1 Introduction

The no-slip boundary situation is acknowledged as the crucial rudiments of the Navier–Stokes theory. But there are circumstances wherein such condition is not proper. Mostly, no-slip condition is deficient for maximum non-Newtonian fluids. For illustration, polymer melts frequently show noticeable wall slip and that is denoted by a nonlinear and monotone fitting together amongst the slip velocity and traction. Moreover, partial velocity slips across a moving surface occur for fluids with particulate, for illustration, polymer solutions, emulsions, foams and suspensions. Firstly, Cohen and Metzner [1] demonstrated that the apparent slip phenomenon is an imperative feature of the flow behaviour of polymer solutions in inhomogeneous stress field. Lawal and Kalyon [2, 3] deliberated the wall slip in their study on the viscoelastic fluid flow amongst two parallel surfaces. They recognized the fact that the slip coefficient weakens the velocity gradient and substantially controls the temperature intensification. By considering stretching sheet, Ariel [4, 5] scrutinized the Newtonian and non-Newtonian fluid flows in the company of partial slip. He perceived that the flow gets hastened for distances close to the sheet upon the raise in viscoelastic fluid parameter. Fang et al. [6, 7] analytically resolved the equations which were raised in the work on the MHD fluid flow over a shrinking/stretching sheet. They revealed that the wall slip velocity upsurges with the rise of magnetic field parameter. Zheng et al. [8] inspected the features of heat transfer on the stagnation point flow of nanofluids across a stretching sheet. One of their verdicts is that the thermophoresis and Brownian motion parameters both display the same impact on temperature. Rosca and Pop [9] did the stability analysis to display the twin solutions in their work on the two-dimensional viscous fluid flow across a vertical shrinking/stretching sheet. Malvandi et al. [10] and Ul Haq et al. [11] elucidated the impacts of (temperature and velocity) slip parameters on the stagnation point flow of nanofluid. Devi and Prakash [12] and Jayachandra Babu and Sandeep [13] analysed different MHD flows in the occurrence of slip parameters. Some of their detections are as follows: (a) temperature jump parameter lowers the temperature, and (b) wall thickness parameter lessens all the profiles (velocity, concentration and temperature). Mustafa [14] numerically deliberates the influence of slip parameters on nanofluid flow over a rotating disc. Usman et al. [15] and Ullah et al. [16] elucidated the slip effects on Casson fluid flow with viscous dissipation. They found the consequences: (a) thermal slip lowers the Nusselt number, and (b) viscous dissipation boosts the temperature. Abbas et al. [17] used Runge–Kutta–Fehlberg system to resolve the equations which were formulated in the work on the micropolar nanofluid flow near stagnation point. Recently, some researchers [18,19,20,21,22] deliberated various MHD flows in the slip regime.

When the plate is cooling or heating, the approximations of Joule heating and viscous dissipation are progressively noteworthy. The noticeable reality of heat transport naturally arises in the procedures of power generation systems, cooling of nuclear reactors, cooling of electronic chips or metallic sheets and liquid metal fluids. Firstly, Javeri [23] examined the effects of Joule heating and viscous dissipation on the MHD fluid flow along with Hall effect and ion slip. Later, Borisevich and Potanin [24] studied the heat transfer characteristics of incompressible fluid flow near a rotating disc. They observed that the temperature upsurges with the rise in the viscous dissipation. Mansour and Rama Subba Reddy [25] and Hakiem et al. [26] analysed the convective micropolar fluid flow with the same effects across different channels. They detected that the magnetic field parameter lowers the friction factor. Yih [27] employed implicit finite difference method to resolve the equations which were raised in the problem on the convective flow of non-Darcy fluid through a permeable sphere immersed in a porous medium. By considering the vertical surface, Chen [28] and Abo-Eldahad and Aziz [29] deliberated the heat transfer characteristics of two-dimensional fluid flow. One of their detections is that the Schmidt number improves the local Scherwood number. Osalusi et al. [30, 31] explained diverse MHD fluid flows through dissimilar channels. Some of their outcomes are as follows: (a) Hall parameter boosts the temperature, and (b) ion-slip current parameter rises the temperature. Later, several researchers [32,33,34] inspected various MHD flows including non-Newtonian fluid flow across various channels including a plane vertical channel. Jafar et al. [35] reported that the fluid temperature and velocity decreased with the increase in the shrinking of the sheet in their problem on the stagnation point flow through a shrinking/stretching surface. By keeping heat flux as constant, Haque et al. [36] and Chakraborty et al. [37] investigated the behaviour of various MHD fluid flows including micropolar fluid flow across narrow flow conduits. They saw that the temperature of the micropolar fluid is less for water contrast to air. Hayat et al. [38] employed HAM to solve the equations which were raised in the work on the unsteady MHD fluid flow across an exponentially stretching surface. Raju et al. [39] and Das et al. [40] deliberated convective MHD flows through porous medium. They revealed that the Prandtl number lowers the mean mixed temperature. Sheikholeslami and Ganji [41] utilized DTM to resolve the equations in the work on the scrutiny of nanofluids flow amongst two vertical permeable plates. They found that the shear stress decreases with the intensification of Lorentz forces. Recently, several researchers [42,43,44] contributed to the work on different flows accompanied with Ohmic (Joule) heating and viscous dissipation.

