2. Mathematical equations for ferrohydrodynamics

Figure 1 is a representation of ferrohydrodynamic flow due to rotation in an oblique, rough magnetized plane. The whole physical phenomenon is demonstrated in two parts in this figure. In the first part of the figure, we consider a rough disc that rotates with uniform angular velocity. At a fixed distance, the disc is controlled by a magnetic dipole. The velocity components $({v_x},{v_y},{v_z})$ of the magnetic fluid are considered along the axes $x,y$ and z, respectively. The actual flow of the present study is represented in the second part of the figure. In this case, we consider that the ferrofluid is deposited through a spraying velocity W over an inclined and rough surface. Let the radius of the rough circular disc be large in comparison with the depth $(t)$ of the deposited film. Under slip conditions, flow occurs over the spinning disc, but the gravitational force causes a drainage flow on the slanted surface. However, the flow is induced in the y direction by surface rotation and gravitational forces. The magnitude of gravity $(\bar{g})$ works in the downward direction, as shown in the figure. Suppose ${T_c}$ denotes the Curie temperature and ${C_t}$ denotes the concentration on the film layer. We use the Cartesian form of coordinates since edge effects are neglected. This assumption presents the three-dimensional flow of ferrofluid film in this domain. However, most of the previous theoretical models of rotating flows were based on two-dimensional axisymmetric flows. The GEs in the three-dimensional Cartesian configuration are given in Wang (Reference Wang2007), Sheikholeslami et al. (Reference Sheikholeslami, Hatami and Ganji2015), Bacri et al. (Reference Bacri, Perzynski, Shliomis and Burde1995) and Andersson & Valnes (Reference Andersson and Valnes1998) as

(2.1)\begin{gather}\frac{{\partial {v_x}}}{{\partial x}} + \frac{{\partial {v_y}}}{{\partial y}} + \frac{{\partial {v_z}}}{{\partial z}} = 0,\end{gather}

(2.2)\begin{gather}\begin{aligned} {v_x}\dfrac{{\partial {v_x}}}{{\partial x}} + {v_y}\dfrac{{\partial {v_x}}}{{\partial y}} + {v_z}\dfrac{{\partial {v_x}}}{{\partial z}} & = \nu \left\{{1 + \dfrac{3}{2}\varphi \dfrac{{\xi - \tanh \xi }}{{\xi + \tanh \xi }}\textrm{si}{\textrm{n}^2}\vartheta } \right\}\left( {\dfrac{{{\partial^2}{v_x}}}{{\partial {x^2}}} + \dfrac{{{\partial^2}{v_x}}}{{\partial {y^2}}} + \dfrac{{{\partial^2}{v_x}}}{{\partial {z^2}}}} \right)\cr &\quad + \bar{g}\sin \beta + \dfrac{{{\mu _0}}}{\rho}M\dfrac{{\partial H}}{{\partial x}}, \end{aligned}\end{gather}

(2.3)\begin{gather}{v_x}\frac{{\partial {v_y}}}{{\partial x}} + {v_y}\frac{{\partial {v_y}}}{{\partial y}} + {v_z}\frac{{\partial {v_y}}}{{\partial z}} = \nu \left\{ {1 + \frac{3}{2}\varphi \frac{{\xi - \tanh \xi }}{{\xi + \tanh \xi }}\textrm{si}{\textrm{n}^2}\vartheta } \right\}\biggl( {\frac{{{\partial^2}{v_y}}}{{\partial {x^2}}} + \frac{{{\partial^2}{v_y}}}{{\partial {y^2}}} + \frac{{{\partial^2}{v_y}}}{{\partial {z^2}}}} \biggr),\end{gather}

(2.4)\begin{gather}\begin{aligned} {v_x}\dfrac{{\partial {v_z}}}{{\partial x}} + {v_y}\dfrac{{\partial {v_z}}}{{\partial y}} + {v_z}\dfrac{{\partial {v_z}}}{{\partial z}} & = \nu \left\{{1 + \dfrac{3}{2}\varphi \dfrac{{\xi - \tanh \xi }}{{\xi + \tanh \xi }}\textrm{si}{\textrm{n}^2}\vartheta} \right\}\left( {\dfrac{{{\partial^2}{v_z}}}{{\partial {x^2}}} + \dfrac{{{\partial^2}{v_z}}}{{\partial {y^2}}} + \dfrac{{{\partial^2}{v_z}}}{{\partial {z^2}}}} \right)\cr &\quad - \dfrac{1}{\rho }\dfrac{{\partial p}}{{\partial z}} - \bar{g}\cos \beta + \dfrac{{{\mu _0}}}{\rho}M\dfrac{{\partial H}}{{\partial z}}, \end{aligned}\end{gather}

