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Abstract

The hydromagnetic dissipative, heat-absorbing, chemically reactive and optically thick radiative flow of nanofluid over a stretchable surface together with heat and mass transport phenomena is investigated. The prevailing partial differential equations (PDE) of the mathematical model are changed by incorporating apt similarity variables in the form of nonlinear similarity equations. Further, these similarity equations are solved numerically utilizing the Runge-Kutta Fehlberg technique in conjunction with the shooting method. For the numerical explorations, three kinds of nanofluids are prepared by disseminating very fine nanoparticles of titanium oxide (TiO₂), aluminium oxide (Al₂O₃) and copper (Cu) into water. The significance of numerous regulatory flow parameters on the nanofluid velocity, nanofluid temperature, local Nusselt number, species concentration, wall velocity gradient and mass flow rates are examined through different graphical results. Additionally, a quadratic regression approximation analysis is accomplished to analyze the connection between the heat transport rate and regulatory flow parameters. The numerical results reveal that the temperature of Cu-water based nanofluid temperature gets enhanced owing to improvement in the strength of radiation, viscous dissipation and magnetic effects. Further, regression approximation analysis unveils that a slight change in the velocity slip parameter leads in the optimal perturbation in both the shear stress values and heat transfer rate at the stretchable surface. Finally, the validation of numerical results and the developed algorithm of employed computational technique have been done by making a comparison of computed results with the available results under restricted situations.

Keywords

Nanofluids chemical reaction thermal radiation magnetic field viscous and joule dissipations

Introduction

Nowadays, energy resources play a significant role in the growth of human society. Researchers are constantly betrothed to the discovery of novel energy resources. In pursuance, many theoretical and experimental analyses have been accomplished to boost the heat transport rate of several fluids. Nanofluid is a colloidal suspension of nano-sized polymeric, non-metallic/ metallic powder in a conventional fluid which are used to augment heat transport rate for numerous applications. In the past few years, numerous ideas of nanofluids have been suggested as a course for augmenting heat transport fluids performance. The idea of achieving high thermal conductivity in a fluid by mixing very fine metal nanoparticles of diameter into a base fluid was initially proposed by Choi¹ and termed as nanofluids. The fluid characteristics of base fluids such as thermal conductivity and heat transport rate etc. can be influenced by disseminating extremely fine metal/metal oxides particles into fluid.² Owing to the novel features, nanofluids are widely used in the nano-medical industry, automobiles, metallurgy, microsystem's cooling, energy conversion, optical and sensor devices.^3,4 Rashidi et al.⁵ reviewed numerically and experimentally both to identify the significance of nanofluids on condensing and evaporating systems’ performance and also, they discussed the disadvantages of using nanofluid in condensing as well as evaporating systems. Subsequently, the impact of nanoparticles on the efficiency of the system of solar desalination incorporating passive as well as active solar distillation systems was made by Rashidi et al.⁶ Ellahi et al.⁷ surveyed the heat transfer significance of water-Al₂O₃ nanofluid stagnation point flow via a permeable wedge. Further, Ellahi et al.⁸ extended their investigation to analyse the particle shape impact on the Marangoni convective nanofluid flow. Sharma et al.⁹ scrutinized the worth of Soret and Dufour impacts on the flow of natural convective nanofluid via a stretched sheet assuming the inclined magnetic field. Moreover, Sheikholeslami et al.¹⁰ observed that the convection mode of nanofluid is strongly affected by the nanoparticle's shapes. Ma et al.¹¹ examined that by enhancing the width of the baffle the average values of Nusselt numbers can be improved. Moreover, Nia et al.¹² noticed that only appropriate baffle arrangements can augment the natural convection in case of high Rayleigh values. Izadi et al.¹³ investigated the melting rate of hydromagnetic ferro phase variation in a cavity due to the location effect of magnetic sources. Hajjar et al.¹⁴ examined the suspension of free convective nano-encapsulated phase alteration materials in the cavity considering a heated wall. Alsabery et al.¹⁵ performed the irreversibility examination of the free convective flow of nanofluids via a wavy cavity.

