Introduction

Ferrofluids are suspensions of nanometer-sized magnetic particles in a suitable carrier liquid. It is synthesized through a chemical process. Ferrofluids is a kind fluid, which works in the zero gravity regions. Ferrofluids can be used in cooling of power electronics, computers and solar cells. Magnetic fluid has been extensively used in varieties of application in leak free seals, lubricants, density separation, ink jet printers, tunable heat transfer fluid, diagnostics in medicine, for heat transport of solar heat, etc. Due to low thermal conductivity of traditional fluid, nanofluid has been used recently in heat transfer applications. Nowadays, most of the heat transfer research has been carried out on Magnetohydrodynamic (MHD) nanofluid flow. MHD is concerned with the study of the interaction between magnetic fields and fluid conductors of electricity. The body force acting on the fluid is the Lorentz force that arises when electric current flows at an angle to the direction of an impressed magnetic field. In the current study, heat transfer analysis has been carried out on ferrohydrodynamic (FHD) nanofluid flow. FHD deals with the mechanics of fluid motion influenced by strong forces of magnetic polarization and there needS to be no electric current flowing in the fluid. Magnetization force works on throughout the volume of the fluid. It changes the viscosity of the magnetic fluid also. In the presence of stationary magnetic field, it creates additional resistance on FHD nanofluid flow. This additional resistance creates rotational viscosity of ferromagnetic fluid. Rotational viscosity depends on the strength of the magnetic field.

Rosensweig has demonstrated that due to the absence of current density in ferrofluid, the magnetic body force per unit volume is \(\mu_{0} \left( {\varvec{M}.\nabla } \right)\varvec{H}\) where \(\mu_{0}\) is the magnetic permeability, \(\varvec{M}\) is the magnetization and \(\varvec{H}\) is the magnetic field intensity [1]. Boundary layer flow of heated ferrofluid along a flat plat and stagnation point flow with variable magnetic field have been investigated [2]. Viscous and non-conducting flow of magnetic fluid over a stretching sheet has been investigated [3]. Heat transfer Jeffery ferrofluid past a lineally stretching sheet by considering the effect of magnetic dipole has been studied [4]. Heat transfer analysis on ferrofluid flow in a miniature channel in the presence of oscillating and uniform magnetic field has been studied [5]. Electrokinetic flow of nanofluid through a porous microtube in the presence of external magnetic field has been studied [6]. Heat transfer on different types of nanofluid has been investigated by finite element method [7]. A brief overview on nanofluid research and heat transfer enhancement has been published in the review article [8]. Convective heat transfer enhancement in nanofluid has been studied [9]. Control volume finite element has been used to study the impact of Coulomb force, entropy analysis and heat transfer analysis on ferromagnetic nanofluid flow [10,11,12]. The behavior of water-based nanofluid in the presence of nonuniform magnetic field has been analyzed numerically [13]. A numerical study presented for ferrofluid in a porous elliptical enclosure [14].

Dynamic viscosity plays an important role in the heat transfer analysis [15]. Convective heat transfer on nanofluid flow has been studied with control finite volume method [16]. Heat transfer characteristics of nanofluid over a inclined vertical plate have been investigated [17]. MHD flow over a stretching sheet has been considered, and different types of stretching have been used by the researchers [18,19,20,21]. Convective heat transfer on magnetic nanofluid with a sinusoidal hot wall has been investigated, and homotopy analysis has been used in the solution of nonlinear-coupled differential equations [22, 23]. Buoyancy and heat transfer characteristics have been investigated analytically [24]. A model for studying chemical reaction on two-dimensional ferrofluid flow due to a stretching sheet has been constructed [25, 26]. Decreasing the coil diameter increases the heat transfer on ferrofluid through a helical tube [27]. Three-dimensional flow of nanofluid containing ferrous nanoparticles over a variable sheet in a slip flow regime have been investigated [28]. Thermomagnetic convection of a ferrofluid flow around a vertical current-carrying wire has been studied [29].

Boundary layer flow of ferrofluid over a stretching surface with the influence of magnetic has been studied [30]. Effect of magnetic dipole on Williamson ferrofluid over a stretching surface has been solved with the help of shooting method [31]. Finite element method has been used to solve a set of nonlinear partial differential equations [32]. Field-dependent viscosity effects on ferrofluid flow in the presence of a disk have studied in the presence of alternating magnetic field and stationary magnetic field [33, 34]. Radiation effects on two-dimensional ferrofluid flow over a stretching surface has been investigated [35]. Effect of magnetic field location on heat transfer has been studied [36].

From above the literature survey, most of the research work has been carried out on the MHD nanofluid flow. However, there are few numbers of research papers, which are published on FHD nanofluid flow. The above literature survey motivates toward the FHD nanofluid flow and its heat transfer characteristics. In the present problem, the two-dimensional incompressible viscous flow of ferrofluid over a stretching sheet has been considered. Rotation of the particles in the ferrofluid is also included in the calculations. Thermal and concentration buoyancy effects, heat absorption and radiation effects, chemical reaction effects, Brownian motion effects, thermophoresis effects and magnetization force effects have been considered. A system of partial differential equation has been transformed to ordinary nonlinear-coupled differential equations. A set of nonlinear-coupled differential equations is solved through finite element method using COMSOL Multiphysics.

Mathematical formulation

A schematic diagram for electrically non-conducting, steady and an incompressible two-dimensional ferrohydrodynamic (FHD) nanofluid flow over a linear stretching sheet is presented in Fig. 1. Stretching in the sheet is along x axis with the velocity \(u_{\mathrm{w}}\) due to the force exerted on the sheet \(y = 0\) and \(y\) axis is taken normal to the sheet. A magnetic dipole is kept in the center of the \(y\) axis and \(a\) unit distance from the sheet. Stretching on the sheet is proportional to the distance from the origin. Wall temperature is assumed as \(T_{\mathrm{w}}\). Curie temperature is denoted by \(T_{\mathrm{c}}\), while ambient temperature is \(T_{\infty } = T_{\mathrm{c}}\). Magnetic effects become zero beyond the Curie temperature. The angular velocity is zero at the surface of the sheet as well as far away from the sheet. Microrotation of the fluid and rotation of the particle are also considered. The motion of the ferrofluid over a stretching sheet is caused by moving sheet. Rotation of the ferrofluid and nanoparticles in the fluid is also considered.