In this paper, the effects of Joule (Ohmic) heating, slip parameters and viscous dissipation on the two-dimensional Carreau fluid flowing through a stretching sheet with variable thickness are examined. Shooting technique is employed in this work to resolve the transmuted equations. To capture the changes and examine the common profiles (velocity, concentration and temperature) against several parameters together with viscous dissipation, plots were presented. And also, we have presented the physical values of friction factor, the rate of transfer (of both mass and heat) in the table for the same parameters. From the outcomes, it is found that velocity slip parameter lowers the velocity but boosts the friction factor, and Weissenberg number improves the velocity though displays reverse behaviour on temperature.

2 Mathematical formulation

A steady, laminar, incompressible, two-dimensional flow of non-Newtonian fluid (Carreau fluid) across a variable thickened stretching sheet is considered. The positive \(x\)-coordinate is measured along the direction of the moving sheet motion, and the positive \(y\)-coordinate is measured perpendicular to the sheet in the outward direction towards the fluid. We have applied a magnetic field of strength \(B_{0}\) normal to the flow as shown in Fig. 1, which produces a magnetic effect in the \(x\)-direction. We assume that the stretching velocity is \(U_{w} \left( x \right) = J_{0} \left( {x + b} \right)^{m}\) (\(J_{0}\) is a constant and \(m\) is the velocity power index parameter). The sheet is not flat and is designated as \(y = A\left( {x + b} \right)^{{{\raise0.7ex\hbox{${\left( {1 - m} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {1 - m} \right)} 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}\), and the coefficient \(A\) is chosen as small for the sheet to be sufficiently thin, to avoid pressure gradient along the sheet \(\left( {\frac{\partial P}{\partial x} = 0} \right)\). As we are considering stretching sheet with non-uniform thickness, this problem is valid for \(m \ne 1.\) The temperature and the concentration at the surface have constant values \(T_{w}\) and \(C_{w}\), respectively. Hydrodynamic, thermal slip conditions are taken into the account at wall and Joule heating, and viscous dissipation effects are incorporated in the energy equation. Here magnetic field parameter is defined as \(B\left( x \right) = B_{0} \left( {x + b} \right)^{{\frac{m - 1}{2}}}\). Magnetic Reynolds number is presumed to be as fewer as possible in order to disregard the induced magnetic field. As the induced magnetic field is assumed to be negligible and since \(B\left( x \right)\) is independent of time, \({\text{curl}}(\vec{E}) = 0\). In the absence of surface charge density, \({\text{div}}(\vec{E}) = 0\). Hence, the external electric field is assumed as negligible.

Fig. 1
figure 1

Physical model of the problem

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With these conditions, the leading equations [45] are

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$
(1)
$$u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = \upsilon \frac{{\partial^{2} u}}{{\partial y^{2} }}\left[ {1 + \varGamma^{2} \left( {\frac{\partial u}{\partial y}} \right)^{2} } \right]^{{\frac{n - 1}{2}}} + \upsilon \left( {n - 1} \right)\varGamma^{2} \frac{{\partial^{2} u}}{{\partial y^{2} }}\left( {\frac{\partial u}{\partial y}} \right)^{2} \left[ {1 + \varGamma^{2} \left( {\frac{\partial u}{\partial y}} \right)^{2} } \right]^{{\frac{n - 3}{2}}} + g\beta_{T} \left( {T - T_{\infty } } \right) + g\beta_{C} \left( {C - C_{\infty } } \right) - \frac{{\sigma B^{2} (x)}}{\rho }u$$
(2)
$$u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} = \frac{k}{{\rho c_{p} }}\frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{\sigma B^{2} \left( x \right)}}{{\rho c_{p} }}u^{2} + \frac{\upsilon }{{c_{p} }}\left( {\frac{\partial u}{\partial y}} \right)^{2}$$
(3)
$$u\frac{\partial C}{\partial x} + v\frac{\partial C}{\partial y} = D_{m} \frac{{\partial^{2} C}}{{\partial y^{2} }} - k_{0} (C - C_{\infty } )$$
(4)