The heat absorption/generation are noteworthy to achieve the optimal heat transport performance in various physical phenomena and industries such as in post-accident heat exclusion, Earth's mantle convection process, combustion and fire modelling, metal waste development from spent nuclear fuel, thermal power generation, automobiles, etc. Following this, Eldahab and Aziz¹⁶ inspected the mixed convective, hydromagnetic, steady laminar flow through a uninterruptedly stretchable surface considering the effect of internal heat absorption/generation. Further, Elsayed et al.¹⁷ illustrated features the dynamics of heat-absorbing and varying heat flux fluid with heat transfer via a time-dependent stretchable surface and observed that the rise in the heat sink/source parameter results in the significant rise of stretchable surface temperature and consequently reduce the rate of heat transport of the fluid. The analytical solution was obtained by Hussain et al.¹⁸ to scrutinize the impact of heat absorption on the dynamics of convective flow through a vertical accelerated and rotating moving surface under the inspirations of chemical reaction and Hall effects. Their findings reveal that both the solutal and thermal buoyancy forces have an accelerating impact on the fluid velocities in both the secondary and primary directions of the flow, whereas the chemical reaction and heat absorption have a reversal impact on the flow field. In countless industrial phenomena such as in nuclear plants, electrical power generation, manufacturing of energy conversion system equipment, designing of steadfast apparatus, missiles, satellites, gas turbines, etc. the properties of thermal radiation are important for the surface heat transfer. Owing to the novel significance of thermal radiation, numerous investigators have inspected the consequence of solar radiation upshot on the boundary layered flow, considering various geometries. Uddin et al.¹⁹ dissected the impact of radiation on the heat transport appearances of the Cu-water conveying nanofluid through a shirking porous surface and found that the rate of heat transference is significantly raised, and the width of boundary layer is diminished with the improvement of thermal effects. Govone et al.²⁰ highlighted the importance of radiation consequence on the micro-reactors thermochemical simulations and concluded that thermal radiation can lead to the evolution of porous solid phases and bifurcation in the nano-liquid. Recently, Qasem et al.²¹ investigated the radiation impact on the natural convective nano-liquid in a heated cavity wherein the lower and upper dividers are considered as thermally protected. They found that the temperature dispersal profiles of the nano-liquid are increased with an upsurge in the thermal radiation parameter. Few prominent research articles emphasizing the importance of thermal radiation are reported by Hussain et al.,^22–24 Hashim et al.,²⁵ Waqas et al.,²⁶ Thriveni et al.,²⁷ Hussain and Jamshed²⁸ and Khan et al.²⁹

In all the above-reported research work, researchers have overlooked the impact of viscous dissipation. Although the influence of viscous dissipation is considered to be weak, but its significance in instrumentations, polymer industry, processing of food items, lubrication, etc., is important as it increases the features of temperature dispersal and consequently enhances the heat transfer rate. Some published research papers exploring the consequence of viscous dissipation on different flow problems are documented by Vajravelu and Hadjinicolaou,³⁰ Partha et al.,³¹ Aziz³² and Shamshuddin et al.³³ Furthermore, Joule dissipation signifies the volumetric heat source features in the flows of hydromagnetic fluids and the unified impact of viscous and Joule dissipations are significant in different heat-treated materials. Owing to this significance, plentiful investigations were carried by the researchers, namely, Seth et al.,³⁴ Daniel³⁵ and Seth and Singh³⁶ considering unified effects of Joule and viscous dissipations on the flow field. The unified upshots of Joule and viscous dissipation on the boundary layer, two-dimensional, radiative, steady, incompressible, heat-absorbing and electrically conducted fluid flow over a stretchable surface considering thermal and velocity slip conditions into account are reviewed by Seth et al.³⁷ The flow features of hydromagnetic Carreau-nanofluid with Joule heating, thermal radiation and viscous dissipation over a stretchy surface was explored by Atif et al.³⁸ Shah et al.³⁹ scrutinized the effects of Hall current, Joule and viscous dissipation on the electrically conducted squeezing nanofluid flow within the rotating parallel plates considering the model of Cattaneo-Christov heat flux. Recently, Tassaddiq⁴⁰ examined the thermal efficiency of hydromagnetic Cattaneo-Christov hybrid micropolar nanofluid flow under the inspirations of Joule and viscous dissipations. In polymer production, biochemicals, chemical treatment of materials, production of insulated cables, catalysis and combustion mechanism chemical reaction occurs naturally. Owing to this reason, Yasin et al.⁴¹ evaluated the chemical reaction bearing on the heat transference features of two-dimensional, hydromagnetic, heat-absorbing and steady flow induced by a permeable shrinking/stretching surface under the stimulus of thermal radiation. Further, Hashmi et al.⁴² considered the noble characteristics of chemical reaction, Joule heating, and viscous dissipation dissipations on dynamics of Oldroyd-B fluid permeated by exothermally and isothermally stretchable disks.

Inspired by the investigations reported above, in this paper, we intended to examine the thermal radiation and heat absorption influences on the hydromagnetic nanofluid flow over a permeable stretching surface. Physical insight of chemical reaction, Joule and viscous dissipations on the flow is captured by disseminating the nanoparticles of copper into water. The temperature and velocity dispersal profiles are compared by considering three kinds of water-nanofluids, namely, water-Cu, water-TiO₂ and water-Al₂O₃ nanofluids. As per our knowledge as well as the present scenario of modern technology, no one has investigated this problem, which may be useful in various metallurgical processes and polymer technology that involves cooling of the uninterrupted strips.

Problem formulation

In this research investigation, the dynamics of steady-state, electrically conducted, radiative, two-dimensional, hydromagnetic and incompressible nanofluid over a stretchy surface is scrutinized. The effects of chemical reaction, heat absorption, Joule and viscous dissipations are also scrutinized on the flow field. The nanofluids are prepared by disseminating very fine spherical particles (radius less than 50 nanometers) of titanium oxide (TiO₂), copper (Cu) and aluminium oxide (Al₂O₃) into the water (H₂O). The thermo-physical properties of spherical particles of TiO₂, Cu, Al₂O₃ and H₂O are given in Table 1.

Table 1.

