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Nanotechnology currently has an important role in reducing engine wear and improving fuel efficiency within engines using nanoparticles in engine oil. Therefore, the work reported in this paper, aims to investigate the magnetohydrodynamic (MHD) flow of Brinkmantype Engine Oilbased Molybdenum disulfide (MoS_{2}) nanofluid (BEOBMN) in a rotating frame along with Hall effect and thermal radiation. The problem is modeled in terms of partial differential equations with physical initial and boundary conditions. The Laplace transform technique is used to evaluate the exact solutions for velocity and temperature profiles. Graphical results are obtained through a computational software Mathcad and discussed for various embedded parameters. The Skinfriction and Nusselt number are computed in the tabular form and it is noticed that the rate of heat trans
BEOBMN, MHD flow, closedform solutions, the laplace transform
Convective heat transfer in nanofluids has gotten great concern in science as well as in engineering. Traditional fluids such as kerosene oil, ethylene glycol, polyethylene glycol, water and engineoil etc. play a vital role in thermal transport application but these fluids have poor thermal conductivities which degrade the heat transfer rate. Despite the substantial attempts to upgrade the rate of heat transfer by the extended surfaces usage, minichannels, and microchannels, more improvement in heating and cooling is always in demand [1]. Early studies, carried out by using millimeter or micrometer particles dispersed in traditional heat transfer fluids. This idea of utilizing smallsized solid particles within the base fluid to enhance their thermal conductivity and heat transfer rate was given by Maxwell [2]. However, there were some limitations of Maxwell’s idea i.e. the faster settling time, clogging microchannels of devices, abrasion of surfaces and erosion of pipelines etc. [3]. In 1995 Choi [4], introduced a new idea of suspension of nanosized particles in the base fluids to improve the thermal conductivity. Alongside higher thermal conductivity, the suspension of nanosized particles instead microsized particles in a traditional base fluid (currently known as nanofluids) is preferred because of some valid scientific reasons such as longer suspension time (more stable), lower clogging and erosion, lower demand for significant energy saving pumping power. Nanofluids, due to high thermal conductivity have several useful and potential applications in physical circumstances. For example, they are used on a large scale in automobiles as a coolant, microelectronics, and microchips in computer, biomedicines, fuel cells, transportation, food processing, and solidstate lighting and manufacturing. In inclusion metals and metallic oxides, carbon nanotubes are used for heat transfer enhancement, biomedical equipment, lubrication in moving parts of machines [56]. Loganathan et al. [7] presented the first exact solution of nanofluids for unsteady free convection flow, taken into account the effect of radiation. Sheikholeslami et al. [8] investigated the MHD flow of nanofluid and heat transfer in a rotating frame along with the effect of Brownian motion. Recently, the effect of gold (Au) nanoparticles on hydromagnetic poiseuille flow of nanofluid embedded in a porous medium have been investigated by Aman et al. [9]. Of course, the list of such studies can continue, but we close it with some of the most interesting analytical and numerical results that have been obtained in [10–16].
Nowadays, different nanoparticles with various shapes are used as additives for improving properties of lubricants. In present work, spherical shape MoS_{2 }nanoparticles have been used in EO because MoS_{2} has low friction properties and robustness which can help in enhancing the heat transfer rate as well as lubricity of engine oil [17 &18]. Molybdenum disulfide (MoS_{2}) is an inorganic compound classified as a metal dichalcogenide, an alternate layer of molybdenum and sulfur atoms. It is a silvery black solid that occurs as the mineral molybdenite. MoS_{2} is relatively unreactive and is unaffected by dilute acids and oxygen. Each molybdenum atom is sandwich between trigonal prisms of sulfur atoms in a hexagonal closed pack arrangement. The sheets of sulfur are loosely bound by van der Waals interactions [19]. The homogenous stable Newtonian molybdenum disulfide MoS_{2} nanofluids were experimentally prepared by Zhang et al. [20]. Similar to graphene, MoS_{2} has single and multilayer sheets with large bandgap structure. MoS_{2} has attracted the interest of researchers due to its promising applications in many areas especially, in twodimensional electronic devices such as Field Effect Transistors (FETs). Due to this special structure, it has been extensively applied to logic circuits and amplifier devices [21 & 22]. Some other attempts on MoS_{2} nanofluids are made by Gul et al. [23], Kato et al. [24] and Mao et al. [25]. The heat capacity, thermal conductivity, and volumetric thermal expansion coefficients were experimentally studied by Liu et al. [26] and Ding and Xiao [27].