and the matching boundary conditions are

$$\left. {\begin{array}{*{20}l} {u\left( {x,y} \right) = U_{w} \left( x \right) + h_{1}^{*} \left( {\frac{\partial u}{\partial y}} \right),v\left( {x,y} \right) = 0,T\left( {x,y} \right) = T_{w} \left( x \right) + h_{2}^{*} \left( {\frac{\partial T}{\partial y}} \right),} \hfill \\ {C\left( {x,y} \right) = C_{w} \left( x \right)\quad {\text{at }}\,\,y = A\left( {x + b} \right)^{{\frac{1 - m}{2}}} } \hfill \\ {u\left( {x,\infty } \right) = 0,T\left( {x,\infty } \right) = T_{\infty } ,C\left( {x,\infty } \right) = C_{\infty } } \hfill \\ \end{array} } \right\}$$
(5)

where \(u,v\) are the velocity components in parallel to x- and y-directions, respectively, \(\upsilon\) is the kinematic viscosity, \(\varGamma\) is the relaxation time of the Carreau fluid, \(n\) is the power law index parameter, g is the gravity field, \(\beta_{T}\) is the volumetric coefficient of thermal expansion, \(T_{w}\) is the surface temperature, \(\beta_{C}\) is the volumetric coefficient of diffusion expansion, \(\sigma\) is the electrical conductivity, \(\rho\) is the density of the fluid, \(B(x)\) is the dimensional magnetic field parameter, \(k\) is the thermal conductivity, \(C_{p}\) is the specific heat capacitance, \(\mu\) is the dynamic viscosity, \(C\) is the dimensional concentration parameter, \(D_{m}\) is the molecular diffusivity, \(k_{0}\) is the dimensional chemical reaction parameter, \(h_{1}^{*}\) is the dimensional velocity slip parameter, and \(h_{2}^{*}\) is the dimensional temperature jump parameter.

Here \(h_{1}^{*} = \left[ {\frac{2}{{f_{1} }} - 1} \right]\xi_{1} \left( {x + b} \right)^{{\frac{ - m + 1}{2}}}\), \(h_{2}^{*} = \left[ {\frac{2}{e} - 1} \right]\xi_{2} \left( {x + b} \right)^{{\frac{ - m + 1}{2}}} ,\xi_{2} = \left( {\frac{2\gamma }{1 + \gamma }} \right)\frac{{\xi_{1} }}{\Pr }\), where \(f_{1}\) is the Maxwell’s reflection coefficient, \(\xi_{1} ,\xi_{2}\) are mean free path constants, \(e\) is the thermal accommodation coefficient, and \(\gamma\) is the ratio of specific heats.

The vital transmutations to alter the leading equations as (nonlinear coupled) ordinary differential equations are specified below:

$$\begin{aligned} \eta &= y\sqrt {\frac{m + 1}{2}J_{0} \frac{{\left( {x + b} \right)^{m - 1} }}{\upsilon }} , \hfill \\ u &= J_{0} (x + b)^{m} F'\left( \eta \right),v = - \sqrt {\frac{m + 1}{2}\upsilon J_{0} \left( {x + b} \right)^{m - 1} } \left[ {F\left( \eta \right) + \left( {\frac{m - 1}{m + 1}} \right)\eta F'\left( \eta \right)} \right], \hfill \\ \varTheta \left( \eta \right) &= \frac{{T - T_{\infty } }}{{T_{w} - T_{\infty } }},\varPhi \left( \eta \right) = \frac{{C - C_{\infty } }}{{C_{w} - C_{\infty } }} \hfill \\ \end{aligned}$$
(6)

