Heat Transfer Characteristics on MHD Oscillatory Radiative Nanofluid with H2O/C2H6O2 (Basefluid): A Comparative Study of Different Nanoparticles of Various Shapes

From the previous few decades, nanofluids have been a major topic of importance among researchers. Nanofluid is an innovative way to improve the characteristics of heat transfer in liquids. It is particularly attractive because of its various industrial, biomedical, electronics, and transportation applications including hybrid engines, advanced nuclear systems, automobiles, biological sensors, drug delivery systems, etc. The average size of nanoparticles used in nanofluids may vary from 1 to 100 nm. Nanoparticles remain suspended almost indefinitely in fluids, reducing erosion and clogging dramatically compared to larger particles in suspension. In MHD (magnetohydrodynamic) nanofluid flow, the presence of magnetic fields can affect the heat transfer properties of the nanofluid. The interaction between the magnetic field and the nanoparticles can cause the particles to form chains or clusters, which can alter the flow behaviour and heat transfer characteristics of the nanofluid. Modeling, preparing, convective heat transfers, and characterization of nanofluids have been studied extensively. Aluminum is the most significant substance in technology. There are wide varieties of crystallographic variations with different properties of these substances. Corundum is the most frequently occurring form which is thermodynamically stable at room temperature. Many metastable polymorphs contain Al₂O_3.because Al₂O₃ has high thermal conductivity. Al₂O₃ nanoparticles are used to improve the heat transfer effect of thermal conductivity. Adding up to 1% Al₂O₃ increases the thermal conductivity of EG to 75%. The efficiency of a nanoparticle is not only influenced by the types of nanoparticles but also the shapes. Al₂O₃ typically spherical shaped nanoparticles, are used by researchers. However spherical-shaped nanoparticles have a restricted range of uses and relevance when compared. This is the reason when non-spherical nanoparticles like brick and platelet have been used in this investigation. The cylinder, platelet, and brick forms of nanoparticles are specifically included in this study. Moreover, square-shaped nanoparticles have a number of essential desired features that make them the major focus of present research, particularly in cancer therapy, according to the literature survey on nanofluids. Choi and Eastman [1] in this valuable research analysed, when nanoparticles are added to regular base fluids, their thermal conductivity and convection heat transfer rate increase significantly; these compositions are referred to as nanofluids. Nanofluids are base fluids suspension of nanoscale particles. Typically, various types of nanoparticles, which include metal, carbides, and oxides, and commonly used base fluids are EG, kerosene and propylene glycol. Aizar et al. [2] investigated the effect of thermal radiative and rate of heat transfer in mixed convection MHD flow of various-shaped nanoparticles of alumina in EG and water-based nanofluid. It was observed that cylindrical and platelet-shaped nanoparticles have lesser velocity than the blade and brick-shaped nanoparticles. This is related to the shear thinning characteristic of cylindrical-shaped nanoparticles in fluid, which causes temperature dependence in viscosity. Timofeeva et al. [3] looked into the effect of Al₂O₃ nanofluids including various shaped nanoparticles, this investigation contains both experimental and theoretical views. More specifically, the authors looked at various alumina-shaped nanoparticles in an EG and water-based fluid mixture of equal quantities. Hamilton and Crosser model has been used and observed that the thermal conductivity is enhanced due to particle shape spherical nanoparticles were taken into account by Loganathan et al. [4] as the effects of thermal radiation on erratic convection flow of microfluids past an infinite vertical plate were investigated. It is observed that spherical-shaped Ag nanofluids are less kinetically active compared with Cu, TiO₂, and alumina nanofluids. Hatami and Ganji [5] and Hatami et al. [6] investigated the natural convection of C₆H₉NaO₇ non-Newtonian nanofluid flow between vertical plates. The GM and LSM have been developed and used in the problem of nanofluid flow and rate of heat transfer between two parallel plates caused by the normal motion of the porous upper plate respectively. Prasad et al. [7] theoretically analysed the heat transfer and MHD flow in a nanofluid over a slender elastic sheet with variable thickness. Ganesh et al. [8] numerically analyses the magnetic field affects the flow of water and ethylene glycol-based Al₂O₃ nanofluids across a flat surface in a Marangoni boundary layer. Sheikholeslami et al. [9] analysed the magnetohydrodynamic nanofluid flow and heat transfer of two horizontal parallel plates. Ilyas Khan and it has been shown by Khan and Khan [10] that viscous dissipation plays a critical role in the MHD Darcy flow in the presence of Cu-Np and that incompressible water-Cu and EG-Cu Newtonian nanofluids transport heat over a flat plate in the presence of Cu-Np. Water-copper nanofluids and EG-copper nanofluids exhibit the same increasing effect of nanoparticle volume fraction. Some relevant studies on the topic can be seen from the studies [11-18].