Fig. 1
figure 1

Geometry of flow configuration. Circles represent magnetic dipole

Full size image

The governing equations for this type of flow based on Shliomis model are as follows [37]:

The equation of continuity

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$
(1)

The equation of motion

$$\begin{aligned} & u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + \lambda_{1} \left( {u^{2} \frac{{\partial^{2} u}}{{\partial x^{2} }} + v^{2} \frac{{\partial^{2} u}}{{\partial y^{2} }} + 2uv\frac{{\partial^{2} u}}{\partial x\partial y}} \right) = \frac{{\mu_{0} }}{\varrho }M\frac{\partial H}{\partial x} + \nu \frac{{\partial^{2} u}}{{\partial y^{2} }} + \frac{k}{\rho }\frac{\partial \varOmega }{\partial y} \\ & \quad + g_{0} \beta_{1} \left( {T - T_{\infty } } \right) + g_{0} \beta_{2} \left( {C - C_{\infty } } \right) + \frac{I}{{2\tau_{\mathrm{s}} }}\left( {\frac{{\partial \varOmega_{\mathrm{p}} }}{\partial y} - \frac{\partial \varOmega }{\partial y}} \right) \\ \end{aligned}$$
(2)

The energy equation

$$\begin{aligned} & u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} + \frac{{\mu_{0} T}}{{\rho C_{{{\uprho }}} }}\frac{\partial M}{\partial T}\left( {u\frac{\partial H}{\partial x} + v\frac{\partial H}{\partial y}} \right) = \frac{k}{{\rho C_{{{\uprho }}} }}\frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{k}{{\rho C_{{{\uprho }}} }}\left[ {\mu \left( {\frac{\partial u}{\partial y}} \right)^{2} + 2\mu \left( {\frac{\partial v}{\partial y}} \right)^{2} } \right] \\ & \quad + \frac{{\left( {\rho c_{{{\uprho }}} } \right)_{\mathrm{p}} }}{{\rho c_{{{\uprho }}} }}\left[ {D_{\mathrm{B}} \frac{\partial C}{\partial y}\frac{\partial T}{\partial y} + \frac{{D_{\mathrm{T}} }}{{T_{\infty } }}\left( {\frac{\partial T}{\partial y}} \right)^{2} } \right] + \frac{{\left( {\rho c_{\mathrm{p}} } \right)_{\mathrm{p}} }}{{\rho c_{\mathrm{p}} }}D_{\mathrm{B}} \frac{{\partial^{2} C}}{{\partial y^{2} }} - \frac{{Q_{0} }}{{\rho c_{{{\uprho }}} }}\left( {T - T_{\infty } } \right) + Q_{1}^{\prime } \left( {C - C_{\infty } } \right) \\ \end{aligned}$$
(3)

The equation of concentration

$$u\frac{\partial C}{\partial x} + v\frac{\partial C}{\partial y} = D_{\mathrm{B}} \frac{{\partial^{2} C}}{{\partial y^{2} }} + + \frac{{D_{\mathrm{T}} }}{{T_{\infty } }}\frac{{\partial^{2} T}}{{\partial y^{2} }} - K_{1} \left( {C - C_{\infty } } \right)$$
(4)

The equation of angular momentum

$$u\frac{\partial \varOmega }{\partial x} + v\frac{\partial \varOmega }{\partial y} = \frac{{\gamma_{\mathrm{F}} }}{j\rho }\frac{{\partial^{2} \varOmega }}{{\partial y^{2} }} + \frac{k}{j\rho }\left( {2\varOmega + \frac{\partial u}{\partial y}} \right)$$
(5)

The boundary conditions for described flow are as follows:

$$y = 0:\quad u = v_{\mathrm{w}} = cx;\quad v = 0;\quad T = T_{\mathrm{w}} = T_{\text{E}} - A\left( {\frac{x}{l}} \right)^{2} ;\quad C = C_{\mathrm{w}} ;\quad \varOmega = 0$$
(6)
$$y \to \infty :\quad u = 0;\quad T = T_{\infty } ;\quad C = C_{\infty } ;\quad \varOmega = 0$$
(7)

The expression \(\frac{I}{{2\tau_{\mathrm{s}} }}\left( {\frac{{\partial \varOmega_{\mathrm{p}} }}{\partial y} - \frac{\partial \varOmega }{\partial y}} \right) = \frac{3}{2}\varPhi_{1} m_{1} \nu \frac{{\partial^{2} u}}{{\partial y^{2} }}\) viscous force due to rotation of the magnetic fluid [33, 38]. \(l = \sqrt {\frac{\nu }{c}}\) is the characteristic length.

There is no current density in ferrofluid. The considered fluid motion is controlled by the applied magnetic field as shown in Fig. 1. The scalar potential for the applied magnetic field is:

$$\varTheta = \frac{{\gamma_{0} }}{2\pi }\left\{ {\frac{x}{{x^{2} + \left( {y + a} \right)^{2} }}} \right\}$$
(8)

The components of magnetic field intensity \(H_{\mathrm{x}}\) and \(H_{\mathrm{y}}\) along x and y axes can be expressed as:

$$H_{\mathrm{x}} = - \frac{\partial \varTheta }{\partial x} = \frac{{\gamma_{0} }}{2\pi }\left[ {\frac{{x^{2} - \left( {y + a} \right)^{2} }}{{\left\{ {x^{2} + \left( {y + a} \right)^{2} } \right\}^{2} }}} \right]$$
(9)
$$H_{\mathrm{y}} = - \frac{\partial \varTheta }{\partial y} = \frac{{\gamma_{0} }}{2\pi }\left[ {\frac{{2x\left( {y + a} \right)}}{{\left\{ {x^{2} + \left( {y + a} \right)^{2} } \right\}^{2} }}} \right]$$
(10)