The thermo-physical properties of spherical nanoparticles of TiO₂, Cu, Al₂O₃.⁴³

	Al₂O₃	TiO₂	Cu	Water (H₂O)
$c_{p}$ (J/kg K)	765	686.2	385	4179
$ρ$ (kg/m³)	3970	4250	8933	997.1
K (W/mK)	40	8.9538	401	0.613
$ϕ$	0.15	0.20	0.05	0.00
$σ$ (Ω · m)⁻¹	1 × 10⁻¹²	1 × 10⁻¹²	1 × 10⁻¹²	1 × 10⁻¹²

For modelling of this problem, the Cartesian coordinate system is selected where the stretching surface is associated with the $\bar{x}$ -axis; the $\bar{y}$ -axis is in a normal direction to the $\bar{x}$ -axis and the fluid flow is limited to $\bar{y}$ > 0. The physical problem is schematically portrayed in Figure 1.

Figure 1.

The schematic depiction of problem.

The below-mentioned assumptions have been made to carry out the present investigation:

The uniform velocity $λ v_{1 w} (\bar{x})$ where $λ > 0$ is considered to induce the nanofluid velocity over the stretching surface.

The permeable stretching surface mass flux velocity is assumed as $β$ wherein $β$ >0 represents the suction and $β$ <0 signifies the injection.

At the stretchable surface, the species concentration and nanofluid temperature are constant and are denoted by ${\bar{C}}_{1 w}$ and ${\bar{T}}_{1 w}$ respectively whereas those of ambient fluid species concentration and temperature are respectively ${\bar{C}}_{1 \infty}$ and ${\bar{T}}_{1 \infty}$ .

An unvarying magnetic field $M_{0}$ is exerted in the vertical direction to induce the nanofluid over the stretching surface.

The nanoparticles of TiO₂, Cu and Al₂O₃ and water are in the equilibrium state and also no-slip exists between them.

The effects of thermal radiation, heat absorption, Joule and viscous dissipations are considered to improve the heat transfer rate of nanofluid.

Because of the non-appearance of a peripheral imposed magnetic field, the significance of polarization is overlooked.⁴⁴

Small values of magnetic Reynolds number are chosen to curtail the upshot of the induced magnetic field.

Between diffusive species and the nanofluid there exist a linear homogeneous chemical reaction one with a fixed rate $k_{0}$ .

According to the above highlighted assumptions, the governing constitutive equations for the nanofluid velocity, temperature and species concentration are represented as (Yasin et al.,⁴¹ Pal,⁴⁵ Rashidi et al.⁴⁶ and Hussain et al.⁴⁷⁾:

\frac{\partial v_{1}}{\partial \bar{x}} + \frac{\partial v_{2}}{\partial \bar{y}} = 0,

(1)

ρ_{n f} (v_{1} \frac{\partial v_{1}}{\partial \bar{x}} + v_{2} \frac{\partial v_{2}}{\partial \bar{y}}) = μ_{n f} \frac{\partial^{2} v_{1}}{\partial {\bar{y}}^{2}} - σ_{n f} M_{0}^{2} v_{1},

(2)

(ρ C_{p})_{n f} (v_{1} \frac{\partial {\bar{T}}_{1}}{\partial \bar{x}} + v_{2} \frac{\partial {\bar{T}}_{1}}{\partial \bar{y}}) = K_{n f} \frac{\partial^{2} {\bar{T}}_{1}}{\partial {\bar{y}}^{2}} + μ_{n f} {(\frac{\partial v_{1}}{\partial \bar{y}})}^{2} - \frac{\partial q_{r}}{\partial \bar{y}} - Q_{0} ({\bar{T}}_{1} - {\bar{T}}_{1 \infty}) + σ_{n f} M_{0}^{2} v_{1}^{2},

(3)

v_{1} \frac{\partial {\bar{C}}_{1}}{\partial \bar{x}} + v_{2} \frac{\partial {\bar{C}}_{1}}{\partial \bar{y}} = D_{n f} \frac{\partial^{2} {\bar{C}}_{1}}{\partial {\bar{y}}^{2}} - k_{0} ({\bar{C}}_{1} - {\bar{C}}_{1 \infty}) .

(4)

The accompanying initial and boundary conditions (Yasin et al.⁴¹ and Mishra et al.⁴⁸⁾ are:

\begin{matrix} v_{1} = λ v_{1 w} (\bar{x}), v_{2} = β, {\bar{T}}_{1} = {\bar{T}}_{1 w}, {\bar{C}}_{1} = {\bar{C}}_{1 w} when \bar{y} = 0, \\ v_{1} \to 0, {\bar{T}}_{1} \to {\bar{T}}_{1 \infty} and {\bar{C}}_{1} \to {\bar{C}}_{1 \infty} for \bar{y} \to \infty, \end{matrix}}

(5)

where

v_{1 x} (\bar{x}) = a \bar{x}

in which the constant a is non-negative.