In 1856 Darcy proposed a model for the fluid that flows over a body with low porosity [28]. The flow passes over a body with high porosity does not obey Darcy’s law. This model was recommended by Brinkman for fluid that passing through high porous media in 1949 [29]. There is a massive use of Brinkmantype flow in many branches of engineering and science e.g., groundwater hydrology, reservoir engineering, soil science, soil mechanics and chemical engineering [30]. Brinkman in his pioneering work [31], presented calculations for the viscous forces exerted by fluid flowing on a dense swarm of particles. Another model, developed by Brinkman [32] for the incompressible viscous fluid passing through a porous media, which incorporates Darcy’s equation as a unique case. Several researchers have done informative works on Brinkmantype fluid, which are listed in the references [3336]. Recently, Ali et al. [37] used the Laplace transform technique to develop several new exact solutions for the flow of Brinkmantype fluid. The solutions were obtained in the simple form of complementary and elementary error functions. In another paper, Ali et al. [38] obtained closedform solutions for unsteady free convection hydromagnetic flow of an incompressible Brinkmantype fluid by using the Laplace transform technique, four different types of flows were discussed in this paper along with thermal radiation. Recently, Ali et al. [39] considered the flow of Brinkmantype fluid with four different types of nanoparticles in a spherical shape. Exact solutions were obtained for velocity, temperature and concentration profiles by using the Laplace transforms.
In recent years, the study of hydromagnetic is under great concerned. It has many important applications in engineering and technology such as hydromagnetic generators (it includes disk system) and MHD flow meters, plasma studies, pumps, bearings, geothermal energy extractions and nuclear reactors, solar energy collection, boundary layer control, aerodynamic heating casting and levitation, extraction of petroleum products and cooling of an infinite metallic plate in a cooling bath. Besides this, there are many applications of hydromagnetic flow of nonNewtonian fluids in a rotating body in metrology, geographic, turbomachinery, astrophysical and several other areas. In addition, it has many applications in biomedical such as blood flow in capillaries, dialysis of blood in artificial kidney and flow in blood oxygenation. Also, it has many applications in engineering such as in transpiration cooling, porous pipe design and design of filters [40].
It is greatly defined that ionized fluid having low density and strong magnetic field, the Hall current become more notable. Hall effects on MHD flow have many applications as recognized by many researchers which are used in engineering which comprises of underground energy storage system, MHD power generation, Hall current accelerators, nuclear power reactors, magnetometers and spacecraft propulsion etc. Some of the important investigations are given in the references [4144]. Furthermore, both Hall current and rotation induce secondary flow in the fluid flow. Some important investigations have been done on the effects of thermal radiation on Hydromagnetic flow and heat transfer problems. It has many applications such as nuclear plants, gas turbines, aircraft, different propulsion devices, satellites, missiles, and space vehicles and at an industrial level as well as in engineering areas thermal radiation effect become important for high operating temperatures [45].
However, none of these papers took into consideration the rotating flow of Brinkmantype nanofluids. Therefore, the objective of this work is to fill this gap. More exactly, our analysis is focused on the study of the impact of Hall current and thermal radiation on the MHD flow of Brinkmantype engine oil based MoS_{2} nanofluid in a rotating disk. The problem is presented in the form of partial differential equations along with physical boundary conditions. The Laplace transform technique has been used to get the closedform solutions for velocity and temperature profiles. The results have been discussed for several embedded parameters. From the present study, it is noticed that the rate of heat transfer enhances 6.35% by adding MoS_{2 }in engine oil which improved its lubrication. Furthermore, the results from the present study for nanofluids may use for industrial and automobile cooling which could result in great energy savings and in emission reduction.
Figure 1.The illustrative diagram of the problem
Consider an incompressible MoS_{2}EObased Brinkmantype nanofluid with spherical shape nanoparticles in a rotating frame. Rotation of the fluid is about $y_{1}$ axis. The fluid is assumed to be electrically conducted, the Hall effect is also taken into account. Heat transfer along with thermal radiation is also considered. Initially, the fluid is at rest with ambient temperature $T_{1 \infty}$ . At the time $t=0^{+}$ the plate starts motion with a variable velocity of constant amplitude $U_{0}$ . The temperature of the plate is raised to $T_{1 w}$ and then remains constant. The geometry of the problem is shown in Figure1.
Under the above assumptions, the governing equations for MHD convective flow of Brinkmantype nanofluid in a rotating frame with Hall effect, taking thermal radiation into account and under Boussinesq’s approximation, are given by:
$\frac{\partial u_{1}}{\partial t_{1}}+\beta u_{1}2 \Omega w_{1}=\frac{\mu_{n f}}{\rho_{n f}} \frac{\partial^{2} u_{1}}{\partial y_{1}^{2}}\frac{\sigma_{n f} B_{0}^{2}\left(u_{1}+m v_{1}\right)}{\rho_{n f}\left(1+m^{2}\right)}$ $+g \beta_{n f}\left(T_{1}T_{1 \infty}\right)$ (1)
$\frac{\partial w_{1}}{\partial t_{1}}+\beta w_{1}+2 \Omega u_{1}=\frac{\mu_{n f}}{\rho_{n f}} \frac{\partial^{2} w_{1}}{\partial y_{1}^{2}}+\frac{\sigma_{n f} B_{0}^{2}\left(m u_{1}w_{1}\right)}{\rho_{n f}\left(1+m^{2}\right)}$ (2)
$\left(\rho c_{p}\right)_{n f} \frac{\partial T_{1}}{\partial t_{1}}=k_{n f} \frac{\partial^{2} T_{1}}{\partial y_{1}^{2}}\frac{\partial q_{1 r}}{\partial y_{1}}$ (3)
subject to the following initial and boundary conditions:
$\left. \begin{matrix} {{u}_{1}}(y,\,0)=0,\,\text{ }{{u}_{1}}(0,\,t)=\Delta \ t,\,\text{ }{{u}_{1}}(\infty ,\,t)=0 \\ {{w}_{1}}(y,\,0)=0,\,\text{ }{{w}_{1}}(0,\,t)=0,\,\text{ }{{w}_{1}}(\infty ,\,t)=0 \\ {{T}_{1}}(y,\,0)={{T}_{1\infty }},\,{{T}_{1}}(0,\,t)={{T}_{1\infty }}+({{T}_{1w}}{{T}_{1\infty }})At,\,{{T}_{1}}(\infty ,\,t)={{T}_{1\infty }} \\\end{matrix} \right\}$ (4)
The temperature on the wall is dependent on time, which means the variable temperature is considered on the boundary.