The resultant magnitude of the magnetic field intensity is:

$$H = \left[ {\left( {\frac{\partial \varTheta }{\partial x}} \right)^{2} + \left( {\frac{\partial \varTheta }{\partial y}} \right)^{2} } \right]^{{\frac{1}{2}}}$$
(11)

The gradient of magnitude of the magnetic field intensity along x and y axes are as follows:

$$\frac{\partial H}{\partial x} = \frac{{\gamma_{0} }}{2\pi }\left\{ {\frac{2x}{{\left( {y + a} \right)^{4} }}} \right\}$$
(12)
$$\frac{\partial H}{\partial y} = \frac{{\gamma_{0} }}{2\pi }\left\{ {\frac{ - 2}{{\left( {y + a} \right)^{3} }} + \frac{{4x^{2} }}{{\left( {y + a} \right)^{5} }}} \right\}$$
(13)

Magnetization is considered as a linear function of temperature as:

$$M = K^{\text{a}} \left( {T_{\mathrm{c}} - T} \right)$$
(14)

Fluid temperature is different from Curie temperature. Once ferrofluid approaches to Curie temperature, there is no furthermore magnetization. It is the situation when all the magnetic particles aligned in the direction of applied magnetic field.

Solution procedure

Using the following similarity transformation, the described model in the above section has now been converted into dimensionless from

$$\psi = \frac{\mu }{\rho }\xi f\left( \eta \right);\quad \theta = \frac{{T_{\mathrm{c}} - T}}{{T_{\mathrm{c}} - T_{\mathrm{w}} }} = \theta_{1} \left( \eta \right) + \xi^{2} \theta_{2} \left( \eta \right);\quad \varphi = \frac{{C - C_{\infty } }}{{C_{\mathrm{w}} - C_{\infty } }};\quad \varOmega = \sqrt {\frac{{c^{3} }}{\nu } xg\left( \eta \right)}$$
(15)
$$T_{\mathrm{c}} - T_{\mathrm{w}} = A\left( {\frac{x}{l}} \right)^{2} ;\quad \xi = \sqrt {c\nu } x;\quad \eta = \sqrt {c\nu } y$$
(16)

The velocity components u and v are related to the stream function as:

$$u = \frac{\partial \psi }{\partial y} = cxf^{\prime } \left( \eta \right);\quad v = - \frac{\partial \psi }{\partial x} = - \sqrt {c\nu } f \left( \eta \right)$$
(17)

Here, prime represents the differentiation with respect to \(\eta\). Using Eqs. (15)–(17) and equating the coefficients of equal powers of \(\xi\) up to \(\xi^{2}\), the equation of motion of motion, the energy equation, the concentration equation and the equation of angular momentum transform into a set of nonlinear-coupled differential equations as:

$$\begin{aligned} & \left( {1 + \frac{3}{2}\phi_{1} m_{1} } \right)\frac{{{\mathrm{d}}^{3} f}}{{{\mathrm{d}}\eta^{3} }} - \gamma_{1} f^{2} \frac{{{\mathrm{d}}^{3} f}}{{{\mathrm{d}}\eta^{3} }} - \left\{ {\left( {\frac{{{\mathrm{d}}f}}{{{\mathrm{d}}\eta }}} \right)^{2} - f\frac{{{\mathrm{d}}^{2} f}}{{{\mathrm{d}}\eta^{2} }}} \right\} \\ & \quad + 2\gamma_{1} f\frac{{{\mathrm{d}}f}}{{{\mathrm{d}}\eta }} \frac{{{\mathrm{d}}^{2} f}}{{{\mathrm{d}}\eta^{2} }} - \frac{2\beta }{{\left( {\eta + \alpha_{1} } \right)^{4} }}\theta_{1} + K\frac{{{\mathrm{d}}g}}{{{\mathrm{d}}\eta }} - {\mathrm{Gr}}\theta_{1} - {\mathrm{Gm}}\varphi = 0 \\ \end{aligned}$$
(18)
$$\begin{aligned} & \frac{{{\mathrm{d}}^{2} \theta_{1} }}{{{\mathrm{d}}\eta^{2} }} + \Pr \left( {f\frac{{{\mathrm{d}}\theta_{1} }}{{{\mathrm{d}}\eta }} - 2\frac{{{\mathrm{d}}f}}{{{\mathrm{d}}\eta }}\theta_{1} } \right) + \frac{{2\lambda \beta \left( {\theta_{1} - \varepsilon } \right)f}}{{\left( {\eta + \alpha_{1} } \right)^{3} }} - 2\lambda \left( {\frac{{{\mathrm{d}}f}}{{{\mathrm{d}}\eta }}} \right)^{2} \\ & \quad - {\mathrm{Nb}}\frac{{{\mathrm{d}}^{2} \varphi }}{{{\mathrm{d}}\eta^{2} }} + {\mathrm{Nb}}\frac{{{\mathrm{d}}\theta_{1} }}{{{\mathrm{d}}\eta }}\frac{{{\mathrm{d}}\varphi }}{{{\mathrm{d}}\eta }} + {\mathrm{Nt}}\left( {\frac{{{\mathrm{d}}\theta_{1} }}{{{\mathrm{d}}\eta }}} \right)^{2} - \chi \theta_{1} + Q_{1} \varphi = 0 \\ \end{aligned}$$
(19)
$$\begin{aligned} & \frac{{{\mathrm{d}}^{2} \theta_{2} }}{{{\mathrm{d}}\eta^{2} }} - \Pr \left( {4\frac{{{\mathrm{d}}f}}{{{\mathrm{d}}\eta }}\theta_{2} - f\frac{{{\mathrm{d}}\theta_{2} }}{{{\mathrm{d}}\eta }}} \right) + \frac{{2\lambda \beta f\theta_{2} }}{{\left( {\eta + \alpha_{1} } \right)^{3} }} - \lambda \beta \left( {\theta_{1} - \varepsilon } \right)\left\{ {\frac{{2\frac{{{\mathrm{d}}f}}{{{\mathrm{d}}\eta }}}}{{\left( {\eta + \alpha_{1} } \right)^{4} }} + \frac{4f}{{\left( {\eta + \alpha_{1} } \right)^{5} }}} \right\} \\ & \quad - \lambda \left( {\frac{{{\mathrm{d}}f}}{{{\mathrm{d}}\eta }}} \right)^{2} + {\mathrm{Nb}}\frac{{{\mathrm{d}}\theta_{2} }}{{{\mathrm{d}}\eta }}\frac{{{\mathrm{d}}\varphi }}{{{\mathrm{d}}\eta }} + {\mathrm{Nt}}\left( {\frac{{{\mathrm{d}}\theta_{2} }}{{{\mathrm{d}}\eta }}} \right)^{2} - \chi \theta_{2} + Q_{1} \varphi = 0 \\ \end{aligned}$$
(20)
$$\frac{{{\mathrm{d}}^{2} \varphi }}{{{\mathrm{d}}\eta^{2} }} + {\mathrm{Sc }}f\frac{{{\mathrm{d}}\varphi }}{{{\mathrm{d}}\eta }} - {\mathrm{Sc}} \gamma \varphi + \frac{\mathrm{Nt}}{\mathrm{Nb}}\frac{{{\mathrm{d}}^{2} \theta_{2} }}{{{\mathrm{d}}\eta^{2} }} = 0$$
(21)
$$\left( {1 + \frac{K}{2}} \right)\frac{{{\mathrm{d}}^{2} g}}{{{\mathrm{d}}\eta^{2} }} + f\frac{{{\mathrm{d}}g}}{{{\mathrm{d}}\eta }} - \frac{{{\mathrm{d}}f}}{{{\mathrm{d}}\eta }}g - K\left( {2g + \frac{{{\mathrm{d}}^{2} f}}{{{\mathrm{d}}\eta^{2} }}} \right) = 0$$
(22)