By using (6), Eqs. (2)–(4) transmuted as the subsequent equations:

$$\frac{m + 1}{2}\left[ {1 + {\text{We}}^{2} F''^{2} } \right]^{{\frac{n - 3}{2}}} \left[ {1 + n{\text{We}}^{2} F''^{2} } \right]F''' + \frac{m + 1}{2}FF'' - mF'^{2} + \chi \varTheta + \chi N^{*} \varPhi - MF' = 0$$
(7)
$$\varTheta '' + \Pr F\varTheta ' + \frac{2}{m + 1}M\Pr {\text{Ec}}F'^{2} + \Pr {\text{Ec}}F''^{2} = 0$$
(8)
$$\varPhi '' + {\text{Sc}}F\varPhi ' - \frac{2}{m + 1}ScKr\varPhi = 0$$
(9)

and the boundary conditions (5) altered as

$$\begin{aligned} F\left( \lambda \right) &= \frac{1 - m}{1 + m}\lambda \left[ {1 + h_{1} F''\left( \lambda \right)} \right],F'\left( \lambda \right) = 1 + F''\left( \lambda \right), \hfill \\ \varTheta \left( \lambda \right) &= 1 + h_{2} \varTheta '\left( \lambda \right),\varPhi \left( \lambda \right) = 1, \hfill \\ F'\left( \infty \right) &= 0,\varTheta \left( \infty \right) = 0,\varPhi \left( \infty \right) = 0 \hfill \\ \end{aligned}$$
(10)

where \(\lambda = A\sqrt {\frac{{J_{0} (m + 1)}}{2\upsilon }}\) is the wall thickness parameter.

Equations (7)–(9) can be transmuted into the system of dimensionless expressions with the help of the following:

$$F\left( \eta \right) = f\left( {\eta - \lambda } \right) = f\left( \zeta \right),\,\,\varTheta \left( \eta \right) = \theta \left( {\eta - \lambda } \right) = \theta \left( \zeta \right),\,\,\varPhi \left( \eta \right) = \phi \left( {\eta - \lambda } \right) = \phi \left( \zeta \right)$$
(11)

Then, the consequential equations are

$$\frac{m + 1}{2}\left[ {1 + {\text{We}}^{2} f''^{2} } \right]^{{\frac{n - 3}{2}}} \left[ {1 + n{\text{We}}^{2} f''^{2} } \right]f''' + \frac{m + 1}{2}ff'' - mf'^{2} + \chi \theta + \chi N^{*} \phi - Mf' = 0$$
(12)
$$\theta '' + \Pr f\theta ' + \frac{2}{m + 1}M\Pr {\text{Ec}}f'^{2} + \Pr {\text{Ec}}f''^{2} = 0$$
(13)
$$\phi '' + {\text{Sc}}f\phi ' - \frac{2}{m + 1}{\text{ScKr}}\phi = 0$$
(14)

and the subsequent boundary conditions are

$$\begin{aligned} f\left( 0 \right) &= \frac{1 - m}{1 + m}\lambda \left[ {1 + h_{1} f''\left( 0 \right)} \right],\;f'\left( 0 \right) = 1 + f''\left( 0 \right), \hfill \\ \theta \left( 0 \right) &= 1 + h_{2} \theta '\left( 0 \right),\;\phi \left( 0 \right) = 1, \hfill \\ f'\left( \infty \right) &= 0,\theta \left( \infty \right) = 0,\;\phi \left( \infty \right) = 0 \hfill \\ \end{aligned}$$
(15)

Here Weissenberg parameter \({\text{We}}\), mixed convection parameter \(\chi\), ratio of the concentration to the thermal buoyancy forces \(N^{*}\), thermal Grashof number \({\text{Gr}}\), mass diffusion Grashof number \({\text{Gc,}}\) Magnetic field parameter \(M,\) Eckert number, Prandtl number \(\Pr ,\) Schmidt number \({\text{Sc,}}\) chemical reaction parameter \({\text{Kr}}\), \(h_{1}\) velocity slip parameter and \(h_{2}\) temperature jump parameter are defined as follows:

