluctuations in entropy stay constant for high values of $\eta$, but they increase for big values of the $Re$ in the domain of 0.0 < $\eta$ $\le$ 0.4.

The effect of the magnetic factor $M$ and Brickman number $Br$ on the entropy-generating function $N_G\left(\eta \right)$ is seen in Figure 24 and Figure 25, respectively. The entropy generation enhances as the values of $M$ and $Br$ are increased. Because of the reduced conduction rate, viscous dissipation creates heat, which increases the creation of entropy.

10. Conclusions

The primary goal of this discussion is to study the velocity of a magnetized thin film second-grade fluid model with heat transfer analysis, as well as the generation of entropy in a vertical stretchable surface. No new research on nanofluid flow across a vertical stretching surface utilizing rheological fluid of second grade has been published to date. Algorithms for entropy creation and boundary-layer thermal expansion across a vertical stretched surface are developed for two-dimensional magnetized thin film nanofluids. The transformed equations are solved numerically and the effect of numerous factors on flow characteristics is observed. The points of this study are as follows:

• The velocity field of the nanoparticle’s fluid film upsurges when the unsteadiness factor $St$ is enhanced, but the velocity profile of the nanofluid film decreases due to increases in the magnetic factor.
• The coefficient of skin friction significantly develops when $M$ and $St$ are increased; however, the coefficient of skin friction reduces as the stretching and thickness parameters are increased.
• The Brownian factor has a direct impact on the temperature profile.
• With rising values of the $Sc$ and $Rd$, the Nusseltnnumber, the thermal boundary-layer thickness reduces.
• With higher Prandtl numbers, the fluid’s surface temperature rises, but for higher values $St$, the temperature field has a reverse effect.
• Observing the impact of the thermophoresis parameter, a similar consequence is observed for the temperature profile.
• The mass flow rate decreases when the Brownian factor is increased, but the thermophoretic factor displays the reverse tendency.
• For the validation of the numerical approach, the implemented technique convergence is quantitatively illustrated. Moreover, the present is compared with the published work reported by Nehad et al. [20], and a good agreement is established.