(2.5)\begin{gather}\begin{aligned} &{v_x}\dfrac{{\partial T}}{{\partial x}} + {v_y}\dfrac{{\partial T}}{{\partial y}} + {v_z}\dfrac{{\partial T}}{{\partial z}} - \dfrac{{{\mu_0}}}{{\rho {c_p}}}T\dfrac{{\partial M}}{{\partial T}}\left( {{v_x}\dfrac{{\partial H}}{{\partial x}} + {v_y}\dfrac{{\partial H}}{{\partial y}} + {v_z}\dfrac{{\partial H}}{{\partial z}}} \right)\cr &\quad = \alpha \left( {\dfrac{{{\partial^2}T}}{{\partial {x^2}}} + \dfrac{{{\partial^2}T}}{{\partial {y^2}}} + \dfrac{{{\partial^2}T}}{{\partial {z^2}}}} \right) + {D_B}\left\{ {\dfrac{{\partial C}}{{\partial x}}\dfrac{{\partial T}}{{\partial x}} + \dfrac{{\partial C}}{{\partial y}}\dfrac{{\partial T}}{{\partial y}} + \dfrac{{\partial C}}{{\partial z}}\dfrac{{\partial T}}{{\partial z}}} \right\}\cr &\qquad + \dfrac{{{D_T}}}{T}\biggl\{{{{\left( {\dfrac{{\partial T}}{{\partial x}}} \right)}^2} + {{\left( {\dfrac{{\partial T}}{{\partial y}}} \right)}^2} + {{\left( {\dfrac{{\partial T}}{{\partial z}}} \right)}^2}} \biggr\} - \dfrac{1}{{\rho {c_p}}}\dfrac{{\partial {q_r}}}{{\partial z}} + \dfrac{Q}{{\rho {c_p}}}(T - {T_c}), \end{aligned}\end{gather}

(2.6)\begin{gather}\begin{aligned}{v_x}\frac{{\partial C}}{{\partial x}} + {v_y}\frac{{\partial C}}{{\partial y}} + {v_z}\frac{{\partial C}}{{\partial z}} &= {D_B}\left( {\frac{{{\partial^2}C}}{{\partial {x^2}}} + \frac{{{\partial^2}C}}{{\partial {y^2}}} + \frac{{{\partial^2}C}}{{\partial {z^2}}}} \right)\cr &\quad + \frac{{{D_T}}}{{{T_0}}}\biggl\{ {{{\left( {\frac{{\partial T}}{{\partial x}}} \right)}^2} + {{\left( {\frac{{\partial T}}{{\partial y}}} \right)}^2} + {{\left( {\frac{{\partial T}}{{\partial z}}} \right)}^2}} \biggr\}\!.\end{aligned}\end{gather}

In the GEs, $\nu $ indicates the kinematic viscosity, $\varphi $ implies the volume fraction, $\xi $ implies the Langevin parameter, $\vartheta $ signifies the obliqueness between the magnetic field and vorticity, ${\mu _0}$ signifies the permeability of free space, $\rho $ indicates density, M implies magnetization, H implies magnetic field intensity, $\alpha $ denotes the thermal diffusivity, T denotes the temperature, p denotes the pressure, $\alpha $ stands for the thermal diffusivity, ${D_B}$ implies the Brownian diffusion coefficient, ${D_T}$ implies thermophoretic coefficient, ${c_p}$ implies the specific heat, Q implies the heat source, C implies the concentration variable and ${T_0}$ denotes the initial temperature.

Figure 1. Demonstration of ferrohydrodynamic flow over an inclined rough surface under a magnetic dipole as per the boundary conditions given in (2.8) and (2.9).

Equation (2.1) is the equation of continuity; (2.2)–(2.4) are the equations of motion. The expression for the energy equation is shown by (2.5), and the concentration equation is represented by (2.6). Except for $(1 + {\textstyle{3 \over 2}}\varphi ((\xi - \tanh \xi )/(\xi + \tanh \xi ))\textrm{si}{\textrm{n}^2}\vartheta )$, $({\mu _0}/\rho )M(\partial H/\partial x)$ and $({\mu _0}/\rho )M(\partial H/\partial z)$, all of the terms in (2.1)–(2.4) are already available in the literature (Wang Reference Wang2007; Sheikholeslami et al. Reference Sheikholeslami, Hatami and Ganji2015) for ordinary viscous flow. When the vorticity is in line with the field, the expression ${\textstyle{3 \over 2}}\varphi ((\xi - \tanh \xi )/(\xi + \tanh \xi ))\textrm{si}{\textrm{n}^2}\vartheta$ becomes zero, indicating that the magnetic field's effect on viscosity is effectively non-existent. The difference between magnetic and viscous torques causes a rotational viscosity effect.

The term in the energy equation that takes into account the magnetic field, temperature and velocities is a representation of the thermo-mechanical interaction because of the magnetism (Neuringer & Rosensweig Reference Neuringer and Rosensweig1964). The coefficient ${D_B}$ in (2.5) and (2.6) is decided by the size of the particles and the properties of fluid and is particularly significant when the particle size is less than 0.1 micrometres. This is a crucial consideration when dealing with ferrofluids, as the nanoparticles they are composed of create additional complexities. The thermo-diffusion coefficient, temperature gradient and radiative heat flux play important roles in the energy equation for research involving ferrofluids. The parameter ${D_T}$ in the energy equation is the thermo-diffusion coefficient, which is the ratio of the thermophoretic velocity to the temperature gradient. Additionally, the variable ${q_r}$ denotes the radiative heat flux, which follows the Stefan–Boltzmann radiative law and can lead to nonlinear radiation effects (Steward & Cannon Reference Steward and Cannon1971; Mahanthesh Reference Mahanthesh2021). Mathematically, this can be evaluated as follows:

(2.7)\begin{equation}{q_r} =- \frac{{4{\sigma ^\ast }}}{{3{k^\ast }}}\frac{{\partial {T^4}}}{{\partial z}} =- \frac{{16{\sigma ^\ast }}}{{3{k^\ast }}}{T^3}\frac{{\partial T}}{{\partial z}}.\end{equation}

In (2.7), ${\sigma ^\ast }$ stands for the Stefan–Boltzmann constant and ${k^\ast }$ implies the average absorption coefficient.