$$\begin{aligned} {\text{We}} &= \left[ {\frac{{J_{0}^{3} \left( {m + 1} \right)\varGamma^{2} \left( {x + b} \right)^{3m - 1} }}{2\upsilon }} \right]^{{\frac{1}{2}}} ,\chi = \frac{Gr}{{\text{Re}_{x}^{2} }},N^{*} = \frac{Gc}{Gr}, \hfill \\ {\text{Gr}}& = \frac{{g\beta_{T} \left( {T_{w} - T_{\infty } } \right)\left( {x + b} \right)^{3} }}{{\upsilon^{2} }},{\text{Gc}} = \frac{{g\beta_{C} \left( {C_{w} - C_{\infty } } \right)\left( {x + b} \right)^{3} }}{{\upsilon^{2} }}, \hfill \\ M &= \frac{{\sigma B_{0}^{2} }}{{\rho J_{0} }},\Pr = \frac{{\mu c_{p} }}{k},{\text{Ec}} = \frac{{U_{w}^{2} }}{{c_{p} \left( {T_{w} - T_{\infty } } \right)}},{\text{Sc}} = \frac{\upsilon }{{D_{m} }},{\text{Kr}} = \frac{{k_{0} }}{{J_{0} (x + b)^{m - 1} }}, \hfill \\ h_{1} &= \left[ {\frac{2}{{f_{1} }} - 1} \right]\xi_{1} \sqrt {\frac{{J_{0} \left( {m + 1} \right)}}{2\upsilon }} ,h_{2} = \left[ {\frac{2}{e} - 1} \right]\xi_{2} \sqrt {\frac{{J_{0} \left( {m + 1} \right)}}{2\upsilon }} \hfill \\ \end{aligned}$$
(16)

The dimensional parameters of interest, local skin friction coefficient \(C_{fx}\), the local Nusselt number \({\text{Nu}}_{x}\) and the local Sherwood number \({\text{Sh}}_{x}\) are defined as the following:

$$C_{{f_{x} }} = \frac{{\tau_{w} }}{{\rho U_{w}^{2} \left( x \right)}},{\text{Nu}}_{x} = \frac{{\left( {x + b} \right)q_{w} }}{{k\left( {T_{w} - T_{\infty } } \right)}},{\text{Sh}}_{x} = \frac{{\left( {x + b} \right)q_{m} }}{{\left( {C_{w} - C_{\infty } } \right)}}$$
(17)

where

$$\tau_{w} = \left. {\mu \frac{\partial u}{\partial y}\left[ {1 + \varGamma^{2} \left( {\frac{\partial u}{\partial y}} \right)^{2} } \right]^{{\frac{n - 1}{2}}} } \right|_{{y = A\left( {x + b} \right)^{{\frac{1 - m}{2}}} }} ,q_{w} = \left. { - k\left( {\frac{\partial T}{\partial y}} \right)} \right|_{{y = A\left( {x + b} \right)^{{\frac{1 - m}{2}}} }} ,q_{m} = \left. { - D_{m} \left( {\frac{\partial C}{\partial y}} \right)} \right|_{{y = A\left( {x + b} \right)^{{\frac{1 - m}{2}}} }}$$

With the help of (6) and (11), they can be transmuted as

$$\begin{aligned} \left( {\text{Re}_{x} } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} C_{fx} &= \sqrt {\frac{m + 1}{2}} f''\left( 0 \right)\left[ {1 + {\text{We}}^{2} \left( {f''\left( 0 \right)} \right)^{2} } \right]^{{\frac{n - 1}{2}}} , \hfill \\ \left( {\text{Re}_{x} } \right)^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}} {\text{Nu}}_{x} &= - \sqrt {\frac{m + 1}{2}} \theta '\left( 0 \right),\left( {\text{Re}_{x} } \right)^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}} Sh_{x} = - \sqrt {\frac{m + 1}{2}} \phi '\left( 0 \right) \hfill \\ \end{aligned}$$
(18)

where \(\text{Re}_{x}\) is defined as \(\text{Re}_{x} = \frac{{U_{w} \left( x \right)\left( {x + b} \right)}}{\upsilon }\)

3 Numerical procedure

Numerical solutions of significantly nonlinear differential Eqs. (12)–(14) connected with the boundary conditions (15) are achieved with the aid of shooting procedure reliant on R–K fourth-order method.