For the nanofluids, the expressions of $\rho_{n f}, \mu_{n f}, \sigma_{n f},(\rho \beta)_{n f}$ and $\left(\rho c_{p}\right)_{n f}$ are given as
$\rho_{n f}=(1\phi) \rho_{f}+\phi \rho_{s}, \mu_{n f}=\mu_{f}(1\phi)^{2.5}$,
$(\rho \beta)_{n f}=(1\phi)(\rho \beta)_{f}+\phi(\rho \beta)_{s}$,
$\left(\rho c_{p}\right)_{n f}=(1\phi)\left(\rho c_{p}\right)_{f}+\phi\left(\rho c_{p}\right)_{s}$,
$\sigma_{n f}=\sigma_{f}\left[1+\frac{3(\sigma1) \phi}{(\sigma+2)(\sigma1) \phi}\right]$,
$\sigma=\frac{\sigma_{s}}{\sigma_{f}}, k_{n f}=k_{f}\left[\frac{k_{s}+2 k_{f}2 \phi\left(k_{f}k_{s}\right)}{k_{s}+2 k_{f}+\phi\left(k_{f}k_{s}\right)}\right]$,
According to Oztop and AbuNada [46], the expressions which are presented above are restricted to spherical shaped nanoparticles and the thermophysical properties of base fluid and nanoparticle are given in Table 1.
Considering small temperature difference between fluid temperature and free stream temperature, can be linearized by using Taylor series about the free stream temperature. Neglecting second and higher order terms in
Table 1. Thermophysical properties of EO and MoS_{2 }[6]

$\rho$ (K gm^{3}) 
c_{p} (JKg^{ 1}K^{1}) 
K (Wm^{1 }K^{1}) 
$\beta$ ×10^{5 }(K^{1}) 
Engine oil 
863 
2048 
0.1404 
0.00007 
MoS_{2} 
5.06×10^{3} 
397.21 
904.4 
2.8424 
We adopt the Roseland’s approximation for radiative flux $q_{1 r}$ namely:
$q_{1 r}=\frac{4 \sigma^{*}}{3 k^{*}} \frac{\partial T_{1}^{4}}{\partial y_{1}}$, (5)
$\left(T_{1}T_{1 \infty}\right)$ , we get:
$T_{1}^{4}=4 T_{1 \infty}^{3} T_{1}3 T_{1 \infty}^{4}$ .
Using above in equation (5), we get
$q_{1 r}=\frac{4 \sigma^{*}}{3 k^{*}} \frac{\partial\left(4 T_{1 \infty}^{3} T_{1}3 T_{10}^{4}\right)}{\partial y_{1}}$
As $T_{1 \infty}^{4}$ is a constant temperature, therefore, $\frac{\partial T_{1 \infty}^{4}}{\partial y_{1}}=0$ , we have
$q_{1 r}=\frac{16 T_{1 \infty}^{3} \sigma^{*}}{3 k^{*}} \frac{\partial T_{1}}{\partial y_{1}}$ (6)
substituting equation (6) into equation (3), we get
$\left(\rho c_{p}\right)_{n f} \frac{\partial T_{1}}{\partial t_{1}}=k_{n f} \frac{\partial^{2} T_{1}}{\partial y_{1}^{2}}+\frac{16 T_{1 \infty}^{3} \sigma^{*}}{3 k^{*}} \frac{\partial^{2} T_{1}}{\partial y_{1}^{2}}$ (7)
introducing the following dimensionless variables
$u=\frac{u_{1}}{u_{0}}$ , $y=\frac{u_{0}}{v} y_{1}$ , $t=\frac{u_{0}^{2}}{v} t_{1}$ , $w=\frac{w_{1}}{u_{0}}$ , $T=\frac{T_{1}T_{1 \infty}}{T_{1 w}T_{1 \infty}}$ ,
into equations (1), (2) and (7) reduce to dimensionless equations (8), (9) and (10) respectively,
$\frac{\partial u}{\partial t}+\beta_{1} u2 \eta w=\frac{1}{\operatorname{Re}} \frac{\partial^{2} u}{\partial y^{2}}M_{0}\left(\frac{u+m w}{1+m^{2}}\right)+G r_{o} T$ (8)
$\frac{\partial w}{\partial t}+\beta_{1} w+2 \eta u=\frac{1}{\operatorname{Re}} \frac{\partial^{2} w}{\partial y^{2}}+M_{0}\left(\frac{m uw}{1+m^{2}}\right)$ (9)
$\frac{\partial T}{\partial t}=\frac{1}{\operatorname{Pr}_{e f f}} \frac{\partial^{2} T}{\partial y^{2}}$ (10)
subject to the dimensionless initial and boundary conditions:
$\begin{aligned} u(y, 0) &=0, u(0, t)=\xi t, u(\infty, t)=0 \\ w(y, 0) &=0, w(0, t)=0, w(\infty, t)=0 \\ T(y, 0) &=0, T(0, t)=t, \quad T(\infty, t)=0 \end{aligned} \}$ (11)
where
$\beta_{1}=\frac{\beta v}{u_{0}^{2}}, \eta=\frac{\Omega w}{u_{0}^{2}}, \operatorname{Re}=(1\phi)^{2.5}\left[(1\phi)+\phi\left(\frac{\rho_{s}}{\rho_{f}}\right)\right]$