2.1. Slip effects of inclined rotating surfaces with roughness

In most cases, it is difficult to achieve a perfectly smooth surface due to the natural imperfections or irregularities that occur on most surfaces. For this reason, a perfectly smooth surface is often hard to attain (Miklavčič & Wang Reference Miklavčič and Wang2004). We use slip conditions with a slip length parameter ${L_1}$ in the boundary conditions to accurately model the surface. This parameter specifies how much liquid is allowed to pass through the surface (Turkyilmazoglu Reference Turkyilmazoglu2010). This slip length parameter measures the length beyond which the liquid velocity linearly extrapolates to zero (Wang et al. Reference Wang2021). The initial and boundary conditions with slip effects are as follows (Mustafa Reference Mustafa2017; Alqarni et al. Reference Alqarni, Alveroğlu, Griffiths and Garrett2019):

(2.8)\begin{gather}\textrm{At}\;z = 0,\quad \left\{ {\begin{array}{*{20}{l}} {{v_x} =- \varOmega y + {L_1}\dfrac{{\partial {v_x}}}{{\partial z}}}\\ {{v_y} = \varOmega x + {L_1}\dfrac{{\partial {v_y}}}{{\partial z}}}\\ {{v_z} = 0}\\ {T = {T_0} + {L_2}\dfrac{{\partial T}}{{\partial z}}}\\ {{D_B}\dfrac{{\partial C}}{{\partial z}} + \dfrac{{{D_T}}}{{{T_c}}}\dfrac{{\partial T}}{{\partial z}} = 0} \end{array}} \right., \end{gather}

(2.9)\begin{gather}\textrm{At}\;z = t,\quad \left\{ {\begin{array}{*{20}{l}} {\dfrac{{\partial {v_x}}}{{\partial z}} = 0}\\ {\dfrac{{\partial {v_y}}}{{\partial z}} = 0}\\ {{v_z} =- W}\\ {T = {T_c}}\\ {C = {C_t}}\\ {p = {P_t}} \end{array}} \right..\end{gather}

In the above equations, the expressions ${L_1}(\partial {v_x}/\partial z)$ and ${L_1}(\partial {v_y}/\partial z)$ represent the slip velocities in the radial and tangential regions of the surface, where ${L_1}$ denotes the slip length which is related to the viscosity of the ferrofluid and the interfacial friction (Nicholson et al. Reference Nicholson, Power, Tammisola, Hibberd and Kay2019). The expression ${L_2}(\partial T/\partial z)$ represents the thermal slip due to a rough surface. Thermal slip in fluid flow is a phenomenon in which the fluid velocity is greater near the boundary of a hot surface than it is further away. This is the result of fluid particles’ interactions with the temperature gradient, which cause an imbalance in the momentum of particles near the hot surface compared with those further away. The parameter ${L_2}$ represents the temperature jump. Based on the wettability of the surface, the slip length can be either positive (+ive) or negative (−ive). These slip conditions, however, are considered for a water-based ferrofluid. As a result, slip parameters can be described using either hydrophobic or hydrophilic surfaces. The temperature jump in ferrofluids could be useful for heat transfer applications and electronic device cooling. Temperature shielding can be accomplished by using large temperature jumps. It can be used to control the heat in electronic devices as well as in medical applications that require localized heating. The parameter ${P_t}$ denotes the pressure on the film.

2.2. Exploring the magnetic effect and the equation of state

In the current problem, the dipole is placed at the centre (middle) of the z-axis; however, it is fixed at a distance $(a)$ below the z-axis. The applied magnetic field $\boldsymbol{H} = {H_x}\hat{i} + {H_y}\hat{j} + {H_z}\hat{k}$ is equal to the scalar potential function's negative gradient $( - \boldsymbol{\nabla }A)$, where ${H_x}$, ${H_y}$ and ${H_z}$ denote the strength of the magnetic field in the x-axis (radial), y-axis (tangential) and z-axis (axial) directions, respectively, and, $\hat{i}$, $\hat{j}$ and $\hat{k}$ represent the corresponding unit vectors. We write it mathematically as $\boldsymbol{H} =- \boldsymbol{\nabla }A$. The magnetic scalar potential function caused by the magnetic dipole is (Rosensweig Reference Rosensweig1985)

(2.10)\begin{equation}A = \frac{{{\gamma _0}}}{{2{\rm \pi} }}\frac{x}{{{x^2} + {{(z + a)}^2}}},\end{equation}

where ${\gamma _0}$ represents the dipole moment per unit length. Using (2.10), the component-wise magnetic field intensity can be calculated as (Andersson & Valnes Reference Andersson and Valnes1998)

(2.11a–c)\begin{gather}\frac{{\partial H}}{{\partial x}} =- \frac{{\partial A}}{{\partial x}} = \frac{{{\gamma _0}}}{{2{\rm \pi} }}\frac{{{x^2} - {{(z + a)}^2}}}{{{{[{x^2} + {{(z + a)}^2}]}^2}}},\quad \frac{{\partial H}}{{\partial y}} = 0,\quad \frac{{\partial H}}{{\partial z}} =- \frac{{\partial A}}{{\partial z}} = \frac{{{\gamma _0}}}{{2{\rm \pi} }}\frac{{2x(z + a)}}{{{{[{x^2} + {{(z + a)}^2}]}^2}}}.\nonumber\\\end{gather}