The transformed boundary conditions from Eqs. (6) and (7) are as follows:

$$f\left( 0 \right) = S;\quad f^{\prime } \left( 0 \right) = 0;\quad \theta_{1} \left( 0 \right) = 1;\quad \theta_{2} \left( 0 \right) = 0;\quad \varphi \left( 0 \right) = 1;\quad g\left( 0 \right) = 0$$
(23)
$$f^{\prime } \left( \infty \right) = 0;\quad \theta_{1} \left( \infty \right) = 0;\quad \theta_{2} \left( \infty \right) = 0;\quad \varphi \left( \infty \right) = 0;\quad g\left( \infty \right) = 0$$
(24)

The dimensionless quantities are presented as follows:

$$\begin{aligned} \beta & = \frac{{\gamma_{0} \rho }}{{2\pi \mu^{2} }}\mu_{0} K^{a} \left( {T_{\mathrm{c}} - T_{\mathrm{w}} } \right);\quad \gamma_{1} = \lambda_{1} c;\quad \lambda = \frac{{c\mu^{2} }}{{\rho k\left( {T_{\mathrm{c}} - T_{\mathrm{w}} } \right)}};\quad \Pr = \frac{{\mu c_{\mathrm{p}} }}{k};\quad \alpha_{1} = \sqrt {\left( {\frac{c\rho }{\mu }} \right)} a; \\ \varepsilon & = \frac{{T_{\mathrm{c}} }}{{T_{\mathrm{c}} - T_{\mathrm{w}} }},{\mathrm{Sc}} = \frac{\mu }{\rho D};\quad K = \frac{k}{\mu };\quad {\mathrm{Gr}} = \frac{{\mu g_{0} \beta_{1} \left( {T_{\mathrm{w}} - T_{\mathrm{c}} } \right)}}{{\rho \nu^{3} }};\quad {\mathrm{Gm}} = \frac{{\mu g_{0} \beta_{2} \left( {C_{\mathrm{w}} - C_{\infty } } \right)}}{{\rho \nu^{3} }};\quad Q_{1} = \frac{{\mu Q_{1}^{'} \left( {C_{\mathrm{w}} - C_{\infty } } \right)}}{{\rho \nu^{2} \left( {T_{\mathrm{w}} - T_{\mathrm{c}} } \right)}}; \\ \chi & = \frac{{Q_{0} \mu }}{{\rho^{2} c_{\mathrm{p}} \nu^{2} }},\gamma = \frac{{K_{1} \mu }}{{\rho \nu^{2} }};\quad {\mathrm{Nb}} = \frac{{\left( {\rho c_{\mathrm{p}} } \right)_{\mathrm{p}} D_{\mathrm{B}} \left( {C_{\mathrm{w}} - C_{\infty } } \right)}}{{\rho c_{\mathrm{p}} \nu }};\quad {\mathrm{Nt}} = \frac{{\left( {\rho c_{\mathrm{p}} } \right)_{\mathrm{p}} D_{\mathrm{T}} \left( {T_{\mathrm{w}} - T_{\mathrm{c}} } \right)}}{{\rho c_{\mathrm{p}} \nu T_{\mathrm{c}} }} \\ \end{aligned}$$
(25)