$\phi_{2}=\frac{(1\phi) \rho_{f}+\phi \rho_{s}\left(\frac{\beta_{s}}{\beta_{f}}\right)}{(1\phi) \rho_{f}+\phi \rho_{s}}, \varphi_{3}=(1\phi)+\phi \frac{\left(\rho c_{p}\right)_{s}}{\left(\rho c_{p}\right)_{f}}$
$M=\frac{\sigma_{f} v B_{0}^{2}}{\rho_{f} u_{0}^{2}}, \quad M_{0}=\frac{M \phi_{1}(1\phi)^{2.5}}{\operatorname{Re}}, G r=\frac{g \beta_{f} \nu\left(T_{1}T_{1 \infty}\right)}{u_{0}^{3}}$
$G r_{0}=G r \phi_{2}, \phi_{1}=1+\frac{3(\sigma1) \phi}{(\sigma+2)(\sigma1) \phi}, \operatorname{Pr}=\frac{\left(\mu c_{p}\right)_{f}}{k_{f}}$
$N r=\frac{16 \sigma^{*} T_{1 \infty}^{3}}{3 k^{*} k_{f}}, \quad \lambda_{n f}=\frac{k_{n f}}{k_{f}}, \quad \operatorname{Pr}_{e f f}=\frac{\operatorname{Pr} \phi_{3}}{\left(\lambda_{n f}+N r\right)}$ (12)
where $\varphi_{i}(i=1,2,3)$ represent the functions which depend on the thermophysical properties of the base fluid and nanoparticles. According to the geometry, equations (8) to (10) are the governing equations of the considered flow regime subjected to the initial and boundary conditions (11). We now take a particular case of interest, namely, uniformly accelerated the movement of the plate i.e. $\Delta(t)=\xi t$ where $\xi$ is dimensionless constant, to understand the flow features of the fluid flow. Equations (8 and 9) can be expressed in more suitable form as
$\frac{\partial F}{\partial t}=\frac{1}{\operatorname{Re}} \frac{\partial^{2} F}{\partial y^{2}}\lambda F+G r_{0} T$ (13)
where $F=u+w i$ is a complex velocity and $\lambda=M_{0} \frac{(1m i)}{1+m^{2}}+\beta_{1}+2 k^{2} i$ is a constant.
The dimensionless initial and boundary conditions are:
$\left. \begin{align} & F\left( y,0 \right)=0,\,\,\,\,T\left( y,0 \right)=0 \\ & F\left( 0,t \right)=\xi t,\,\,\,T\left( 0,t \right)=t \\ & F\left( \infty ,t \right)=0,\,\,\,\,T\left( \infty ,t \right)=0 \\\end{align} \right\}$ (14)
The Laplace transform technique has been used to obtain the closedform solutions for the partial differential equations (10 and 13).
3.1 Calculation of temperature distribution
Apply the Laplace transform technique to equation (10) using the corresponding initial and boundary conditions from equation (14), we get
$\overline{T}(y, s)=\frac{1}{s^{2}} e^{y \sqrt{\operatorname{Pr}_{\mathrm{dff}} s}}$ (15)
Now by taking Laplace inverse of equation (15), we get
$T(y, t)=F_{4}\left(y \sqrt{\operatorname{Pr}_{e f f}}, t\right)$ (16)
3.2 Calculation of velocity distribution
Apply the Laplace transform technique to equation (13) and using corresponding initial conditions from equation (14), we get
$\frac{d^{2} \overline{F}}{d y^{2}}\operatorname{Re}(\lambda+s) \overline{F}(y, s)+\operatorname{Re} G r_{0} \overline{T}(y, s)=0$ (17)
Now substituting equation (15) into equation (17) and using the corresponding boundary conditions from equation (14), the solution of equation (17) becomes
$\overline{F}(y, s)=\left[\frac{\xi}{s^{2}}\frac{G r_{2}}{s^{2}}+\frac{G r_{2}}{b_{1}\left(sb_{1}\right)}\frac{G r_{2}}{b_{1} s}\right] e^{y \sqrt{\operatorname{Re}} \sqrt{\lambda+s}}$ $\left[\frac{G r_{2}}{b_{1}\left(sb_{1}\right)}\frac{G r_{2}}{s^{2}}\frac{G r_{2}}{b_{1} s}\right] e^{y \sqrt{\operatorname{Pr}_{\mathrm{E} f} s}}$ (18)
where
$b_{1}=\frac{\operatorname{Re} \lambda}{\operatorname{Pr}_{e f f}\operatorname{Re}}$ , $G r_{1}=\frac{\operatorname{Re} G r_{0}}{\operatorname{Pr}_{e f f}\operatorname{Re}}$ and $G r_{2}=\frac{G r_{1}}{b_{1}}$.