The magnetic body force due to magnetic dipole is ${\mu _0}M\boldsymbol{\nabla }H$ and the resultant determinable magnetic strength (Andersson & Valnes Reference Andersson and Valnes1998)

(2.12)\begin{equation}H = {[{({H_x})^2} + {({H_y})^2} + {({H_z})^2}]^{1/2}}.\end{equation}

Using (2.11a–c) and (2.12), we can compute the parts of the magnetic field gradient (Neuringer Reference Neuringer1966). Thereafter, these expressions can be used in the momentum equations (2.2) and (2.4). These components are

(2.13a,b)\begin{equation}\frac{{\partial H}}{{\partial x}} = \frac{{\dfrac{{\partial A}}{{\partial x}}\; \dfrac{{{\partial ^2}A}}{{\partial {x^2}}} + \dfrac{{\partial A}}{{\partial z}}\; \dfrac{{{\partial ^2}A}}{{\partial x\; \partial z}}}}{{{{\biggl[ {{{\left({\dfrac{{\partial A}}{{\partial x}}} \right)}^2} + {{\left({\dfrac{{\partial A}}{{\partial z}}} \right)}^2}} \biggr]}^{1/2}}}},\quad \frac{{\partial H}}{{\partial z}} = \frac{{\dfrac{{\partial A}}{{\partial x}}\; \dfrac{{{\partial ^2}A}}{{\partial x\; \partial z}} + \dfrac{{\partial A}}{{\partial z}}\; \dfrac{{{\partial ^2}A}}{{\partial {z^2}}}}}{{{{\biggl[{{{\left({\dfrac{{\partial A}}{{\partial x}}} \right)}^2} + {{\left({\dfrac{{\partial A}}{{\partial z}}} \right)}^2}} \biggr]}^{1/2}}}}.\end{equation}

The component of the gradient in the y direction $\partial H/\partial y$ becomes automatically zero because the scalar potential function is considered to be composed of two variables x and z. We consider the second-order approximation of the variable x. This approximation reduces the components of the magnetic field gradient as (Andersson & Valnes Reference Andersson and Valnes1998)

(2.14a,b)\begin{equation}\frac{{\partial H}}{{\partial x}} =- \frac{{{\gamma _0}}}{{2{\rm \pi} }}\frac{{2x}}{{{{(z + a)}^4}}},\quad \frac{{\partial H}}{{\partial z}} = \frac{{{\gamma _0}}}{{2{\rm \pi} }}\left[ { - \frac{2}{{{{(z + a)}^3}}} + \frac{{4{x^2}}}{{{{(z + a)}^5}}}} \right]\!.\end{equation}

Temperature shifts influence the magnetic force of ferrofluid, and temperature variations can alter ferrofluid magnetism (Zablotsky, Mezulis & Blums Reference Zablotsky, Mezulis and Blums2009). At average temperature ${T_a}$ the magnetic field ${H_0}$ follows an equation of state, or the equilibrium state at temperature ${T_a}$ is determined by the equation of state for ${H_0}$ as Finlayson (Reference Finlayson1970), Zebib (Reference Zebib1996)

(2.15)\begin{equation}M = {M_0} + {\left( {\frac{{\partial M}}{{\partial H}}} \right)_{{H_0},{T_a}}}(H - {H_0}) + {\left( {\frac{{\partial M}}{{\partial T}}} \right)_{{H_0},{T_a}}}(T - {T_a}).\end{equation}

When the magnetism is sufficient to saturate the ferrofluid within the boundary layer, it could be described by an equation (linear) of the state between the extreme temperatures experienced by the fluid (Neuringer Reference Neuringer1966). The approximation of the linear equation of state is (Neuringer & Rosensweig Reference Neuringer and Rosensweig1964)

(2.16)\begin{equation}M =- \left|{\frac{{\partial M}}{{\partial T}}} \right|(T - {T_c}),\end{equation}

where $|\partial M/\partial T|= K$ represents the pyro-magnetic coefficient and ${T_c}$ implies the Curie temperature.

3. Normalizing the thin-film flow equations

The PDEs concerning ferrohydrodynamic thin-film flow can be simplified into a set of nonlinear ODEs. Wang (Reference Wang2007) suggested the following similarity transformation:

(3.1)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {{v_x} =- \varOmega yg(\eta ) + \varOmega xf^{\prime}(\eta ) + \dfrac{{\bar{g}\sin \beta h(\eta )}}{\varOmega },\quad {v_y} = \varOmega xg(\eta ) + \varOmega yf^{\prime}(\eta ) + \dfrac{{\bar{g}\sin \beta k(\eta )}}{\varOmega },}\\ {{v_z} =- 2\sqrt {\varOmega \nu } \; f(\eta ),\quad T = {T_c} - {T_1}(\eta )({T_c} - {T_0}),\quad C = {C_t} + {C_1}(\eta )({C_0} - {C_t}),}\\ {\eta = z{{\left( {\dfrac{\varOmega }{\nu }} \right)}^{1/2}},\quad \zeta = x{{\left( {\dfrac{{c\rho }}{\mu }} \right)}^{1/2}}.} \end{array}} \right\}\end{equation}

By satisfying (2.1) and degenerating (2.2)–(2.6) and (2.8) and (2.9), this transformation produces a set of nonlinear coupled differential equations:

(3.2)\begin{gather}Rf^{\prime\prime\prime} + 2ff^{\prime\prime} - {(\,f^{\prime})^2} + {g^2} = 0,\end{gather}