The expressions of $μ_{n f}, ϕ_{0}, ρ_{n f}, σ_{n f} and (ρ C_{p})_{n f}$ for the considered nanofluid are articulated in equation (6).

\begin{matrix} ϕ_{0} = (1 - ϕ), ρ_{n f} = ϕ_{0} ρ_{f} + ϕ ρ_{s}, μ_{n f} = ϕ_{0}^{- 2.5} μ_{f}, (ρ C_{p})_{n f} = ϕ_{0} (ρ C_{p})_{f} + ϕ_{0} (ρ C_{p})_{s}, \\ σ_{0} = (σ - 1), σ_{n f} = σ_{f} [1 + \frac{3 σ_{0} ϕ}{(σ_{0} + 3) - σ_{0} ϕ}], σ = \frac{σ_{s}}{σ_{f}} . \end{matrix}}

(6)

It is worthy to remark here that all the defined expressions reported in equation (6) are valid for spherical nanoparticles and are not applicable for other shape nanoparticles.⁴³ Further, the $K_{n f}$ (effective thermal conductivity) for the nanofluids prepared by disseminating the spherical nanoparticles, is defined as:

K_{n f} k_{f}^{- 1} = \frac{k_{s} + 2 k_{f} - 2 ϕ (k_{f} - k_{s})}{k_{s} + 2 k_{f} + ϕ (k_{f} - k_{s})} .

(7)

The Rosseland approximation⁴⁹ is employed for the considered optically thick radiative nanofluid to convert radiative heat flux

q_{r}

in the below mentioned form:

q_{r} = - \frac{4 σ^{*}}{3 k^{*}} \frac{\partial {\bar{T}}_{1}^{4}}{\partial \bar{y}} .

(8)

Furthermore, the temperature variances inside the flow are supposed to be adequately small. Therefore, the term

{\bar{T}}_{1}^{4}

is linearized by expressing it as a Taylor's series about

{\bar{T}}_{1 \infty}

(free stream temperature). This Taylor's series after overlooking the terms having higher degrees is expressed as:

{\bar{T}}_{1}^{4} ≅ 4 {\bar{T}}_{1 \infty}^{3} {\bar{T}}_{1} - 3 {\bar{T}}_{1 \infty}^{4} .

(9)

When, the equations (8) and (9) are used in the equation (3), the result is

(σ C_{p})_{n f} (v_{1} \frac{\partial {\bar{T}}_{1}}{\partial \bar{x}} + v_{2} \frac{\partial {\bar{T}}_{1}}{\partial \bar{y}}) = [K_{n f} + \frac{16 σ * {\bar{T}}_{1} {_{\infty}}^{3}}{3 k^{*}}] \frac{\partial^{2} {\bar{T}}_{1}}{\partial {\bar{y}}^{2}} + μ_{n f} {(\frac{\partial v_{1}}{\partial \bar{y}})}^{2} - Q_{0} ({\bar{T}}_{1} - {\bar{T}}_{1 \infty}) + σ_{n f} M_{0}^{2} v_{1}^{2} .

(10)

The mathematical model represented by the equations (2), (4) and (10) are in terms of nonlinear partial differential equations. Thus, to obtain the analytical solution is quite challenging. Owing to this reason, following similarity variable

η

and stream function

ψ

are introduced to convert these equations in the nonlinear ordinary differential equations forms:

ψ (\bar{x}, \bar{y}) = \bar{x} {(\frac{a}{υ_{f}})}^{1 / 2}, η = \bar{y} {(\frac{a}{υ_{f}})}^{1 / 2} f (η), θ (η) = \frac{{\bar{T}}_{1} - {\bar{T}}_{1 \infty}}{{\bar{T}}_{1 w} - {\bar{T}}_{1 \infty}} and ε (η) = \frac{{\bar{C}}_{1} - {\bar{C}}_{1 \infty}}{{\bar{C}}_{1 w} - {\bar{C}}_{1 \infty}},

(11)

where,

v_{1} = \frac{\partial ψ}{\partial \bar{y}} and v_{2} = - \frac{\partial ψ}{\partial \bar{x}} .

(12)

The relations of equation (11) are used in equation (12), we get

v_{1} / \bar{x} a = f^{'} (η) and v_{2} (a υ_{f})^{- 1 / 2} = - f (η),

(13)

here, differentiation with regard to

η

is indicated by primes.

The equations (2), (10) and (4) are changed to three ordinary differential equations using the equations (11) and (13) and are represented as:

f^{″'} (η) + K_{1} f (η) f^{″} (η) - M K_{2} f^{'} (η) - K_{1} f^{'} (η)^{2} = 0,

(14)

[\frac{K_{3} + R}{Pr ϕ_{4}}] θ^{″} (η) - \frac{1}{ϕ_{4}} Q θ (η) + f (η) θ^{'} (η) + \frac{E c}{ϕ_{4}} [ϕ_{1} {(f^{″} (η))}^{2} + f^{'} {(η)}^{2} M ϕ_{3}] = 0,

(15)

ε^{″} (η) + S c f (η) ε^{'} (η) - S c K ε (η) = 0.