Now finally applying the inverse Laplace transform to equation (18), we get
$F(y, t)=\xi F_{1}(y \sqrt{\operatorname{Re}}, t, \lambda)G r_{2} F_{1}(y \sqrt{\operatorname{Re}}, t, \lambda)$ $+\frac{G r_{2}}{b_{1}} F_{2}\left(y \sqrt{\operatorname{Re}}, t, b_{1}, \lambda\right)\frac{G r_{2}}{b_{1}} F_{3}(y \sqrt{\operatorname{Re}}, t, \lambda)$ $\frac{G r_{2}}{b_{1}} F_{6}\left(y \sqrt{\operatorname{Pr}_{e f f}}, t, b_{1}\right)+G r_{2} F_{4}\left(y \sqrt{\operatorname{Pr}_{e f f}}, t\right)$ $+\frac{G r_{2}}{b_{1}} F_{5}\left(y \sqrt{\operatorname{Pr}_{e f f}}, t\right)$ (19)
The expressions for $F_{i}(i=1,2, \ldots 6)$ are given in Appendices.
3.3 Special case
By taking $\beta_{1} \rightarrow 0$ , $\lambda=M_{0} \frac{(1m i)}{1+m^{2}}+2 k^{2} i$ , equation (18) is reduced to the following form
$F(y, t)=\xi F_{1}(y \sqrt{\operatorname{Re}}, t, \lambda)G r_{2} F_{1}(y \sqrt{\operatorname{Re}}, t, \lambda)$ $+\frac{G r_{2}}{b_{1}} F_{2}\left(y \sqrt{\operatorname{Re}}, t, b_{1}, \lambda\right)\frac{G r_{2}}{b_{1}} F_{3}(y \sqrt{\operatorname{Re}}, t, \lambda)$ $\frac{G r_{2}}{b_{1}} F_{6}\left(y \sqrt{\operatorname{Pr}_{e f f}}, t, b_{1}\right)+G r_{2} F_{4}\left(y \sqrt{\operatorname{Pr}_{e f f}}, t\right)$ $+\frac{G r_{2}}{b_{1}} F_{5}\left(y \sqrt{\operatorname{Pr}_{e f f}}, t\right)$ (20)
which is the solution for Newtonian nanofluid in a rotating frame.
3.4 Skin friction and Nusselt number
The skinfriction and Nusselt number are defined as:
The expression for dimensionless skin friction is given as
$C_{f}=\frac{1}{(1\phi)^{2.5}}\left.\frac{\partial F(y, t)}{\partial y}\right_{y=0}=\frac{1}{(1\phi)^{25}} L^{1}\left\{\lim _{y \rightarrow 0} \frac{\partial F(y, s)}{\partial y}\right\}$ (21)
The dimensionless Nusselt number is given by
$\mathrm{Nu}=\frac{k_{n f}}{k_{f}}\left.\frac{\partial T(y, t)}{\partial y}\right_{y=0}$ $=\left[\frac{k_{s}+2 k_{f}2 \phi\left(k_{f}k_{s}\right)}{k_{s}+2 k_{f}+\phi\left(k_{f}k_{s}\right)}\right] L^{1}\left\{\lim _{y \rightarrow 0} \frac{\partial T(y, s)}{\partial y}\right\}$. (22)
Figure 2. Velocity profile of MoS_{2}EObased Brinkman type nanofluid for different values of $\beta_{1}$ when $\phi=0.03, G r=0.4$ , $\mathrm{Pr}=1000, m=0.5, \eta=0.2, M=1, t=1, N r=0.2$
In this section, the analytical results are plotted graphically for embedded parameters using computational software MATHCAD. Different graphs are shown for this purpose. These graphs show the influence of Brinkman parameter $\beta_{1}$ , magnetic parameter $M$ , volume fraction $\phi$ , Hall effect $m$ , Grashof number $G r$ , thermal radiation $N r$ and rotation parameter $\eta$ on primary and secondary velocities. Additionally, the effects of thermal radiation $N r$ and volume fraction $\phi$ are shown in temperature distribution.
Figure2 has been plotted to show the influence of Brinkman parameter $\beta_{1}$ on fluid velocities (Primary and Secondary), it is observed that $\beta_{1}=0.2$ have greater fluid velocity than $\beta_{1}=0.5$ . Physically, it is true because Brinkman parameter $\beta_{1}$ is a ratio between drag forces and density, so by increasing $\beta_{1}$ drag forces increased which cause a decrease in velocity.