(3.3)\begin{gather}Rg^{\prime\prime} - 2g\kern0.06em f^{\prime} + 2fg^{\prime} = 0,\end{gather}

(3.4)\begin{gather}Rh^{\prime\prime} - hf^{\prime} + kg + 2fh^{\prime} - {F_m}\frac{\zeta}{{{{(\eta + {\alpha _1})}^4}}}\theta + 1 = 0,\end{gather}

(3.5)\begin{gather}Rk^{\prime\prime} + 2fk^{\prime} - kf^{\prime} - hg = 0,\end{gather}

(3.6)\begin{gather}\begin{aligned} \dfrac{1}{{Pr}}&\left[ {{T^{\prime\prime}_1}}\left\{ {1 + \dfrac{4}{3}N{{(1 + {T_1}({T_R} - 1))}^3}} \right\} + \dfrac{4}{3}N\{ 3{{(1 + {T_1}({T_R} - 1))}^2}({T_R} - 1){{({T^{\prime}_1})}^2}\} \right]\nonumber\\ & \qquad +\, 2f{T^{\prime}_1} + Nb{C^{\prime}_1}{T^{\prime}_1} + Nt{({T^{\prime}_1})^2} + \lambda {T_1} + {F_1}\dfrac{\zeta}{{{{(\eta + {\alpha_1})}^4}}}({T_1} - \varepsilon )h\nonumber\\ & \qquad -\, {F_2}(\theta - \varepsilon )f\left[ {\dfrac{2}{{{{(\eta + {\alpha_1})}^3}}} - \dfrac{{4{\zeta^2}}}{{{{(\eta + {\alpha_1})}^5}}}} \right] = 0, \end{aligned}\end{gather}

(3.7)\begin{gather}{C^{\prime\prime}_1} + \frac{{Nt}}{{Nb}}{T^{\prime\prime}_1} + Scf{C^{\prime}_1} = 0.\end{gather}

In (3.2)–(3.7), the parameter $R = 1 + {\textstyle{3 \over 2}}\varphi \xi ((\xi - \tanh \xi )/(\xi + \tanh \xi ))\textrm{si}{\textrm{n}^2}{\alpha _m}$ represents the rotational viscosity parameter. This parameter depends on the parameter $\xi $, which is the index of magnetic energy (and thermal energy) and the volume fraction $(\varphi )$ of the magnetic particles. However, if the field is parallel to the vorticity, then this parameter becomes zero. The parameter ${F_m} = {\mu _0}K{\gamma _0}\textrm{(}{T_c} - {T_w}\textrm{)}{\varOmega ^2}/{\rm \pi} \bar{g}\; \rho {c^{1/2}}{\nu ^{3/2}}\sin \beta $ represents the influence of the magnetization force on the flow. The parameter ${\alpha _1} = a{(\varOmega /\nu )^{1/2}}$ represents the effect of the distance of the dipole from the surface of the plate. The parameter $Pr = \nu /\alpha $ represents the Prandtl number and it is a dimensionless number used to express the proportion between thermal diffusivity and momentum diffusivity. The parameter $N = 4{\sigma ^\ast }T_c^3/3{k^\ast }k$ denotes the radiation parameter and it describes the rate of radiation-based heat transfer in a homogeneous medium and is a function of the temperature, absorption coefficient and the Stefan–Boltzmann constant. The parameter ${T_R} = {T_0}/{T_c}$ denotes the temperature ratio parameter. The parameters ${F_1} = {\mu _0}K{\gamma _0}\bar{g}\sin \beta {\varOmega ^2}/2{\rm \pi} \; {c_p}\rho {c^{1/2}}{\nu ^{3/2}}$ and ${F_2} = {\mu _0}K{\gamma _0}\varOmega /2{\rm \pi} {c_p}\rho \nu $ represent the ferromagnetic interaction numbers and they measure the relative strengths of ferromagnetic interactions between particles and fluid. The parameter $\varepsilon = {T_c}/({T_c} - {T_0})$ denotes the dimensionless Curie temperature and it is used to describe the temperature at which a material undergoes a phase change from one magnetic state to another. The strength of the Brownian motion in comparison with other fluid motions is expressed by the Brownian motion parameter $Nb = {D_B}({C_0} - {C_t})/\nu $. The relative strength of thermophoretic forces in a fluid is measured using the thermophoresis parameter $Nt = {D_T}({T_c} - {T_0})/\nu {T_c}$. The strength of the heat source or sink in relation to the fluid's actual thermal diffusivity is described by the heat source/sink parameter $\lambda = Q/\varOmega \rho {c_p}$. How well different molecules transfer heat and momentum in a laminar flow is determined by the Schmidt number $Sc$.