(16)

The resulting allied conditions are:

\begin{matrix} f^{'} = λ, f = S, θ = 1 and ε = 1 when η = 0, \\ θ \to 0, f^{'} \to 0, and ε \to 0 for η \to \infty . \end{matrix}}

(17)

where,

\begin{matrix} ϕ_{1} = (1 - ϕ)^{- 2.5}, ϕ_{2} = (1 - ϕ) + ϕ \frac{ρ_{s}}{ρ_{f}}, ϕ_{3} = 1 + \frac{3 (σ - 1) ϕ}{(σ + 2) - (σ - 1) ϕ}, ϕ_{4} = (1 - ϕ) + ϕ \frac{{(ρ C_{p})}_{s}}{{(ρ C_{p})}_{f}}, \\ K_{1} = \frac{ϕ_{2}}{ϕ_{1}}, K_{2} = \frac{ϕ_{3}}{ϕ_{1}}, K_{3} = \frac{K_{n f}}{k_{f}}, M = \frac{σ_{f}}{ρ_{f}} \frac{M_{0}^{2}}{a}, R = \frac{16 σ^{*} {\bar{T}}_{1 \infty}^{3}}{k_{f} k^{*}}, Pr = \frac{{(ρ C_{p} υ)}_{f}}{k_{f}}, \\ Q = \frac{Q_{0}}{a {(ρ C_{p})}_{f}}, E c = \frac{a^{2} {\bar{x}}^{2}}{{(C_{p})}_{f} ({\bar{T}}_{1 w} - {\bar{T}}_{1 \infty})}, S c = \frac{υ_{f}}{D_{n f}}, K = \frac{k_{0}}{a} and S = - \frac{v_{0}}{\sqrt{a υ_{f}}} . \end{matrix}}

(18)

The negative and positive values of S indicated the injection and suction respectively while Q > 0 indicates the source and Q < 0 signifies the sink.

Wall velocity gradient, local nusselt number and mass flow rate

From engineering and practical applications viewpoint, the skin friction coefficient $C f_{x}$ , local Sherwood number $S h_{x}$ , local Nusselt number $N u_{x}$ are analyzed to capture the shear stress function, mass flow and heat transport rates respectively; which are expressed in dimension form as under:

C f_{x} = τ_{w} (ρ v_{1 w}^{2} (\bar{x}))^{- 1}, N u_{x} = \bar{x} q_{w} (k ({\bar{T}}_{1 w} - {\bar{T}}_{1 \infty}))^{- 1} and S h_{x} = \bar{x} q_{m} (D_{n f} ({\bar{C}}_{1 w} - {\bar{C}}_{1 \infty}))^{- 1},

(19)

where

τ_{w} = μ_{n f} (\partial v_{1} / \partial \bar{y})_{\bar{y} = 0}

q_{m} = - D_{n f} (\partial {\bar{C}}_{1} / \partial \bar{y})_{\bar{y} = 0}

and

q_{w} = - (\partial {\bar{T}}_{1} / \partial \bar{y})_{\bar{y} = 0} + (q_{r})_{\bar{y} = 0}

are the wall shear stress, mass and wall heat fluxes respectively.

The expressions of $C f_{x}$ , $N u_{x}$ and $S h_{x}$ in dimensionless forms are obtained as:

C f_{x} = {Re}_{x}^{- 1 / 2} f^{″} (0), N u_{x} = - {(4 R + 3) / 3} {Re}_{x}^{- 1 / 2} θ^{'} (0) and S h_{x} = - ε^{'} (0) {Re}_{x}^{- 1 / 2},

(20)

where

R e_{x} = \bar{x} {\bar{u}}_{w} / υ

is the local Reynolds number.

Numerical solution

The analytical solutions of governing equations (14)-(16) with the accompanying conditions (17) are not possible owing to the high nonlinearity and intricacy. Therefore, the numerical solution of these equations satisfying the accompanying conditions asymptotically are obtained by utilizing the Runge-Kutta-Fehlberg technique (RKF45) in conjunction with the shooting method (Gerald and Wheatley⁵⁰ and Kundu and Sarkar.⁵¹⁾ Further, the Runge-Kutta-Fehlberg technique is accomplished to solve these resulting differential equations wherein the values of $θ^{'} (0), f^{″} (0),$ and $ε^{'} (0)$ are analyzed through the shooting technique. The details of Shooting method and RKF45 technique are mentioned below:

Shooting method

This method is implemented to solve a boundary value problem (BVP) by converting it to the form of an initial value problem. To execute this technique, foremost, from the dimensionless equations (14)-(16), five extremely coupled, nonlinear and first-order differential equations are derived. Further, the random value (guess) at one end of the initial value problem (IVP) is chosen and at the other end, we try to obtain a solution by implementing RKF45 technique by considering appropriate step length l. If we get the convergent solution then we assume that our guess is accurate and correct otherwise we use the Secant method to re-guess. This process is repeated until to get the desired convergent and accurate solution at the other end.

RKF45 technique (Runge-Kutta-Fehlberg technique)

The reliable and convergent solution of an initial value problem can be obtained by solving the problem two times by considering the step lengths l and $l / 2$ respectively and the results are compared at the mess points to the higher step lengths. But these processes involve a substantial amount of numerical computation corresponding to smaller step lengths and the processes are repeated in the case of non-convergent solutions. This problem can be resolved by implementing the Runge-Kutta-Fehlberg technique (RKF45) by considering the appropriate step length l. To execute this method, two approximations are being considered for the solution and compared at each step. The considered approximations are accepted if the two solutions are in close agreement. If the solution does not converge to the desired accuracy, then the step lengths are reduced. If the solution converges to more significant digits than the required value then the step lengths are enhanced.