Figure 3. Velocity profile of MoS_{2}EObased Brinkman type nanofluid for different values of $G r$ when$\phi=0.03, \beta_{1}=0.5$ , $\mathrm{Pr}=1000, m=0.5, \eta=0.2, M=1, t=1, N r=0.2$
Figure 4. Velocity profile of MoS_{2}EObased Brinkman type nanofluid for different values of $m$ when $\phi=0.03, \beta=0.5$ , $\mathrm{Pr}=1000, G r=0.4, \eta=0.2, M=1, t=1, N r=0.2$
The effect of thermal Grashof number $G r$ on secondary and primary velocities has been portrayed in figure 3. It is noticed that both velocities raised by increasing the values of $G r$ . This behavior is obvious because it is the ratio of buoyancy forces and viscous forces so by increasing the values of $G r$ from 0.1 to 0.4, it increases the buoyancy forces and consequently velocities increase.
Figure 4 depicts the effect of Hall current parameter $m$ on primary and secondary velocities. The figure illustrated that for $m=0$ the velocity is greater than $m=0.6$ , which shows that Hall current decelerates the primary fluid velocity. Whereas, it has an opposite effect on the secondary fluid velocity. Secondary fluid velocity enhanced for higher values of $m$ .
Figure 5. Velocity profile of MoS_{2}EObased Brinkman type nanofluid for different values of $\eta$ when$\phi=0.03, \beta_{1}=0.5$ , $\mathrm{Pr}=1000, G r=0.4, m=0.5, M=1, t=1, N r=0.2$
Figure6 is plotted to present the effect of $M$ on primary and secondary velocity profiles. The results show that by applying high magnetic field strength, both the primary and secondary velocities become decreased, this is an obvious result because magnetic field raises the Lorentz forces of an electrically conducted fluid which tends to slow down the motion of the fluid in the boundary layer.
The outcomes for rotation parameter $\eta$ can be seen in figure 5, it is observed that the primary and secondary velocities decrease for higher values of $\eta$ . Although it is the fact that Coriolis forces have a tendency to decrease the primary flow and increases the secondary flow, the behavior of secondary flow is not obvious so in the present case, the secondary flow is decreased by increasing the value of $\eta$ .
The influence of volume fraction $\phi$ on the velocity of fluid flow has been illustrated in figure 7. The range of nanoparticles volume fraction is taken between $0 \leq \phi \leq 0.04$ because if the volume fraction exceeds from 0.04 then the problem of sedimentation arises. From figure 7, it is observed that when $\phi$ is increased from 0.01 to 0.04, both the primary and secondary velocities are decreased. Large values of $\phi$ enhancing the viscous forces due to which fluid velocity is decreased. A decrease in the velocity means that the viscosity of EO is increased which consequently increase the boiling and freezing point of the EO due to which the lubrication properties of EO become more effective and efficient.
Figure 6. Velocity profile of MoS_{2}EObased Brinkman type nanofluid for different values of $M$ when $\phi=0.03, \beta_{1}=0.5$ , $\operatorname{Pr}=1000, G r=0.4, \eta=0.2, m=0.5, t=1, N r=0.2$
Figure 7. Velocity profile of MoS_{2}EObased Brinkman type nanofluid for different values of $\phi$ when $\beta_{1}=0.5,$ Pr $=1000$ , $m=0.5, \quad G r=0.4, \eta=0.2, M=1, t=1, N r=0.2$
Figure. 8 highlights the influence of thermal radiation $N r$ on the fluid velocity, which shows that by increasing the values of $N r$ both primary and secondary velocities are enhanced.
Figure 8. Velocity profile of MoS_{2}EObased Brinkman type nanofluid for different values of $N r$ when $\phi=0.03, \beta_{1}=0.5$ , $\operatorname{Pr}=1000, m=0.5, G r=0.4, \eta=0.2, M=1, t=1$
Figure 9. Temperature profile of MoS_{2}EObased Brinkman type nanofluid for different values of $\phi$ when $\mathrm{Pr}=1000, t=1, N r=0.2$
Figure 10. Temperature profile of MoS_{2}EObased Brinkman type nanofluid for different values of $N r$ when $\mathrm{Pr}=1000, t=1, \phi=0.03$
In figure 9 the effect of nanoparticle volume fraction is noticed on the temperature profile, that is, an increase in nanoparticle volume fraction causes an increase in temperature due to shear thinning behaviour.
The influence of thermal radiation parameter $N r$ on temperature has been illustrated in figure 10, which shows the rise in velocity as we increase the value of $N r$ .
Table 2. shows the effect of various parameters on skin friction, by taking higher values of $M, \eta, t, G r$ and $N r$ , skin friction increases.