Equations (18)–(22) with the help of Eqs. (23) and (24) are solved using finite element method in COMSOL Multiphysics. For use system of differential equation tool in COMSOL, these equations are converted to second order differential equations as:

$$f = p;\quad \frac{{{\mathrm{d}}f}}{{{\mathrm{d}}\eta }} = q$$
(26)
$$\frac{{{\mathrm{d}}p}}{{{\mathrm{d}}\eta }} = q$$
(27)
$$\left( {1 + \frac{3}{2}\phi_{1} m_{1} } \right)\frac{{{\mathrm{d}}^{2} q}}{{{\mathrm{d}}\eta^{2} }} - \gamma_{1} p^{2} \frac{{{\mathrm{d}}^{2} q}}{{{\mathrm{d}}\eta^{2} }} - \left\{ {q^{2} - p\frac{{{\mathrm{d}}q}}{{{\mathrm{d}}\eta }}} \right\} + 2\gamma_{1} pq \frac{{{\mathrm{d}}q}}{{{\mathrm{d}}\eta }} - \frac{2\beta }{{\left( {\eta + \alpha_{1} } \right)^{4} }}\theta_{1} + K\frac{{{\mathrm{d}}g}}{{{\mathrm{d}}\eta }} - {\mathrm{Gr}}\theta_{1} - {\mathrm{Gm}}\varphi = 0$$
(28)
$$\begin{aligned} & \frac{{{\mathrm{d}}^{2} \theta_{1} }}{{{\mathrm{d}}\eta^{2} }} + { \Pr }\left( {p\frac{{{\mathrm{d}}\theta_{1} }}{{{\mathrm{d}}\eta }} - 2q\theta_{1} } \right) + \frac{{2\lambda \beta \left( {\theta_{1} - \varepsilon } \right)p}}{{\left( {\eta + \alpha_{1} } \right)^{3} }} \\ & \quad - 2\lambda q^{2} - {\mathrm{Nb}}\frac{{E^{2} \varphi }}{{{\mathrm{d}}\eta^{2} }} + {\mathrm{Nb}}\frac{{{\mathrm{d}}\theta_{1} }}{{{\mathrm{d}}\eta }}\frac{{{\mathrm{d}}\varphi }}{{{\mathrm{d}}\eta }} + {\mathrm{Nt}}\left( {\frac{{{\mathrm{d}}\theta_{1} }}{{{\mathrm{d}}\eta }}} \right)^{2} - \chi \theta_{1} + Q_{1} \varphi = 0 \\ \end{aligned}$$
(29)
$$\begin{aligned} & \frac{{{\mathrm{d}}^{2} \theta_{2} }}{{{\mathrm{d}}\eta^{2} }} - { \Pr }\left( {4q\theta_{2} - p\frac{{{\mathrm{d}}\theta_{2} }}{{{\mathrm{d}}\eta }}} \right) + \frac{{2\lambda \beta p\theta_{2} }}{{\left( {\eta + \alpha_{1} } \right)^{3} }} - \lambda \beta \left( {\theta_{1} - \varepsilon } \right)\left\{ {\frac{2q}{{\left( {\eta + \alpha_{1} } \right)^{4} }} + \frac{4p}{{\left( {\eta + \alpha_{1} } \right)^{5} }}} \right\} \\ & \quad - \lambda q^{2} + {\mathrm{Nb}}\frac{{{\mathrm{d}}\theta_{2} }}{{{\mathrm{d}}\eta }}\frac{{{\mathrm{d}}\varphi }}{{{\mathrm{d}}\eta }} + {\mathrm{Nt}}\left( {\frac{{{\mathrm{d}}\theta_{2} }}{{{\mathrm{d}}\eta }}} \right)^{2} - \chi \theta_{2} + Q_{1} \varphi = 0 \\ \end{aligned}$$
(30)
$$\frac{{{\mathrm{d}}^{2} \varphi }}{{{\mathrm{d}}\eta^{2} }} + {\mathrm{Sc }}p\frac{{{\mathrm{d}}\varphi }}{{{\mathrm{d}}\eta }} - {\mathrm{Sc }}\gamma \varphi + \frac{\mathrm{Nt}}{\mathrm{Nb}}\frac{{{\mathrm{d}}^{2} \theta_{2} }}{{{\mathrm{d}}\eta^{2} }} = 0$$
(31)
$$\left( {1 + \frac{K}{2}} \right)\frac{{{\mathrm{d}}^{2} g}}{{{\mathrm{d}}\eta^{2} }} + p\frac{{{\mathrm{d}}g}}{{{\mathrm{d}}\eta }} - qg - K\left( {2g + \frac{{{\mathrm{d}}q}}{{{\mathrm{d}}\eta }}} \right) = 0$$
(32)

Dirichlet boundary conditions used to solve Eqs. (27)–(32) are as follows:

$$p\left( 0 \right) = S;\quad q\left( 0 \right) = 1;\quad \theta_{1} \left( 0 \right) = 1;\quad \theta_{2} \left( 0 \right) = 0;\quad \varphi \left( 0 \right) = 1;\quad g\left( 0 \right) = 0$$
(33)
$$q\left( \infty \right) = 0;\quad \theta_{1} \left( \infty \right) = 0;\quad \theta_{2} \left( \infty \right) = 0;\quad \varphi \left( \infty \right) = 0;\quad g\left( \infty \right) = 0$$
(34)

The physical quantities of practical interest are skin friction coefficient, heat transfer rate and Sherwood number, which can be expressed as:

$$C_{{{\text{f}}_{\text{x}} }} = - \frac{{\tau_{\text{w}} }}{{\rho \left( {cx} \right)^{2} }};\quad Nu_{\text{x}} = - \frac{{xq_{\text{w}} }}{{k\left( {T_{\text{c}} - T_{\text{w}} } \right)}};\quad Sh_{\text{x}} = \frac{{xJ_{\text{w}} }}{{D_{\text{f}} \left( {C_{\text{w}} - C_{\infty } } \right)}}$$
(35)
$$\tau_{\mathrm{w}} = \mu \left( {1 + \gamma_{1} } \right)\left( {\frac{\partial u}{\partial y}} \right)_{{{\mathrm{y}} = 0}} ;\quad q_{\mathrm{w}} = - \left( {\frac{\partial T}{\partial y}} \right)_{{{\mathrm{y}} = 0}} ;\quad J_{\mathrm{w}} = - D_{\mathrm{f}} \left( {\frac{\partial C}{\partial y}} \right)_{{{\mathrm{y}} = 0}}$$
(36)