Reduce Eqs. (12)–(14) into system of first-order equations by using sensible replacements described as follows:

$$\begin{aligned} & f_{1} = f, \\ & f_{2} = \frac{{{\text{d}}f_{1} }}{{{\text{d}}\varsigma }} = \frac{{{\text{d}}f}}{{{\text{d}}\varsigma }}, \\ & f_{3} = \frac{{{\text{d}}f_{2} }}{{{\text{d}}\varsigma }} = \frac{{{\text{d}}^{2} f}}{{{\text{d}}\varsigma^{2} }}, \\ & f_{4} = \theta \left( \varsigma \right), \\ & f_{5} = \frac{{{\text{d}}f_{4} }}{{{\text{d}}\varsigma }} = \frac{{{\text{d}}\theta }}{{{\text{d}}\varsigma }}, \\ & f_{6} = \phi \left( \varsigma \right), \\ & f_{7} = \frac{{{\text{d}}f_{6} }}{{{\text{d}}\varsigma }} = \frac{{{\text{d}}\phi }}{{{\text{d}}\varsigma }} \\ \end{aligned}$$
(19)

By using this, we can write

$$\begin{aligned} & \frac{{{\text{d}}f_{1} }}{{{\text{d}}\zeta }} = f_{2} , \\ & \frac{{{\text{d}}f_{2} }}{{{\text{d}}\zeta }} = f_{3} , \\ & \frac{{{\text{d}}f_{3} }}{{{\text{d}}\zeta }} = - \frac{2}{{\left[ {m + 1} \right]\left[ {1 + {\text{We}}^{2} f_{3} } \right]^{{\frac{n - 3}{2}}} \left[ {1 + n{\text{We}}^{2} f_{3}^{2} } \right]}}\left[ {\frac{m + 1}{2}f_{1} f_{3} - mf_{2}^{2} + \chi f_{4} + \chi N^{*} f_{6} - Mf_{2} } \right] \\ & \frac{{{\text{d}}f_{4} }}{{{\text{d}}\zeta }} = f_{5} , \\ & \frac{{{\text{d}}f_{5} }}{{{\text{d}}\zeta }} = - \left[ {\Pr f_{1} f_{3} + \frac{2}{m + 1}M\Pr {\text{Ec}}f_{2}^{2} + \Pr {\text{Ec}}f_{3}^{2} } \right], \\ & \frac{{{\text{d}}f_{6} }}{{{\text{d}}\zeta }} = f_{7} , \\ & \frac{{{\text{d}}f_{7} }}{{{\text{d}}\zeta }} = \frac{2}{m + 1}{\text{Sc}}\;{\text{Kr}}f_{6} - {\text{Sc}}f_{1} f_{7} \\ \end{aligned}$$
(20)

We can use an analogous approach for the boundary conditions stated in (15). After the alteration, we can employ the shooting procedure dependent on R–K fourth-order method to get the reasonable results. The boundary conditions \(f{\prime }\left( 0 \right),\theta \left( 0 \right)\) and \(\phi \left( 0 \right)\) for \(\zeta \to \infty\) are altered into finite interval length (\(\zeta = 5\)). Insert three primary suppositions to \(f^{\prime\prime}\left( 0 \right),\theta^{\prime}\left( 0 \right)\) and \(\phi '\left( 0 \right)\) for probable solution. Here the step size and convergence criteria are chosen to be \(0.001\) and \(10^{ - 6}\) (in all cases).

4 Results and Discussion

The set of (ordinary differential) Eqs. (12)–(14) with respect to the boundary conditions (15) is solved numerically by applying R–K fourth-order-based shooting procedure. We exhibited the impact of non-dimensional parameters, for example \({\text{We}},{\text{Ec}},h_{1}\) on velocity, concentration and temperature, using plots and the impact of the same parameters on friction factor, rate of transfer (both mass and heat) using tables.