At each step, the following six values are being used:

\begin{matrix} h_{1} = l f_{1} (t_{h}, y_{h}), \\ h_{2} = l f_{2} (t_{h} + \frac{1}{4} l, y_{h} + \frac{1}{4} h_{1}), \\ h_{3} = l f_{3} (t_{h} + \frac{3}{8} l, y_{h} + \frac{3}{32} h_{1} + \frac{9}{32} h_{2}), \\ h_{4} = l f_{4} (t_{h} + \frac{12}{13} l, y_{h} + \frac{1932}{2197} h_{1} - \frac{7200}{2197} h_{2} + \frac{7296}{2197} h_{3}), \\ h_{5} = l f_{5} (t_{h} + l, y_{h} + \frac{439}{216} h_{1} - 8 h_{2} + \frac{3680}{513} h_{3} - \frac{845}{4104} h_{4}), \\ h_{6} = l f_{6} (t_{h} + \frac{1}{2} l, y_{h} - \frac{8}{27} h_{1} + 2 h_{2} - \frac{3544}{2565} h_{3} + \frac{1859}{4104} h_{4} - \frac{11}{40} h_{5}) . \end{matrix}}

(21)

Then following the 4^th order Runge-Kutta method, an approximate numerical solution of the IVP is obtained using the below-mentioned relation:

y_{h + 1} = y_{h} + \frac{25}{216} h_{1} + \frac{1408}{2565} h_{3} + \frac{2197}{4101} h_{4} - \frac{1}{5} h_{5} .

(22)

From above relation (22), one point to be remarked that the functional values

f_{1}, f_{3}, f_{4} and f_{5}

have been used to analyze the approximate value of

y_{h + 1}

while

f_{2}

has not been used. Further, the improved approximate solution is obtained with the help of 5^th order Runge-Kutta method:

z_{h + 1} = y_{h} + \frac{16}{135} h_{1} + \frac{6656}{12825} h_{3} + \frac{28561}{56430} h_{4} - \frac{9}{50} h_{5} + \frac{2}{55} h_{6} .

(23)

It is worthy to remark here that the ideal step length

(s l)

can be obtained by multiplying the current step length l with the scalar quantity s. The value of s is determined by using the following relation:

s = {(\frac{tol l}{2 | z_{h + 1} - y_{h + 1}})}^{1 / 4} \approx 0.84 {(\frac{tol l}{2 | z_{h + 1} - y_{h + 1}})}^{1 / 4},

(24)

where tol represents the error control tolerance.

One can found the above-mentioned formula (24) in any advanced text book of numerical analysis. It is significant to mention here that the consideration of fixed step length is not the best strategy even though we get the better convergent solution

Results, discussion and validation

Owing to the intricacy and highly nonlinear nature, the numerical solutions of governing similarity equations (14)-(16) composed with the accompanying conditions (17) are obtained executing the Runge-Kutta-Fehlberg technique in conjunction with the shooting method. The significance of prompting flow parameters on the nanofluid velocity $f^{'} (η)$ , dispersal profiles of temperature $θ (η)$ , nanofluid species concentration $ε (η)$ , shear stress function $C f_{x},$ wall temperature gradients $N u_{x}$ and mass flow rate $S h_{x}$ are analyzed using numerous graphical results. Also, the physical explanations of effective flow parameters characteristics are provided. In this examination, three different nanofluids are prepared by dispersing spherical particles having a radius less than 50 nanometers of TiO₂, Cu and Al₂O₃ into the water (H₂O). In the process of numerical computations, following the research works of Seth et al.,³⁷ Pal,⁴⁵ Rashidi et al.,⁴⁶ Hussain et al.,^47,52 Kumar et al.,⁵³ Waini et al.,⁵⁴ Hussain⁵⁵ and Bhattacharyya et al.,⁵⁶ the fixed values of various flow parameters are chosen as $M = 0.2, ϕ = 0.05,$ $S = 0.1, λ = 0.3, E c = 0.1$ , $R = 0.1$ , $Q = 0.1$ , $S c = 1.1$ , $K = 1.1$ and $Pr = 6.2$ (as water is base fluid) unless specified otherwise. The computational results and executed numerical technique are authenticated by the numerical results comparison of local Nusselt number $N u_{x}$ with the results of Pal⁴⁵ and Rashidi et al.⁴⁶ under special assumptions. The compared results show an outstanding agreement between the results as is noticed from Table 2. This suggests that the numerical results of the paper and the executed numerical technique are valid and acceptable.

Table 2.

The values of $N u_{x}$ in numerical form computed in present study are compared with those of (i) Pal⁴⁵ computed by considering S = $λ$ = R = $ϕ$ = 0, Q = 0.1, P_r = 6.2, E_C = 0.1 and (ii) Rashidi et al.⁴⁶ analyzed by assuming S = E_C = 0, R = 0.1, $λ$ = 1.3.

M	Pal⁴⁵	Rashidi et al.⁴⁶	Present result
0.2	−1.635433	−1.635431	−1.635436
2.2	−1.495125	−1.495120	−1.495129
4.2	−1.394246	−1.394251	−1.394243
6.2	−1.314157	−1.314154	−1.314159

The distribution profiles of velocity and temperature are compared for three water-driven nanofluids, prepared by mixing very fine spherical particles of TiO₂, Al₂O₃ and Cu into the water through the depicted Figures 2 and 3. This comparison of profiles reveals that TiO₂-water nanofluid has higher temperature dispersal and velocity profiles followed by Al₂O₃-water nanofluid while Cu-water nanofluid has the lowest temperature circulation and velocity profiles. Further, for the particular values of controlling flow parameters, these distribution profiles are nearly overlapped with each other and approach to free stream value.