Using Eqs. (15)–(17), the above physical quantities can be written as:

$$C_{\mathrm{f}} \sqrt {{\mathrm{Re}}_{\mathrm{x}} } = - \left( {1 + \gamma_{1} } \right)f^{\prime}\left( 0 \right);\quad \frac{{{\mathrm{Nu}}_{\mathrm{x}} }}{{\sqrt {{\mathrm{Re}}_{\mathrm{x}} } }} = - \left\{ {\theta_{1}^{'} \left( 0 \right) + \xi^{2} \theta_{2}^{'} \left( 0 \right)} \right\};\quad \frac{{{\mathrm{Sh}}_{\mathrm{x}} }}{{\sqrt {{\mathrm{Re}}_{\mathrm{x}} } }} = - \varphi^{\prime}\left( 0 \right)$$
(37)

For modeling of Eqs. (26)–(34) in COMSOL Multiphysics, PDE interface has been used in the solution. The transformed differential equations are one-dimensional; therefore a line geometry has been drawn. In coefficient form PDE, Dirichlet boundary conditions are introduced in COMSOL. For solving this one-dimensional problem, the maximum element size in the solution is taken 0.001 and the minimum element size is 0.00008. Maximum element growth rate is 1.1. Resolution for narrow region is 1. For the coupling of the differential equations, nonlinear method has been selected as automatic Newton method; initial damping factor is taken 1. Minimum damping factor is selected \(10^{ - 4}\). Recovery damping factor is 0.75. Maximum number of iterations are selected 50. Residual factor is 1000. The error in the solution is of \(10^{ - 7}\) order. The flowchart for the solution used in COMSOL is as follows:

figure a

Results and discussion

Results for velocity distributions \(\left( {f^{\prime}} \right)\), temperature distributions \(\left( {\theta_{1} } \right)\), concentration distributions \(\left( \varphi \right)\) and angular velocity distributions \(\left( g \right)\) are presented in graphical form for different values of Maxwell parameter \(\left( {\gamma_{1} } \right)\), ferromagnetic interaction number \(\left( \beta \right)\), thermal Grashof number \(\left( {\mathrm{Gr}} \right)\), solutal Grashof number \(\left( {\mathrm{Gm}} \right)\), Brownian motion parameter \(\left( {\mathrm{Nb}} \right)\), thermophoresis number \(\left( {\mathrm{Nt}} \right)\), chemical reaction parameter \(\left( \gamma \right)\), radiation absorption coefficient \(\left( {Q_{1} } \right)\), heat generation/absorption parameter \(\left( \chi \right)\), Prandtl number \(\left( { \Pr } \right)\) and Schmidt number \(\left( {\mathrm{Sc}} \right)\). The default values of the parameters in the present work are considered as \(m_{1} = 0.4\), \(\phi_{1} = 0.4\), \(\gamma_{1} = 0.1\), \(K = 0.2\), \({\mathrm{Gr}} = 2\), \({\mathrm{Gm}} = 2\), \(\beta = 0.1\), \(\alpha_{1} = 1\), \({ \Pr } = 10\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Nb}} = 0.1\), \({\mathrm{Nt}} = 0.1\), \(\chi = 2\), \(Q_{1} = 2\), \({\mathrm{Sc}} = 0.2\), \(\gamma = 2\). In the present study, viscous dissipation has been neglected. Therefore, at \(\lambda = 0\) gives the trivial solution \(\theta_{2} = 0\).

Figure 2a–d represents the velocity, temperature, concentration and angular velocity distributions of different values of ferromagnetic interaction number. Increasing values of ferromagnetic interaction number decreases the velocity distribution and concentration distribution. Increasing the values of \(\beta\), temperature distribution increases in the fluid. Ferromagnetic interaction number impedes the rotation of the particles in the fluid. These results indicate that magnetization force creates an additional resistance on the flow which reduces the velocity in the flow. There are magnetic nanoparticles in ferrofluid which aligned in the direction of the applied magnetic field. This intervention in the flow decreases the velocity distribution. Figure 3a–d shows the velocity, temperature, concentration and angular velocity distributions, respectively, for different values of Maxwell parameter \(\gamma_{1}\). Increasing the values of \(\gamma_{1}\), the velocity distribution decreases. Figure 3d shows that the angular velocity distribution of the fluid increases for increasing values of Maxwell parameter. Maxwell parameter favors the rotation of the particle in the fluid. There is no significant impact of \(\gamma_{1}\) on the temperature distribution. However, increasing \(\gamma_{1}\) decreases the concentration distribution.

Fig. 2
figure 2

a Velocity distribution, b concentration distribution, c concentration distribution, d angular velocity distribution for different values of \(\beta\) at \(m_{1} = 10\), \(\phi_{1} = 0.4\), \(K = 0.2\), \({\mathrm{Gr}} = 2\), \({\mathrm{Gm}} = 2\), \(\gamma_{1} = 0.1\), \(\alpha_{1} = 1\), \({ \Pr } = 10\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Nb}} = 0.1\), \({\mathrm{Nt}} = 0.1\), \(\chi = 2\), \(Q_{1} = 2\), \({\mathrm{Sc}} = 0.2\), \(S = 0.1\)

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Fig. 3
figure 3

a Velocity distribution, b temperature distribution, c concentration distribution, d angular velocity distribution for different values of \(\gamma_{1}\) at \(m_{1} = 10\), \(\phi_{1} = 0.4\), \(K = 0.2\), \({\mathrm{Gr}} = 2\), \({\mathrm{Gm}} = 2\), \(\beta = 0.1\), \(\alpha_{1} = 1\), \({ \Pr } = 10\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Nb}} = 0.1\), \({\mathrm{Nt}} = 0.1\), \(\chi = 2\), \(Q_{1} = 2\), \({\mathrm{Sc}} = 0.2\), \(S = 0.1\)