The similarity transformation simplifies the initial and boundary conditions by transforming them into a normalized form

(3.8)\begin{gather}\left. {\begin{array}{*{20}{c@{}}} {f(0) = 0,\quad f^{\prime}(0) = \gamma f^{\prime\prime}(0),\quad g(0) = 1 + \gamma g^{\prime}(0),\quad h(0) = \gamma h^{\prime}(0)}\\ {k(0) = \gamma k^{\prime}(0),\quad {T_1}(0) = 1 + \alpha {T^{\prime}_1}(0),\quad {C^{\prime}_1}(0) - \dfrac{{Nt}}{{Nb}}{T^{\prime}_1}(0) = 0} \end{array}} \right\}\!,\end{gather}

(3.9)\begin{gather}f^{\prime\prime}(\delta ) = 0,\quad g^{\prime}(\delta ) = 0,\quad h^{\prime}(\delta ) = 0,\quad k^{\prime}(\delta ) = 0,\quad {T_1}(\delta ) = 0,\quad {C_1}(\delta ) = 0\ \textrm{at}\ \eta = \delta .\end{gather}

The parameters $\gamma = {L_1}{(\varOmega /\nu )^{1/2}}$ and ${\gamma _1} = {L_2}{(\varOmega /\nu )^{1/2}}$ in the equations above stand for the velocity slip parameter and thermal slip parameter, respectively. The symbol denotes the normalized and constant film thickness $\delta $. Utilizing the equation below, it can be calculated

(3.10)\begin{equation}\delta = t{(\varOmega /\nu )^{1/2}}.\end{equation}

3.1. Local heat transfer and mass transport quantity

The speed at which heat moves from the surface of a slanted plate to a thin film is known as the local rate of heat transfer. The rate of convective heat transfer between two surfaces in a flow is determined using the dimensionless Nusselt number (Nu). The quantity is calculated by dividing the convective heat transfer by the conductive heat transfer

(3.11)\begin{equation}Nu = \frac{{t[ - \kappa {{({T_z})}_{z = 0}} + {{({q_r})}_{z = 0}}]}}{{\kappa ({T_c} - {T_w})}} =- \left[ {1 + \frac{{4N}}{3}{{\{ 1 + ({T_R} - 1){T_1}(0)\} }^3}} \right]\delta {T^{\prime}_1}(0).\end{equation}

The ratio of diffusive mass flux to convective mass flux determines the mass transport quantity. The Sherwood number $(Sh)$ is a dimensionless number that is used to calculate the mass transport rate in a particular flow system

(3.12)\begin{equation}Sh = \frac{{ - t{C_z}}}{{({C_w} - {C_0})}} =- \delta {C^{\prime}_1}(0).\end{equation}

4. Analysis and convergence of numerical solutions

We employ the finite element technique in the COMSOL Multiphysics program to numerically compute (3.2)–(3.7) and the boundary conditions are given in (3.8) and (3.9) numerically. We set the relative tolerance to 0.0000001. When solving a system of equations, the relative tolerance determines the extent of convergence. It is defined as the ratio of the absolute and relative errors and is set to a specific value before solving. When the relative error exceeds the tolerance, the solver will stop and report an error. The convergence plot can be seen in figure 2 for the results of the normalized equations of the system. The convergence plot in COMSOL Multiphysics is a graphical representation of the solution's progress toward convergence. It displays the residual errors (in blue) as the number of iterations increases. The plot's lines should bend downward, indicating that the errors are decreasing over time. This shows that the solution is approaching a valid result.

Figure 2. Convergence of numerical solutions using finite element procedures.

During the numerical solution, we take the largest length of the element as 0.0001, the highest element enlargement scale as 1.1 and the resolution of a narrow region as 1. The largest size an element can be before the mesh becomes too coarse is referred to as the maximum element size. This value should be set so that the mesh has enough resolution to capture the solutions but is not too fine, as this can cause the mesh to become too large and slow down the solution process. When optimizing the mesh size and distribution, the maximum element growth rate is an important parameter to adjust because it allows the user to control the maximum size of elements while ensuring that the mesh does not become too coarse. The procedure for utilizing COMSOL Multiphysics software to numerically solve the NDE is shown in figure 3.

Figure 3. Solution procedure for the NDEs in COMSOL Multiphysics software.

4.1. Validation of solution

The NDEs reduce to the previous theoretical model by excluding the terms containing $Nb$ and $Nt$ and considering the values of the parameters $R = 0$, ${F_m} = 0$, $N = 0$, ${F_1} = 0$, ${F_2} = 0$, $\lambda = 0$. In this case, with no-slip effects, our results for the radial (linear) velocity $f^{\prime}$, the tangential (angular) velocity g, the axial (vertical) velocity f and the fluid temperature $({T_1})$ profile are in agreement with Wang's numerical solution (Wang Reference Wang2007). Figure 4 shows the profiles of the axial, radial and tangential velocities, respectively. These results illustrate the special case of the current model, in which the normalized unitless equations are in the form of Wang's normalized case. The findings support the earlier numerical findings (Wang Reference Wang2007). Figure 5 displays the temperature profile for varied values of the Prandtl number, which are adopted from the previous results (Wang Reference Wang2007) for comparison. The temperature profile has excellent correspondence to the past solution (Wang Reference Wang2007). Omitting the abovementioned parameters along with the inclination, the current problem can be simplified to the Kármán swirling problem (Miklavčič & Wang Reference Miklavčič and Wang2004; Turkyilmazoglu Reference Turkyilmazoglu2010). In our case, $f^{\prime\prime}(0)$ and $g^{\prime}(0)$ are 0.5102134911 and −0.6159097466, respectively. These values match with the solution of the Kármán swirling flow problem obtained by researchers (Benton Reference Benton1966; Miklavčič & Wang Reference Miklavčič and Wang2004; Turkyilmazoglu Reference Turkyilmazoglu2010).

Figure 4. Validation for velocity profiles through finite element procedure exactly matches with Wang's solution (Wang Reference Wang2007).

Figure 5. Validation for temperature profile through finite element procedure exactly matches with Wang's solution (Wang Reference Wang2007).