Since porous media is more permissible than non-porous media, the research on fluid flow and heat transfer in porous media is studied in the current literature. Heat transfer is directly proportional to the Hartmann number but inversely proportional to the Rayleigh number. Makinde and Mhone [19] who carried out an investigation of the unsteady flow field of a thin fluid through a channel filled with a porous medium when there is a combined effect of a transverse magnetic field and a radiative heat transfer. The results demonstrate that increase in magnetic field decreases the quality of wall shear stress while increasing the radiation parameter through heat absorption increases the amount of wall shear stress. Abbas et al. [20] analysed the oscillatory slip flow of electrically conducting viscous nanofluids through saturated porous media which are exposed to thermal radiation. In this work, it is observed that enhancement in SWCNTs and MWCNTs volume fractions increases the heat transfer rate at the boundaries and declines the temperature at the center of the channel. Waqas et al. [21] did a comparative study and analysed the numerical outcomes of hybrid nanofluid flow with heat radiation effect with cu, aluminum oxide, graphene oxide and ethylene nanoparticles over a stretching sheet. According to the study, the velocity profile increases with an increase in Deborah’s number, whereas it decreases with an increase in magnetic parameter estimations. Das et al. [22] analysed the problem of free convection fluid flow through a vertical porous plate with heat and mass transfer through a porous media, the time-dependent permeability of a viscous flow incompressible electrically conducting fluid, whereas after a few years, Mishra et al. [23] modified the type of fluid to viscoelastic and analysed and it is noted that boundary layer flow is reduced by the heavier species with low conductivity. One can find more remarkable examples of the rate of heat transfer flow in studies [24-28] and the references therein.

Furthermore, reviews of the literature show that the majority of the previous studies in fluid flow are based on constant physical values, an although in a few cases, researchers investigate varying physical properties. Due to the significant influence of magnetic fields on channel flow, researchers have focused on MHD flow with heat and mass transfer in permeable media. This has numerous applications, which include MHD power generators, nuclear reactor cooling, plasma studies and geothermal energy extraction. Additionally, the study of oscillatory flow is crucial for solving many engineering and physiological issues, like blood flow through arteries, water pumps and fuel pumps, in view of these applications. Narayana et al. [29], the knowledge from the current study will be helpful for many industrial applications (which are like cooling of electronic devices) and scientific, where it is essential to avoid disturbance in the assembly chips by oscillations. It is noted that the oscillatory flow enhances the heat transfer rate. The heat is transported axially because the fluid oscillates at high frequency with large total displacement, so larger quantities of heat are transported. Srinivasacharya and Mendu [30] have taken into account how the first-order chemical reaction, radiation and magnetic field effects affect the free convection heat and mass transfer over a vertical plate with irregular wall conditions for temperature and concentration and it is found that increasing radiation parameter leads to an increase in Nusselt number, Sherwood number, skin friction coefficient and a fall in wall couple stress. A homogeneous chemical reaction of magneto micropolar fluids, which pass through vertical plates with slip flow, is examined by Chaudhary and Jha [31]. The author concludes that translational velocity increases with increasing values of the permeability parameter, but the microrotation profile tends to decrease with an increase in various values of permeability. Hamilton and Crosser [32] both theoretically and experimentally analysed the thermal conductivities of various types of two components mixture in which heterogeneous were correlated. The author concluded by giving an equation that determines the empirical shape factor based on the thermal conductivities of the phases and the shape of the particles can be used to calculate thermal conductivities for heterogeneous two-component systems. Jafar et al. [33] consider the combined effect of various parameters on the MHD flow of nanofluid through a nonlinear stretching sheet in porous media. The study reveals that the raise in the volume fraction parameters enhance the velocity and temperature. Shoaib et al. [34] analysed a significant amount of research that has been conducted on the production, classification, and uses of various types of Nps. However, hybrid nanofluids are of more recent type that is produced using two or more types of nanoparticles, either in mixed or compound form. The main goal of this technique is to make the ideal simultaneous combination of several materials’ chemical and physical characteristics in a special building. Authors have also paid close attention related to this investigation [35-37].

Sedki [38] examines the influence of internal heat with permeable stretching/shrinking wedges variable $k_{n f}$ and $\mu_{\mathrm{nf}}$ on unsteady MHD induces in convective boundary wall flow of various incomprehensible electron conducting fluid in the presence of internal heat. The author observed that as the surface nonlinearity, Brownian motion, chemical reaction and permeability raises, the heat transfer rate increases but the reverse occurs with the volume fraction.