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Figure 4a–d shows the velocity, temperature, concentration and angular velocity distribution for different values of thermal Grashof number. For increasing the values of thermal Grashof number decreases the velocity and concentration distributions. However, it enhances the temperature distribution. The thermal Grashof number favors the rotational velocity distribution. In case of ordinary viscous flow, thermal Grashof number increases the velocity and temperature distribution. Microrotation of the fluid and the rotation of the particles are also considered in this study. The rotation of the fluid particles in the fluid decreases the velocity distribution. Figure 5a–d indicate the velocity, temperature, concentration and angular velocity distribution for different values of thermal Solutal number. Increasing Solutal Grashof number increases the angular velocity distribution on the as shown in Fig. 5d. Solutal Grashof number favors the rotation of the magnetic fluid. It decreases the velocity distribution in ferrofluid. A decrease in the velocity distribution increases the thickness of boundary layer. However, the thermal boundary layer thickness increases for increasing values of \({\mathrm{Gm}}\) as shown in Fig. 5b.

Fig. 4
figure 4

a Velocity distribution, b temperature distribution, c concentration distribution, d angular velocity distribution for different values of \({\mathrm{Gr}}\) at \(m_{1} = 10\), \(\phi_{1} = 0.4\), \(K = 0.2\), \(\gamma_{1} = 0.1\), \({\mathrm{Gm}} = 2\), \(\beta = 0.1\), \(\alpha_{1} = 1\), \({ \Pr } = 10\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Nb}} = 0.1\), \({\mathrm{Nt}} = 0.1\), \(\chi = 2\), \(Q_{1} = 2\), \({\mathrm{Sc}} = 0.2\), \(S = 0.1\)

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Fig. 5
figure 5

a Velocity distribution, b temperature distribution, c concentration distribution, d angular velocity distribution for different values of \({\mathrm{Gm}}\) at \(m_{1} = 10\), \(\phi_{1} = 0.4\), \(K = 0.2\), \(\gamma_{1} = 0.1\), \({\mathrm{Gr}} = 2\), \(\beta = 0.1\), \(\alpha_{1} = 1\), \({ \Pr } = 10\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Nb}} = 0.1\), \({\mathrm{Nt}} = 0.1\), \(\chi = 2\), \(Q_{1} = 2\), \({\mathrm{Sc}} = 0.2\), \(S = 0.1\)

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Figure 6a–d represents the velocity, temperature, concentration and angular velocity distribution for different values of Brownian motion parameter \({\mathrm{Nb}}\). Brownian motion parameter decreases the velocity distribution in the flow. Increasing the Brownian motion parameter enhances the angular velocity distribution. It happens in the presence of magnetic dipole. Magnetic field creates the additional viscosity in the fluid. Therefore, rotation of nanoparticles increases rather than velocity distribution. However, heat transfer and mass transfer increase for increasing values of Brownian motion parameter. Magnetic field plays an important role in changing heat and mass transfer characteristics in this flow. Figure 7a–d shows the velocity, temperature, concentration and angular velocity profiles for different values of thermophoresis parameter \({\mathrm{Nt}}\). Increasing the values of thermophoresis parameter enhances the velocity distribution and reduces the angular velocity distributions. However, it decreases the heat transfer and concentration distribution in the ferrofluid flow.

Fig. 6
figure 6

a Velocity distribution, b temperature distribution, c concentration distribution, d angular velocity distribution for different values of \({\mathrm{Nb}}\) at \(m_{1} = 10\), \(\phi_{1} = 0.4\), \(K = 0.2\), \(\gamma_{1} = 0.1\), \({\mathrm{Gr}} = 2\), \(\beta = 0.1\), \(\alpha_{1} = 1\), \({ \Pr } = 10\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Gm}} = 2\), \({\mathrm{Nt}} = 0.1\), \(\chi = 2\), \(Q_{1} = 2\), \({\mathrm{Sc}} = 0.2\), \(S = 0.1\)

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Fig. 7
figure 7

a Velocity distribution, b temperature distribution, c concentration distribution, d angular velocity distribution for different values of \({\mathrm{Nt}}\) at \(m_{1} = 10\), \(\phi_{1} = 0.4\), \(K = 0.2\), \(\gamma_{1} = 0.1\), \({\mathrm{Gr}} = 2\), \(\beta = 0.1\), \(\alpha_{1} = 1\), \({ \Pr } = 10\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Gm}} = 2\), \({\mathrm{Nb}} = 0.1\), \(\chi = 2\), \(Q_{1} = 2\), \({\mathrm{Sc}} = 0.2\), \(S = 0.1\)

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Figure 8a–d represents the effect of chemical reaction parameter on the velocity, temperature, concentration and angular velocity distributions. Chemical reaction parameter in ferrofluid impedes the rotation of ferrofluids and speed up the velocity distribution. It also decreases the heat transfer and concentration distribution. Chemical reaction parameter has higher impact on mass transfer than heat transfer. Figure 9a, b depicts the effect of radiation parameter \(\left( {Q_{1} } \right)\) on the temperature and concentration distributions. There is no significant impact of radiation parameter on the velocity and angular velocity distributions, and therefore these results are not presented in figures. However, increasing the radiation parameter increases the heat transfer distribution and decreases the concentration distribution. Figure 10a, b indicates the temperature and concentration distribution for different values of Prandtl number. The present results are obtained for different types of water-based ferrofluid. Increasing Prandtl number decreases the heat transfer distribution and increases the concentration in the fluid.