Figure 2.

Comparison of $f^{'} (η)$ profiles for different nanofluids.

Figure 3.

Comparison of $θ (η)$ profiles for different nanofluids.

The effects of volume fraction of nanoparticles $ϕ$ on the Cu and water-nanofluid velocity $f^{'} (η)$ and temperature $θ (η)$ distribution profiles are captures via Figures 4 and 5. These figures portray that the velocity of the nanofluid can be reduced whereas its temperature can be escalated by improving the volume fraction of copper nanoparticles. These findings seem to be justified as the thermal conductivity features of nanofluid are reduced with the rise in Cu-nanoparticles volume fraction; which diminishes the boundary layer thickness and improves the viscosity of the fluid and consequently, the nanofluid temperature gets enhanced while velocity is retarded. As revealed in Figures 6 and 7, an upsurge in the magnetic effect indicates a reduction in the nanofluid wall velocity while it augments the fluid temperature. This behavior of the magnetic field is valid because the magnetic field strength induces an opposing force, which resists the fluid motion and hence diminishes the nanofluid wall velocity and induces the fluid temperature.

Figure 4.

Dispersal profiles of $f^{'} (η)$ versus $ϕ$ .

Figure 5.

Distribution profiles of $θ (η)$ versus $ϕ$ .

Figure 6.

Dispersal profiles of $f^{'} (η)$ versus M.

Figure 7.

Distribution profiles of $θ (η)$ versus M.

It is enumerated from the supplementary Figures 8S and 9S that the nanofluid temperature and velocity are enhanced owing to the increment in the velocity slip parameter $λ .$ Generally, the stretching surface and fluid velocities are different when the slip phenomenon occurs also pulling off the stretching surface is moderately transmitted to the fluid. Due to this reason the nanofluid temperature and velocity are heightened with the rise in the values of $λ .$ The supplementary Figures 10S and 11S signify the reducing characteristics of the suction parameter $S (> 0)$ on the nanofluid temperature dispersal and velocity profiles whereas the profiles are overturned for the injection parameter $S (< 0)$ . These characteristics of profiles are due to the reason that under the occurrence of suction, the momentum boundary layer sticks to the stretching surface compactly which in turn demolish the momentum of flow and hence, the nanofluid velocity is retarded. In contrast, injection induces the fluid momentum through its lateral mass flux over the stretching surface and subsequently the nanofluid velocity is enhanced. The significance of viscous dissipation on the Cu-water nanofluid temperature is captured through the supplementary Figure 12S. A rise in the Eckert number $E c$ indicates an augmentation in the temperature dispersal profiles, which reveals that the presence of viscous dissipation effect is significant to upsurge the nanofluid temperature. The bearings of heat absorption and radiation effect are captured through the graphical illustrations reported in the supplementary Figures 13S and 14S. These graphical illustrations show that increasing the heat absorption parameter Q reasons the nanofluid temperature to fall down due to enrichment in the heat-absorbing capacity of the nanofluid. Further, the temperature distribution profiles get boosted due to rising values of radiation parameter R. Generally, the radiation effect enhances the conduction properties of the nanofluid which expand the thickness of the boundary layer and consequently the temperature of nanofluid gets elevated. The effects of Sc (Schmidt number) and K (chemical reaction parameter) on the species concentration profiles of the nanofluid are displayed in the supplementary Figures 15S and 16S. It is apparent from these depicted figures that as the values of S_C and K increase the species concentration profiles are reduced which disclose that the chemical reaction has a reducing impact on the species concentration though the mass diffusion leaves a reversed effect on it. The supplementary Figures 17S and 18S show the impressions of magnetic parameter M and Cu-nanoparticle volume fraction $ϕ$ on the wall velocity gradient $C f_{x}$ and the Nusselt number $N u_{x}$ . It is evident from both the figures that as the values of M and $ϕ$ increase the values of $| C f_{x} |$ are reduced while $| N u_{x} |$ are increased. This implies that the wall velocity gradient is reduced while the heat transport rate is augmented due to enhancement in the magnetic field strength and volume fraction of Cu-nanoparticles. The features of injection/suction and velocity slip parameters i.e., S and $λ$ are scrutinized through the supplementary Figures 19S and 20S. It is exemplified that shear stress over the stretchable surface is declined due to an increase in the suction and velocity slip factor while it is diminished owing to a rise in injection. Further, the heat transport rate of the fluid is improved owing to the inspirations of velocity slip factor and suction whereas injection has the opposite characteristics. The comparisons of the wall velocity gradient $C f_{x}$ and the Nusselt number $N u_{x}$ profiles for the TiO₂-water, Cu-water and Al₂O₃-water have been presented in supplementary Figures 21S and 22S. It is observed from these comparisons that the shear stress and the heat transference rate of TiO₂-water nanofluid are maximum at the stretching surface followed by Cu-water and Al₂O₃-water nanofluids. The significance of mass diffusion and linear chemical reaction on the mass flow rate is examined via the supplementary Figure 23S. It is professed that the Sherwood number values are reduced with the rising values of Sc and K. This finding discloses that the mass flow rate can be augmented due to the inclusion of mass diffusion while the chemical reaction is responsible to reduce the mass flow rate.