5. Description and analysis of results

This manuscript describes the influence of magnetism on the ferrohydrodynamics upon a slanted rotating surface under an applied field of a dipole. The earlier section presents the validation of the present numerical solution. However, in this work, the constant parameters in the solution are the volume fraction $(\varphi ) = 0.05$, magnetic torque-induced parameter $(\xi ) = 1$, ferromagnetic interaction number due to a magnetic dipole in the momentum equation $({F_m}) = 0.5$, ferromagnetic interaction numbers due to magnetic energy $({F_1},\; {F_2}) = 0.5,0.5$, dimensionless Curie temperature$(\varepsilon ) = 2$, the distance of the dipole from the surface of the plate $({\alpha _1}) = 1$, normalized radial distance $(\zeta ) = 0.1$, Prandtl number for a water-based ferrofluid $(Pr) = 7$, velocity slip parameter due to a rough surface $(\gamma ) = 0.1$, thermal slip parameter $({\gamma _1}) = 0.1$, Brownian motion parameter$(Nb) = 0.4$, thermophoresis parameter $(Nt) = 0.1$, heat source parameter $(\lambda ) = 0.2$, Schmidt number $(Sc) = 0.08$, radiation parameter $(N) = 0.5$ and temperature ratio parameter ${T_R} = 1.5$. From previously published literature (Cowley Reference Cowley1989; Blūms, Cebers & Maĭorov Reference Blms, Cebers and Maĭorov1997; Wang Reference Wang2007; Acharya Reference Acharya2021; Bhandari Reference Bhandari2022), these parameter values are taken into account. However, in the previous study of the flow over an inclined rotating surface, the researchers have considered the large range of Schmidt numbers for which the flow becomes turbulent (Sheikholeslami et al. Reference Sheikholeslami, Hatami and Ganji2015; Acharya Reference Acharya2021). However, the equations consider the laminar flow dynamics.

Figures 6(a)–6(c) represent the velocity profiles to measure the influence of the coefficient of velocity slippage $(\gamma )$ on the ferrofluid thin film. The speed at which the fluid is rotating around the centre diminishes when the slip velocity parameter is increased. Because more particles move away from the centre, the radial velocity increases. At the same time, because of the slower rotational speed, the tangential velocity decreases. Axial velocity increases in rotational flow after increasing the slip velocity parameter because, as the slip velocity increases, the flow becomes more intense, causing the axial velocity to increase. This is due to momentum conservation, as increased fluctuation causes the axial velocity to increase while the total flow rate remains constant. Figures 7(a) and 7(b) show the drainage velocity in the linear direction due to inclination and induced velocity in the angular region due to nonlinearities. The draining velocity of an inclined surface is the speed at which liquid flows down the inclined surface due to gravity. The draining velocity is determined by the surface slope, liquid viscosity and gravitational acceleration. Escalating $\gamma $ causes a higher gradient in the velocity distribution, which increases the draining velocity. This occurs because a higher slip velocity causes more fluid motion, which increases the force exerted on the liquid due to pressure gradients. This compressive stress gradient causes a faster draining velocity. The Coriolis force, which acts on a moving object and is vertical to the motion of fluid, causes an induced stream in the y direction due to a rotating inclined surface. The speed of the surface's rotation and the angle of inclination both affect the amount of flow that is induced as the surface rotates in the y direction. More flow fluctuation from increased velocity slip leads to an imbalance in the stress at the surface. This increased stress causes a greater force on the liquid particles, pushing them along the line of the flow, resulting in a higher induced flow velocity.

Figure 6. Role of the slip factor $\gamma $ in the velocity profiles due to inclined flow of ferrofluid thin film.

Figure 7. Role of the slip factor in the drainage flow velocity and induced flow velocity of ferrofluid thin film.

Figures 8(a) and 8(b) show the behaviour of temperature and concentration for different ranges of $\gamma $. Because of the increased fluid velocity, escalating $\gamma $ can cause the temperature to drop. The turbulent flow raises the kinetic energy of the fluid particles, which results in a drop in the fluid's internal energy and a lowering of its temperature. Because of the increased fluid velocity, increasing the velocity slip parameter can increase the concentration of the ferrofluid. The increased fluctuation causes more particle mixing, resulting in a higher concentration of ferrofluid. Furthermore, the increased kinetic energy of the particles aids in the breakdown of larger particle clusters, increasing the particle concentration. Figures 9(a) and 9(b) depict how the temperature and concentration of a ferrofluid thin liquid film are affected by the thermal slip parameter $({\gamma _1})$. The thermal slip parameter measures the thermal difference between the wall and the liquid. The localized temperature gradient at the wall divided by the overall temperature gradient of the bulk fluid yields the thermal slip parameter. It is employed to compute the impact of frictional heating, brought on by the fluid's viscosity, on the wall's temperature. Increasing ${\gamma _1}$ can lower the temperature by diminishing thermal energy transfer from the wall to the fluid. This is because the temperature gradient between the fluid and the wall has decreased, so has the overall rate of heat transfer. Additionally, the increased viscous heating brought on by the elevated thermal slip parameter may help to further lower the wall's temperature, lowering the system's overall temperature. Increasing ${\gamma _1}$ reduces wall-to-fluid thermal energy transfer, lowering the bulk temperature and raising ferrofluid concentration. This aids in producing an environment that is more favourable to particle cluster formation, which ultimately leads to higher concentration. Furthermore, the larger particle clusters can be broken up even more by the increased thermal slip parameter. It rises with particle concentration due to the increased viscous heating caused by the increased thermal slip parameter.