Fig. 8
figure 8

a Velocity distribution, b temperature distribution, c concentration distribution, d angular velocity distribution for different values of \(\gamma\) at \(m_{1} = 10\), \(\phi_{1} = 0.4\), \(K = 0.2\), \(\gamma_{1} = 0.1\), \({\mathrm{Gr}} = 2\), \(\beta = 0.1\), \(\alpha_{1} = 1\), \({ \Pr } = 10\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Gm}} = 2\), \({\mathrm{Nb}} = 0.1\), \(\chi = 2\), \(Q_{1} = 2\), \({\mathrm{Sc}} = 0.2\), \({\mathrm{Nt}} = 0.1\), \(S = 0.1\)

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Fig. 9
figure 9

a Temperature distribution, b concentration distribution for different values of \(Q_{1}\) at \(m_{1} = 10\), \(\phi_{1} = 0.4\), \(K = 0.2\), \(\gamma_{1} = 0.1\), \({\mathrm{Gr}} = 2\), \(\beta = 0.1\), \(\alpha_{1} = 1\), \({ \Pr } = 2\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Gm}} = 2\), \({\mathrm{Nb}} = 0.1\), \(\chi = 2\), \({\mathrm{Sc}} = 0.2\), \({\mathrm{Nt}} = 0.1, \gamma = 2\), \(S = 0.1\)

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Fig. 10
figure 10

a Temperature distribution, b concentration distribution for different values of \({\mathrm{Pr }}\) at \(m_{1} = 10\), \(\phi_{1} = 0.4\), \(K = 0.2\), \(\gamma_{1} = 0.1\), \({\mathrm{Gr}} = 2\), \(\beta = 0.1\), \(\alpha_{1} = 1\), \({\mathrm{Sc}} = 0.2\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Gm}} = 2\), \({\mathrm{Nb}} = 0.1\), \(Q_{1} = 2\), \(\chi = 2\), \({\mathrm{Nt}} = 0.1, \gamma = 2\), \(S = 0.1\)

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Figure 11a–d shows the velocity, temperature, concentration and angular velocity profiles for different values of heat absorption parameter \(\chi\). It increases the velocity distribution and decreases the angular velocity distribution in the flow. Increasing heat absorption parameter decreases the thermal boundary layer thickness and increases the concentration boundary layer. This parameter transforms the thermal energy into concentration energy. Figure 12a–d shows the velocity, temperature, concentration and angular velocity profiles for different values of Schmidt number \(\left( {\mathrm{Sc}} \right)\). In the current problem, the Schmidt has less impact on velocity, temperature, concentration and angular velocity distribution as compared to other physical parameter considered in the flow. In this flow, magnetization force, rotation of the nanofluid and rotational viscosity due to magnetic field play key role in changing the flow characteristics, heat transfer and mass transfer. The Schmidt number increases the velocity distribution and decreases the angular velocity of the ferromagnetic nanofluid. Far from the sheet, there is a small decrease in the heat and mass transfer in the considered flow.

Fig. 11
figure 11

a Velocity distribution, b temperature distribution, c concentration distribution, d angular velocity distribution for different values of, \(\chi\) at \(m_{1} = 10\), \(\phi_{1} = 0.4\), \(K = 0.2\), \(\gamma_{1} = 0.1\), \({\mathrm{Gr}} = 2\), \(\beta = 0.1\), \(\alpha_{1} = 1\), \({ \Pr } = 10\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Gm}} = 2\), \({\mathrm{Nb}} = 0.1\), \(Q_{1} = 2\), \({\mathrm{Sc}} = 0.2\), \({\mathrm{Nt}} = 0.1, \gamma = 2\), \(S = 0.1\)

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Fig. 12
figure 12

a Velocity distribution, b temperature distribution, c concentration distribution, d angular velocity distribution for different values of \({\mathrm{Sc }}\) at \(m_{1} = 10\), \(\phi_{1} = 0.4\), \(K = 0.2\), \(\gamma_{1} = 0.1\), \({\mathrm{Gr}} = 2\), \(\beta = 0.1\), \(\alpha_{1} = 1\), \({ \Pr } = 2\), \(\lambda = 0.01\), \(\varepsilon = 0.1\), \({\mathrm{Gm}} = 2\), \({\mathrm{Nb}} = 0.1\), \(Q_{1} = 2\), \(\chi = 2\), \({\mathrm{Nt}} = 0.1, \gamma = 2\)

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Conclusions

In the present work, FHD nanofluid flow over a stretching surface considering the rotation of the particles has been investigated in the presence of a stationary magnetic field. The problem is solved through finite element method through COMSOL Multiphysics. Maxwell parameter, ferromagnetic interaction number, rotation of the fluid and rotational viscosity plays important role in velocity distribution, heat transfer and mass transfer. Physical parameter considered in FHD nanofluid flow has some interesting characteristics. Some important observations based on the present investigation for FHD flow are as follows:

  • Increasing the values of ferromagnetic interaction number \(\left( \beta \right)\), Maxwell parameter \(\left( {\gamma_{1} } \right)\), thermal Grashof number \(\left( {\mathrm{Gr}} \right)\), solutal Grashof number \(\left( {\mathrm{Gm}} \right)\) and Brownian motion Parameter \(\left( {\mathrm{Nb}} \right)\) decreases the velocity distribution in the flow. However, for increasing values of thermophoresis parameter \(\left( {\mathrm{Nt}} \right)\), chemical reaction parameter \(\left( \gamma \right)\) and heat absorption parameter \(\chi\) increases the velocity distribution. When velocity decreases in the flow, some of the momentum force has been transformed into rotational force which increases the angular velocity distribution.

  • Heat transfer in the magnetic fluid increases for increasing values of ferromagnetic interaction number, thermal Grashof number, solutal Grashof number, Brownian motion parameter and radiation absorption coefficient. It decreases for increasing the values of thermophoresis number, chemical reaction parameter, Prandtl number, heat absorption parameter and Schmidt number. However, concentration distribution increases only for increasing the values of Prandtl number and heat absorption parameter. For other parameters, the concentration decreases in the fluid.

  • These results are presented in the composite effects of magnetization force, magnetic field-dependent viscosity and rotation of ferromagnetic fluid. Magnetic energy distributes into kinetic energy, thermal boundary layer and concentration boundary layer in the presence of considered physical parameters.