Quadratic regression approximations

In this section, a study of quadratic regression approximation has been accomplished on the numerical findings of the wall temperature gradient and the shear stress function to understand how the properties of dimensionless flow parameters change owing to the slight variation of other flow parameters. During the regression approximation analysis, the remaining flow parameters are considered to be fixed. The approximated numerical entities of $C f_{x}$ and $N u_{x}$ are figured for 110 groups of different values of S and $ϕ$ which are considered randomly within the intervals [0.05, 0.5] and [0.05,0.20] respectively, while fixed values of other dimensionless parameters are selected as highlighted in previous section. The regression models for the reduced $C f_{x}$ and $N u_{x}$ are presented below:

C f_{q r e} = C f_{x} + d_{1} S + d_{2} ϕ + d_{3} S^{2} + d_{4} ϕ^{2} + d_{5} S ϕ,

(21)

N u_{q r e} = N u_{x} + e_{1} S + e_{2} ϕ + e_{3} S^{2} + e_{4} ϕ^{2} + e_{5} S ϕ,

(22)

where d₁, d₂, d₃, d₄, d₅ are the approximated skin friction coefficients and

e_{1}, e_{2}, e_{3}, e_{4}, e_{5}

are the estimated values of the coefficients of reduced local Nusselt number.

The approximated values of the coefficients have been obtained from the quadratic regression approximation analysis corresponding to the estimated $C f_{x}$ and $N u_{x}$ for enhancing values of velocity slip parameter $λ$ and magnetic parameter M. These approximated values have been presented in the supplementary Tables 3S and 4S. In addition, the optimal relative error bounds for the estimated $C f_{x}$ and $N u_{x}$ have been also computed using the relations $ε_{c} = | C f_{q r e} - C f_{x} | / C f_{x}$ and $ε_{n} = | N u_{q r e} - N u_{x} | / N u_{x}$ respectively. It is perceived from the supplementary Tables 3S and 4S that coefficients of magnetic parameter M are smaller as compared to the coefficients of velocity slip parameter $λ$ . This indicates that a slight difference in the parameter of velocity slip factor causes a maximum perturbation in both the wall velocity gradient and heat transference rate in contrast to the magnetic effect. Further, it is remarked that the optimal relative error of approximated skin friction coefficients is nearly zero and the convergent rate to this accurateness is quicker as compared to the quadratic regression approximated values of the wall temperature gradient.

Conclusions

In this research study, the influence of heat absorption and thermal radiation on the hydromagnetic nanofluid flow via a permeable stretching surface has been examined. Physical insight of chemical reaction, Joule and viscous dissipations on the flow is captured by disseminating the spherical nanoparticles of copper into the water. The important findings of the explorations are summarized below:

The TiO₂-water nanofluid temperature and velocity profiles are higher as compared to Al₂O₃ and Cu-water nanofluids.

The nanofluid velocity are strongly retarded owing to augmentation in volume fractions of Cu-nanoparticle, velocity slip factor and magnetic effects while these parameters have reversal impacts on the temperature dispersal profiles.

The Cu-water fluid temperature and velocity are upsurged due to the injection while suction parameter has adverse impact.

The viscous dissipation is accountable to improve the temperature of Cu-water nanofluid.

The wall velocity gradient is reduced due to the enhancement in the values of suction parameter and velocity slip parameter whereas it gets heightened owing to the injection parameter.

The heat transport rate over the surface of stretching sheet can be increase by improving the effects of thermal radiation, thermal slip factor and viscous dissipation.

The shear stress and the heat transference rate of TiO₂-water nanofluid are maximum at the stretching surface followed by Cu-water and Al₂O₃-water nanofluids.

The chemical reaction declines the mass transport rate of Cu-water nanofluid whereas mass diffusion has an opposite influence.

A minor change in the velocity slip parameter reasons a maximum perturbation in both the wall velocity gradient and heat transfer rate in contrast to the magnetic effect.

The optimal relative error of approximated skin friction coefficients is nearly zero and the convergent rate to this accuracy is more as compared to quadratic regression approximated values of the wall temperature gradient.

Supplemental Material

sj-docx-1-pie-10.1177_09544089221096103 - Supplemental material for Dynamics of heat absorbing and radiative hydromagnetic nanofluids through a stretching surface with chemical reaction and viscous dissipation

Supplemental material, sj-docx-1-pie-10.1177_09544089221096103 for Dynamics of heat absorbing and radiative hydromagnetic nanofluids through a stretching surface with chemical reaction and viscous dissipation by Syed M. Hussain, M. R. Mishra, G. S. Seth and Ali J. Chamkha in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Footnotes

Acknowledgement

Authors are obliged to the Deanship of Scientific Research, Islamic University of Madinah, Ministry of Education, KSA for providing the support to execute their research work via research project grant: Tamayuzz Program/2/490.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Syed M. Hussain

Supplemental material

Supplemental material for this article is available online.

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