Figure 8. Role of the slip factor in the temperature flow and concentration profile.

Figure 9. Role of the thermal slip factor in the temperature and concentration profiles.

Figure 10(a) illustrates how the concentration profile is affected by the Brownian motion parameter $(Nb)$. The concentration flow decreases as this parameter's value rises. By raising $Nb$, one can reduce the concentration in the flow over a rotating surface because it decreases the particle diffusion rate. This indicates that the particles are moving more slowly, which lowers the concentration overall. More Brownian motion can also make particles more evenly distributed, resulting in lower concentrations in particular flow regions. Figures 10(b) and 10(c) display the temperature profile as well as the concentration profile for different values of the thermophoresis parameter $(Nt)$. This expedites the wall to fluid heat transfer. This is due to a greater temperature difference between the wall and the liquid, which enhances the overall rate of heat transfer. Furthermore, increased thermophoresis can cause particles to become more evenly distributed, resulting in a faster heat transfer rate in specific regions of the flow. Because it increases the rate of particle diffusion, the thermophoresis parameter $(Nt)$ increases the concentration near the surface and diminishes it beyond the surface in the stream triggered by a slanted rotating plate. The increased thermal gradient between the wall and the fluid contributes to a faster overall particle diffusion rate. As a result, particles near the wall are more likely to diffuse away from it, causing the concentration near the surface to rise. Particles far from the wall, on the other hand, are more likely to be drawn towards it, resulting in a decrease in concentration far from the surface.

Figure 10. Role of the Brownian motion parameter in the concentration profile and role of the thermophoresis parameter in the temperature and concentration profiles.

Figures 11(a) and 11(b) illustrate how the temperature and concentration profile are affected by the heat source parameter $(\lambda )$. Because it accelerates the rate of heat transfer from the wall to the fluid, the heat source parameter$(\lambda )$ raises the temperature in the fluid film over an angled spinning surface. Furthermore, the increased heat source can cause the particles to become more evenly distributed, resulting in a higher heat flux rate in a ferrofluid film flow. This variable accelerates the transfer of heat from the surface to the film. As an outcome of the particle's decreased solubility, the temperature of the fluid rises, resulting in a decrease in concentration. Furthermore, the increased heat source can cause the particles to become more evenly distributed, resulting in lower concentrations in specific regions of the flow.

Figure 11. Role of the heat source parameter in the temperature and concentration profiles.

Table 1 shows how the Nusselt number (Nu) and Sherwood number (Sh) are affected by $\gamma $, ${\gamma _1}$, $Nb$, $Nt$ and $\lambda $. Increasing the velocity slip parameter in an inclined narrow film passing over a spinning surface raises the Nusselt number by lowering the velocity gradient and thus increasing the shear rate and convective heat transfer rate. The Nusselt number consequently increases. Increasing the velocity slip parameter in a fluid flow raises the Sherwood number because it reduces the viscous drag between the two surfaces and makes diffusive mass transfer less dominant, resulting in a higher mass transport rate. Increasing the thermal slip parameter reduces Nu and Sh in the fluid flow because it reduces the temperature gradient between wall and fluid and thus the rate of convective heat transfer and diffusive mass transfer. Consequently, the mass transport rate is reduced, as are the Nusselt and Sherwood numbers. A similar conclusion can be reached for the other parameters listed in table 1. The surface stress in the x and y directions is represented by $h^{\prime}(0)$ and $k^{\prime}(0)$, respectively, in the normalized form. Due to a magnetic dipole, it is displayed in table 2 for various values of the velocity slip parameter $(\gamma )$ and ferromagnetic interaction number $({F_m})$. Stress in the x direction is reduced and stress in the y direction is increased by the velocity slip parameter. On the other hand, stress on the thin film's surface decreases as the ferromagnetic interaction number increases.

Table 1. Nusselt numbers and Sherwood numbers for different values of the parameters of thermal slip, Brownian motion, thermophoresis and velocity slip.

Table 2. Stress on the surface for different values of the velocity slip parameter and ferromagnetic interaction number.

6. Outcomes

In the current work, we examined the effects of temperature and velocity slip on ferrofluid thin-film flow across an inclined, rough–rotating surface when a magnetic dipole is present. In situations where the axis of rotation is not parallel to the gravitational vector, such as those found in space or on Earth, this problem is significant. However, controlling the ferrofluid flow in a zero-gravity environment is possible when a magnetic dipole is there. The following are the primary conclusions of the current investigation:

• • The draining speed in the x direction, the induced speed in the y direction, the radial velocity and the axial velocity all rise as the velocity slip parameter is increased. On the other hand, the local heat and mass transport rates are increased by this parameter.

• • Increasing the velocity slip parameter and thermal slip parameter can cause the temperature to drop and the concentration of ferrofluid to increase. The heat and mass transfer rates in the flow are decreased by the thermal slip parameter.

• • The concentration is lowered by raising the thermophoresis and Brownian motion parameters.

• • The thermophoresis parameter increases the mass transfer rate while decreasing the heat transfer rate, and the Brownian motion parameter decreases the mass transfer rate.

• • Due to reduced solubility, heat source parameters can decrease concentration while increasing the temperature in the ferrofluid film.

• • The findings and understanding gained from this work could be used in many industrial processes, such as coating, lubrication and heat transfer, as well as a variety of medical domains such as targeted medicine therapy and delivery, tissue engineering, etc.