Kataria and Patel [39] analysed the radiation, chemical reaction, Soret and heat generation effects on MHD Casson nanofluid in oscillating vertical channel through porous media. It is noticed that with an increase in Casson parameter, the velocity decreases, this is due to the viscous force in the fluid which drags the fluids whereas Patel [40] investigated a similar kind of study without chemical reaction parameters and concluded that the concentration of the fluid is enhanced when the chemical reaction parameter is raised. Sasikumar et al. [41] analysed the rate of heat transfer, mass transfer and chemical reaction effect with slip in the asymmetric channel. It is noted that the heat transfer rate rises in both the channel walls with varying values of time. Sasikumar et al. [42] studied the impact of heat and mass transfer rate on MHD oscillatory flow in an asymmetric tapered channel. The study investigates various parameters that influence the flow and heat transfer characteristics. It is observed that as the Peclet number rises, the temperature tends to decrease, while the temperature tends to increase with higher radiation parameters.

The effects of chemical reactions and viscous dissipation on MHD free convection flow are studied analytically by Pal and Biswas [43] in the presence of heat sources and sinks, thermal radiation, and variable surface temperatures and concentrations and it is observed that enhancement in magnetic field parameter decreases the skin friction coefficient. Gupta et al. [44] analyses a model of the 3-D flow of Cu-water and alumina water Nf over an exponentially stretching sheet. Cu-water and Al₂O₃-water simultaneous effects for convective conditions of magnetic field radiation over 3-D flow. It is found that the alumina water nanofluid has a thicker momentum boundary layer when compared with copper-water nanofluid. Zeeshan et al. [45] investigate the non-Newtonian couple stress fluid over the upper horizontal surface of the parabola of the reacting catalytic surface of the double-diffusive stretching fluid and the velocity and temperature profile are generated graphically for various parameters. Modather et al. [46] investigated the impact of a chemical reaction on the heat and mass transfer of micropolar fluids over an infinitely moving permeable plate in a saturated porous medium. It has been analysed that the chemical reaction parameter or permeability parameter when increased the velocity also increases whereas the temperature falls. References [47-51] provide a list of some recent researchers who have studied the MHD oscillatory flow in various fields.

It is evident from the above literature survey that how the addition of nanoparticles to the base fluid can significantly increase its thermal conductivity and convective heat transfer coefficient, resulting in more efficient heat transfer. Overall, these studies demonstrate the importance of considering the size and shape of nanoparticles in MHD nanofluid flow and heat transfer analysis. The results suggest that the choice of nanoparticle size and shape can significantly affect the flow behaviour, heat transfer characteristics, and pressure drop of the nanofluid, which has important implications for the design and optimization of MHD nanofluid-based heat transfer systems.

From the above literature survey, we observe that there is no problem analysis on MHD nanofluid combined with oscillatory flow types, thermal radiation effect and specific geometries like asymmetric wavy and tapered wavy channels. It is evident that there is no report on heat transfer characteristic of oscillatory nanofluid flow of various shapes and different types of nanoparticles. This creates scope for new research and motivated the present work analysing the effect of various shapes and types of nanoparticles on heat transfer characteristics of oscillatory magnetic nanofluid flow through asymmetric wavy channel in the presence of thermal radiation.

As a result, the goal of the present study is planned to design and solve the research issues listed below:

• What is the influence of copper (Cu), gold (Au), silver (Ag), and alumina (Al₂O₃) nanoparticles on the velocity field, temperature field and heat transfer regime?
• How do the shear stress and heat transfer rate change with the rising thermal radiation parameter?
• What is the significance of various shapes of nanoparticles on the velocity field?

The remaining part of the paper is organized into various sections as follows: In Section 2, the mathematical model for the fluid flow problem is formulated in the form of governing equations with a set of boundary conditions. The non-dimensional quantities are defined by non-similarity transformation through which the governing equations and boundary conditions are obtained in non-dimensional form. Section 3, includes a solution procedure for solving the governing equations subject to boundary conditions. The exact solutions of the velocity profile, temperature profiles, skin friction coefficient and Nusselt number are obtained. In Section 4, the graphs are generated using MATLAB software for the velocity field, temperature field, skin friction coefficient and heat transfer rate. The discussion on the results obtained by analysing graphs and tables of various flow profiles is presented. Conclusions with limitations and scope for future research work are drafted in Section 5.