Figure 2 shows the impact of \(M\) on velocity profile. It is evident from the figure that \(M\) lowers the velocity. This occurs because of Lorentz force emerging from the collaboration of electric and magnetic fields during the movement of an electrically conducting fluid. The produced Lorentz force controls the fluid movement in boundary layer area and thus lessening the thickness of the momentum boundary layer. So, velocity diminishes with the rise in \(M\). Because of the mounting of Lorentz force, thickness of the thermal boundary layer upsurges. As a consequence, temperature increases with the increase in \(M\) which can be observed in Fig. 3. When we increase \({\text{We}}\), there is a diminution in the viscosity of the fluid. So, with the rise in \({\text{We}}\), there is an enhancement in the velocity (Fig. 4) and drop in the temperature (Fig. 5). From the Fig. 6, we can observe that \({\text{Ec}}\) raises the temperature. Generally, when we increase \({\text{Ec}}\), additional heat will be added to the fluid flow. As a result, temperature increases. Figure 7 explained the fact that \(\chi\) boosts the velocity profile. This may be due to the fact that the increase in \(\chi\) leads to the diminution in the density of the fluid. We know the fact that \({\text{Sc}}\) is inversely proportional to the mass diffusion. So, when we increase \({\text{Sc}}\), there is a reduction in the concentration profile (Fig. 8). Figure 9 shows the natural behaviour of \({\text{Kr}}\) on the concentration. Figures 10 and 11 revealed the influence of \(\lambda\) on velocity and temperature profiles. From these, we can notice that \(\lambda\) lessens all profiles. Reasons behind this behaviour may be the following: (a) increasing \(\lambda\) barricades the flow motion to slow down the flow field. Eventually, the boundary layer thickness gets thinner for increasing values of \(\lambda\), and (b) thicker surfaces slowly emit the heat into the flow. So, when we increase \(\lambda\), there is a reduction in the heat dissipation towards the flow which causes the reduction in temperature. From Fig. 12, we can notice that \(h_{1}\) lessens the velocity. When we increase \(h_{1}\), initially velocity gets decreased near the surface the sheet and ultimately, in the entire fluid flow, velocity decreases. When we increase \(h_{2}\), thermal accommodation coefficient upsurges which lowers the thermal diffusion towards the flow. So, temperature decreases with the rise in \(h_{2}\) (Fig. 13). Table 1 displays the values of local skin friction coefficient and rate of transfers (heat and mass) against the aforesaid parameters. \(We\) and \(h_{1}\) are beneficial to improve the friction factor, rate of transfers (heat and mass). \(M\) and \({\text{Ec}}\) both lessen the heat transfer rate. \(\lambda\) and \({\text{Kr}}\) both improve the rate of mass transfer. Table 2 displays the assessment of the present result with the existing results in the restrictive case. We found decent agreement.

Fig. 2
figure 2

Influence of \(M\) on velocity profile

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Fig. 3
figure 3

Influence of \(M\) on temperature profile

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Fig. 4
figure 4

Influence of \({\text{We}}\) on velocity profile

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Fig. 5
figure 5

Influence of \({\text{We}}\) on temperature profile

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Fig. 6
figure 6

Influence of \({\text{Ec}}\) on temperature profile

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Fig. 7
figure 7

Influence of \(\chi\) on velocity profile

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Fig. 8
figure 8

Influence of \({\text{Sc}}\) on concentration profile

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Fig. 9
figure 9

Influence of \({\text{Kr}}\) on concentration profile

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Fig. 10
figure 10

Influence of \(\lambda\) velocity on profile

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Fig. 11
figure 11

Influence of \(\lambda\) temperature on profile

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Fig. 12
figure 12

Influence of \(h_{1}\) on velocity profile

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Fig. 13
figure 13

Influence of \(h_{2}\) on temperature profile

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Table 1 Amounts of skin friction coefficient and transfer rates (heat and mass) for various parameters
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Table 2 Comparison of local Nusselt number for various values of \(\Pr\) when \(M = {\text{Ec}} = h_{1} = h_{2} = {\text{Le}} = \chi = N^{*} = 0,{\text{We}} = 3,m = 1,n = 0.5\)
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5 Conclusions

When the plate is cooling or heating, the estimations of Joule heating and viscous dissipation are progressively significant. The noticeable reality of heat transport typically emerges in the procedures of power generation systems, cooling of nuclear reactors, cooling of electronic chips or metallic sheets and liquid metal fluids. Keeping these facts in the mind, the current study addresses the convective flow of Carreau fluid across a slendering stretching sheet in the presence of Joule heating, viscous dissipation and slip parameters. Numerical solutions of altered equations are achieved via shooting technique. The impacts of various parameters including viscous dissipation on the usual profiles (velocity, concentration and temperature) deliberated and demonstrated by means of plots. The effects of same parameters on the friction factor, rate of transfers (heat and mass) are conversed via Table 1. We also utilized special occurrences of the present model to approve our outcomes with past inquiries (Table 2). The subsequent deductions can be drawn from the present study.

  • \({\text{We}}\) improves the velocity and lessen the temperature

  • All boundary layers are observed to be thinner against \(\lambda\). So, it benefits to improve both transfer rates (heat and mass).

  • \(h_{1}\) lowers the velocity and boosts the friction factor.

  • The concentration boundary layer is observed to be thinner during chemical reaction.

  • \(M\) and \(Ec\) both reduce the local Nusselt number. \(\lambda\) and \({\text{Kr}}\) both improve the rate of mass transfer.