Table 2. Effect of various parameters on skin friction coefficient ( $C_{f}$ )
$\phi$ 
$M$ 
$\mathrm{Pr}$ 
$m$ 
$\eta$ 
$t$ 
$G r$ 
$N r$ 
$\beta_{1}$ 
$C_{f}$ 
0.03 
1 
1000 
0.5 
0.2 
1 
5 
0.2 
0.5 
10.747 
0.04 
1 
1000 
0.5 
0.2 
1 
5 
0.2 
0.5 
10.572 
0.03 
1.5 
1000 
0.5 
0.2 
1 
5 
0.2 
0.5 
24.204 
0.03 
1 
1500 
0.5 
0.2 
1 
5 
0.2 
0.5 
10.724 
0.03 
1 
1000 
0.7 
0.2 
1 
5 
0.2 
0.5 
5.768 
0.03 
1 
1000 
0.5 
0.4 
1 
5 
0.2 
0.5 
93.253 
0.03 
1 
1000 
0.5 
0.2 
1.5 
5 
0.2 
0.5 
20.052 
0.03 
1 
1000 
0.5 
0.2 
1 
6 
0.2 
0.5 
10.712 
0.03 
1 
1000 
0.5 
0.2 
1 
5 
0.4 
0.5 
10.757 
0.03 
1 
1000 
0.5 
0.2 
1 
5 
0.2 
0.7 
10.592 
Table 3 shows that Nusselt number is increased by increasing the values of $\phi$ . Whereas $\operatorname{Pr}$ increases the Nusselt number while decreases the skin friction.
Table 3. Effect of various parameters on Nusselt number ( $N_{\mathcal{U}}$ )
$\phi$ 
$N r$ 
$\mathrm{Pr}$ 
$t$ 
$N u$ 
0.03 
0.2 
1000 
1 
34.151 
0.03 
0.4 
1000 
1 
31.618 
0.03 
0.2 
1500 
1 
38.662 
0.03 
0.2 
1000 
1.5 
41.826 
Table 4. Impact of volume fraction on nusselt number and percent enhancment
$\phi$ 
$N r$ 
$\mathrm{Pr}$ 
$t$ 
$N u$ 
% Enhancement 
0 
0.2 
1000 
1 
32.603 
 
0.01 
0.2 
1000 
1 
33.115 
1.57 
0.02 
0.2 
1000 
1 
33.631 
3.15 
0.03 
0.2 
1000 
1 
34.151 
4.74 
0.04 
0.2 
1000 
1 
34.674 
6.35 
Table 4 shows the impact of volume fraction on the Nusselt number. From Table 4 it observed that at zero volume fraction the Nusselt number is 32.603 and by increasing the volume fraction up to 0.04 the rate of heat transfer enhances up to 6.35 % .from here it can be concluded that the engine oilbased MoS_{2} nanofluid is thermally more stable, has more effectitive lubrication properties and has efficient heat transfer rate as compare to the regular engine oil.
In recent years, many lubricants with different properties are produced to be used in various mechanical systems including internal combustion engines. Friction in mechanical systems is the main factor for energy loss and lubrication is one of the most effective ways of reducing friction. Engine oil (EO) is one of the most used lubricants in internal combustion engines, which power cars, motorcycles, enginegenerators, and many other machines. Hence, improving lubricant properties of EO is always been in demand. From last few decades, nanofluids are categorized as a new low friction technology, and a method to reduce wear properties. Therefore, in the present work, Spherical shaped MoS_{2} nanoparticles are used in EO, which is taken as a base fluid. More exactly, this work investigates for the first time the MHD flow of Brinkmantype nanofluid in a rotating frame with Halleffect. Molybdenum disulfide nanoparticles of spherical shape are dispersed in EO to enhance its thermal conductivity. Exact solutions are developed for velocity and temperature via the Laplace transform technique. The important points from this study are summarized in the following:
· By increasing the values of $\phi, \beta_{1}, \eta$ and $M$ decreased the nanofluid primary and secondary velocities.
· With increasing values of $m$ , the primary velocity is decreased whereas the secondary velocity is increased.
· Temperature is increased with increasing values of $\phi$ and $N r$ .
· Skin friction is found as the increasing function of $M, \eta, t, G r$ and $N r$ .
· The Nusselt number/heat transfer rate increased with increasing values of $\phi$ .
· $\mathrm{Pr}$ increases the Nusselt number while decreased the skin friction.
· Heat transfer rate of EObased nanofluid is 6.35% greater as compared to regular EO fluid.