The partial differential Eq. (13), Eq. (14) converted to ordinary differential equation by taking the pressure gradient purely oscillatory, velocity and temperature for oscillatory flow as (see Refs. [23, 26, 41, 47]):

$\begin{gathered}\mathrm{u}(\mathrm{y}, \mathrm{t})=\mathrm{u}_0(\mathrm{y}) {e}^{i \omega t} \\ \theta(y, t)=\theta_0(y) e^{i \omega t} \\ -\frac{\partial p}{\partial x}=\lambda e^{i \omega t}\end{gathered}$            (16)

3.1 Calculation

On substituting (16) on (13) and (14), we obtain the below the ordinary differential equations,

$\frac{d^2 u_0}{d y^2}-m_1^2 u_0=\frac{-\lambda}{\phi_2}-\frac{a_3 \theta_0}{\phi_2}$              (17)

$\frac{d^2 \theta_0}{d y^2}+m_2^2 \theta_0=0$            (18)

where, $m_1^2=\frac{m_0^2-a_0 i \omega}{\phi_2}, m_2^2=b_3^2-b_0 i \omega$

The below equations are the boundary conditions:

$\begin{aligned} & u_0=0, \theta_0=1 \text { on } y=h_1 \\ & u_0=0, \theta_0=0 \text { on } y=h_2\end{aligned}$             (19)

The solution is obtained for u(y, t) (velocity) and θ(y, t) (temperature) by solving Eq. (17), Eq. (18) under the boundary conditions (19).

$u(y, t)=\left(1-e^{m 1\left(y-h_2\right)}\right) \frac{\lambda}{\phi_2 m_1} e^{i \omega t}+\frac{a_1}{\phi_2\left(m_2-m_1\right)} \frac{\sin m_2\left(y-h_2\right)}{\sin m_2\left(h_1-h_2\right)} e^{i \omega t}$

$-\left[\frac{\lambda}{\phi_2 m_1}+\frac{a_1}{\phi_2\left(m_2-m_1\right)}-\frac{\lambda}{\phi_2 m_1} e^{m_1\left(h_1-h_2\right)}\right]\left[\frac{\sinh m_1\left(y-h_2\right)}{\sinh m_1\left(h_1-h_2\right)}\right] e^{i \omega t}$                (20)

$\theta(y, t)=\frac{\sin m_2\left(y-h_2\right)}{\sin m_2\left(h_1-h_2\right)} e^{i \omega t}$                  (21)

The Shear Stress (τ) along the walls of the channel is given by:

$\tau=\mu\left\{\frac{\partial u}{\partial y}\right\}_{y=h_1, h_2}=\mu\left\{\left(-e^{m 1\left(y-h_2\right)}\right) \frac{\lambda}{\phi_2} e^{i \omega t}\right\}+\mu\left\{\frac{a_1}{\phi_2\left(m_2-m_1\right)}\right\} \frac{m_2 \cos m_2\left(y-h_2\right)}{\sin m_2\left(h_1-h_2\right)} e^{i \omega t}$

$-\mu\left\{\left[\frac{\lambda}{\phi_2 m_1}+\frac{a_1}{\phi_2\left(m_2-m_1\right)}-\frac{\lambda}{\phi_2 m_1} e^{m_1\left(h_1-h_2\right)}\right]\left[\frac{m_1 \cosh m_1\left(y-h_2\right)}{\sinh m_1\left(h_1-h_2\right)}\right]\right\} e^{i \omega t}$                   (22)

The rate of heat transfer along the walls of the channel is given by

$N u=-\left(\frac{\partial \theta}{\partial y}\right)_{y=h_1, h_2}=\frac{-m_2 \cos m_2\left(y-h_2\right)}{\sin m_2\left(h_1-h_2\right)} e^{i \omega t}$             (23)

We have investigated the impact of heat transfer on MHD oscillatory nanofluid flow with thermal radiation for three various shapes and four different nanoparticles in this research. The nonlinear governing equations are solved by analytical method for velocity and temperature profiles.

The following are the major conclusions:

• The cylindrical shaped and platelet shaped nanoparticle have lesser velocity when compared to brick shaped nanoparticle Al₂O₃-water nanofluids.
• It is noted that Au has higher velocity followed by Ag, Cu and Al₂O₃ of platelet shaped water based nanofluid.
• Velocity of Al₂O₃ nanofluid of cylindrical shape are compared for water and EG base fluid. It is observed that the Al₂O₃-water nanofluid has higher velocity when compared with Al₂O₃ -EG nanofluid.
• The fluid velocity decreases with increase in Hartmann number or volume fraction or oscillatory parameter whereas velocity increases with the increase in permeability parameter or Grashof number or thermal radiation.
• The fluid temperature increases with increase in thermal radiation and decreases with the increase in Peclet number or volume fraction.
• The Al₂O₃ water nanofluid has higher temperature profile followed by Cu, Ag and Au water based nanofluid of platelet shape.
• The heat transfer rate falls at the channel wall y=h₁ and increases at the other wall y = h₂ with the increase of thermal radiation.

In the current research, the effect of variable viscosity, entropy generation, chemical reaction etc. on the problem of MHD nanofluid flow is not taken into consideration. The future research for this study can be modelled for the above mentioned ignored quantities with hybrid or ternary nanofluids. Numerical schemes such as RK4, shooting technique, and finite difference schemes can be used to solve the governing equations.