A 
constant with dimension $\frac{1}{s}$ and shows the amplitude 
$B_{0}$ 
induced magnetic field; 
$\mathcal{g}$ 
acceleration due to gravity (m s^{2}); 
$u^{\prime}$ 
nanofluid velocity in $x^{\prime}$direction (m/s); 
$w^{\prime}$ 
nanofluid velocity in $z^{\prime}$direction (m/s); 
m 
Hall current parameter 
$G r$ 
Grashof number 
$k_{f}$ 
the thermal conductivity of the base fluid (Wm^{1}K^{1}); 
$k_{n f}$ 
the thermal conductivity of nanofluid (Wm^{1}K^{1}); 
$k_{s}$ 
the thermal conductivity of nanoparticles (Wm^{1}K^{1}); 
$k^{*}$ 
mean absorption coefficient; 
$\mu_{n f}$ 
dynamic viscosity of nanofluid (kg m^{1} s^{1}); 
$q_{1 r}$ 
radiative heat flux (=W m^{2}); 
$T_{1}$ 
fluid temperature (K); 
$\beta_{f}$ 
the thermal expansion coefficient of base fluid (K^{1}); 
$\beta_{n f}$ 
the thermal expansion coefficient of nanoparticle (K^{1}); 
$\beta_{s}$ 
the thermal expansion coefficient of nanofluid (K^{1}); 
$\Omega$ 
rotation parameter; 
$\mu_{f}$ 
viscosity of base fluid (kg m^{1} s^{1}); 
$\phi$ 
the solid volume fraction of the nanoparticle; 
$\rho_{f}$ 
the density of the base fluid (kg m^{3}); 
$\rho_{n f}$ 
the density of nanofluid (kg m^{3}); 
$\left(\rho c_{p}\right)_{f}$ 
heat capacitance of base fluid; 
$\left(\rho c_{p}\right)_{n f}$ 
heat capacitance of nanofluid; 
$\left(\rho c_{p}\right)_{s}$ 
heat capacitance of nanoparticle; 
$\sigma^{*}$ 
StefanBoltzmann constant $\left(=5.67 \times 10^{8} \mathrm{W} / \mathrm{m}^{2} \mathrm{K}^{4}\right)$; 
$\sigma_{f}$ 
electrical conductivity of base fluid (=s^{3} A^{2} m^{3}kg^{1}); 
$\sigma_{n f}$ 
electrical conductivity of nanofluid (=s^{3} A^{2} m^{3}kg^{1}); 
$\sigma_{s}$ 
electrical conductivity of nanoparticle (=s^{3} A^{2} m^{3}kg^{1}). 
$v_{f}$ 
dynamic viscosity coefficient of base fluid; 
$\beta_{1}$ 
Brinkman parameter; 
$\eta$ 
rotation parameter; 
$M$ 
magnetic parameter; 
$G r$ 
Grashof number; 
Pr 
Prandtl number; 
Nr 
radiation parameter; 
$\eta$ 
Dimensionless rotation parameter; 
$\overline{F_{1}}(y, s, H)=L^{1}\left[\frac{1}{s^{2}} e^{y \sqrt{H+s}}\right]=F_{1}(y, t, H)=\left[\begin{array}{c}{e^{y \sqrt{H}} \operatorname{erfc}\left(\frac{y}{2 \sqrt{t}}\sqrt{H t}\right)\left(\frac{t}{2}\frac{y}{4 \sqrt{H}}\right)} \\ {+e^{y \sqrt{H}} \operatorname{erfc}\left(\frac{y}{2 \sqrt{t}}+\sqrt{H t}\right)\left(\frac{t}{2}+\frac{y}{4 \sqrt{H}}\right)}\end{array}\right]$
$\overline{F_{2}}(y, s, b, M)=L^{1}\left[\frac{1}{sb} e^{y \sqrt{M+s}}\right]=F_{2}(y, t, b, M)=\frac{e^{b t}}{2}\left[\begin{array}{c}{e^{y \sqrt{M+b}} \operatorname{erfc}\left(\frac{y}{2 \sqrt{t}}\sqrt{(b+M) t}\right)} \\ {+e^{y \sqrt{M+b}} \operatorname{erfc}\left(\frac{y}{2 \sqrt{t}}+\sqrt{(b+M) t}\right)}\end{array}\right]$
$\overline{F_{3}}(y, s, M)=L^{1}\left[\frac{1}{s} e^{y \sqrt{M+s}}\right]=F_{3}(y, t, M)=\frac{1}{2}\left[\begin{array}{c}{e^{y \sqrt{M}} \operatorname{erfc}\left(\frac{y}{2 \sqrt{t}}\sqrt{M t}\right)} \\ {+e^{y \sqrt{M+b}} \operatorname{erfc}\left(\frac{y}{2 \sqrt{t}}+\sqrt{M t}\right)}\end{array}\right]$
$\overline{F_{4}}(y, s)=L^{1}\left[\frac{1}{s^{2}} e^{y \sqrt{s}}\right]=F_{4}(y, t)=\left[\left(t+\frac{y^{2}}{2}\right) \operatorname{erfc}\left(\frac{y}{2 \sqrt{t}}\right)\frac{y \sqrt{t}}{\sqrt{\pi}} e^{\frac{y^{2}}{4 t}}\right]$
$\overline{F_{5}}(y, s)=L^{1}\left[\frac{1}{s} e^{y \sqrt{s}}\right]=F_{5}(y, t)=\operatorname{erfc}\left(\frac{y}{2 \sqrt{t}}\right)$
$\overline{F_{6}}(y, s, b)=L^{1}\left[\frac{1}{sb} e^{y \sqrt{s}}\right]=F_{6}(y, t, b)=\frac{e^{b t}}{2}\left[\begin{array}{c}{e^{y \sqrt{b}} \operatorname{erfc}\left(\frac{y}{2 \sqrt{t}}\sqrt{b t}\right)} \\ {+e^{y \sqrt{b}} \operatorname{erfc}\left(\frac{y}{2 \sqrt{t}}+\sqrt{b t}\right)}\end{array}\right]$
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