Amplitude and Phase Angle of Oscillatory Heat Transfer and Current Density along a Nonconducting Cylinder with Reduced Gravity and Thermal Stratification Effects

Abstract

Due to excessive heating, various physical mechanisms are less effective in engineering and modern technologies. The aligned electromagnetic field performs as insulation that absorbs the heat from the surroundings, which is an essential feature in contemporary technologies, to decrease high temperatures. The major goal of the present investigation is to use magnetism perpendicular to the surface to address this issue. Numerical simulations have been made of the MHD convective heat and amplitude problem of electrical fluid flow down a horizontally non-magnetized circular heated cylinder with reduced gravity and thermal stratification. The associated non-linear PDEs that control fluid motion can be conveniently represented using the finite-difference algorithm and primitive element substitution. The FORTRAN application was used to compute the quantitative outcomes, which are then displayed in diagrams and table formats. The physical features, including the phase angle, skin friction, transfer of heat, and electrical density for velocity description, the magnetic characteristics, and the temperature distribution, coupled by their gradients, have an impact on each of the variables in the flow simulation. In the domains of MRI resonant patterns, prosthetic heartvalves, interior heart cavities, and nanoburning devices, the existing magneto-hydrodynamics and thermodynamic scenario are significant. The main findings of the current work are that the dimensionless velocity of the fluid increases as the gravity factor $R_g$ decreases. The prominent change in the phase angle of current density ${\alpha }_m$ and heat flux ${\alpha }_t$ is examined for each value of the buoyancy parameter at both $\alpha =\pi /6$ and $\pi$ angles. The transitory skin friction and heat transfer rate shows a prominent magnitude of oscillation at both $\alpha =\pi /6$ and $\pi /2$ positions, but current density increases with a higher magnitude of oscillation.

1. Introduction

The properties of viscous fluid are causes of variations under the influence of temperature-dependent gravity. The variation in the thermal stratification and reduced gravity of fluid has been illustrated by large numbers of researchers due to their applications in a wide range of science and engineering processes. From the perspective of both research and application, this current intensity and heat rate process of free/forced convection through a non-magnetized material is essential to the disciplines of metals, polymeric materials, and nuclear technology. Engineers have used magnetic hydrodynamic and free/forced convection processes in a variety of sectors, including liquid metallic materials, astrophysical research, geophysical sciences, and nuclear energy. MHD may additionally be utilized directly to modify or enhance the transfer of heat in a variety of circumstances. This includes applications such as stirring molten iron, reducing instability in inductive furnaces, reducing neutrally buoyant driven motion in crystallization, and forming slabs for continuous processes of casting with molten steel. Magneto-hydrodynamics, which Maxwell invented for electromagnetism, is the dynamic study of electromagnetic fluid. As a result, in this branch of study, consolidation of the findings has taken precedence. Evaluation of the current progress in the study of heat transport in a magnetic environment and the provision of a concise assessment of the situation are the study’s secondary priorities.

The current work comprises theoretical and numerical analysis of the amplitude of oscillating heat transfer and current density along a nonconducting heated cylinder. The numerous engineering and geophysical applications of simultaneous heat transfer from circular geometries with reduced gravity and thermal stratification effects include artificial heart valves, internal heart cavities, nanoburning technologies, geothermal reservoirs, drying of porous solids, thermal insulation, enhanced oil recovery, packed-bed catalytic reactors, cooling of nuclear reactors, and underground energy transport [1]. Mechanical, civil, chemical, and manufacturing procedures, geothermal power, and convective heat-driven flow through horizontal surfaces have been enhanced by this phenomenon. Significant illustrations of the fluid motions that occur in the magnetized environment additionally involve the circulation of the helium in pebble-bed reactors, the underground storage of nuclear or non-nuclear waste, the preparation and preservation of food, the processing of petroleum products, the flow in glaucoma patients’ eyes, and the movement within a filtered medium. The Lorentz force interactions give constraints on the fluid movement because a fluid’s viscosity differs from that of a magnetic field. In addition, the present issues have substantial implications for polymeric fields, such as in paper manufacturing, crystal-fiber fabrication, crystal-liquid crystallization, petroleum development, manufacturing of typical lubricants, suspension solutions, wiring sculpting, continuous cooling, material spinning, plastic film manufacturing, polymeric paper extraction, fault zones, petroleum recycling, catalytic reactors, and the fabrication of technological machines.

Using the computational Karman inertia approach, Millsaps and Pohlhausen [1] tackled the issue of the free convection laminar flow field from the external wall of a vertical cylinder. Chawla [2] used the series solution and the Karman–Pohlhausen procedure to resolve a technological issue regarding hydro-magnetic transport across a magnet surface. He concluded that the amplitude increases with frequency but the current surface phase angle decreases from $90°$ to $45°$. The Williamson nanofluid steady free convective Darcy–Forchheimer flow above a linearly extensible surface was computationally approximated by Tamilzharasan et al. [3]. Based on a boundary layer assumption for a heat transport simulation with stratified and thermo-radiating processes, Rehman et al. [4] looked into the mixed convection in the Sutterby flow field. For a multi-mixed-convection flow of second-grade fluid via an asymmetric cylinder in a porous medium, thermodynamic and solute stratifications were shown in Salahuddin et al. [5]. The effects of heating, mass, and skin drag factors on boosting the values of various quantities were explored. Shah et al. [6] numerically analyzed the hydro-magnetic mixed convective Maxwell fluid movement on a vertically stretched surface with a varying thickness close to a stagnation point. It was discovered that at larger values of the Maxwell and thermal conductivity factors, the temperature rises. Hayat et al. [7] looked at the impacts of two stratification flows in a multi-hybrid-convection flow of an Oldroyd-B nanofluid via a reversible stretched surface. The influences of temperature and concentration stratification were used in quantitative equations. The technique of finite elements was computationally used by Farhany et al. [8] to investigate the convective thermal transfer performance in a Cu-water nanofluid-filled enclosure using a baffle. The flow of a magnetic nanopolymer covering across a cylindrical structure in the presence of irregular turbulence was estimated via a Keller box algorithm by Mdallal et al. [9]. Sofiadis and Sarris [10] numerically analyzed Reynolds effects on micropolar flow in a channel.

Measurements have been made by Maranna et al. [11] of the mass transport of a chemically reactive material and the distribution of MHD through an expanding sheet subjected to a magnetic force. A two-dimensional transparent stretchable sheet provided by a Jeffrey nanofluid subjected to multiple factors was computationally researched by Mabood et al. [12] for the production of entropy. The thermally stratified based nanofluid flow caused by a transparent stretching/shrinking cylinder was highlighted by Khashi’ie et al. [13]. Within the range of the input variables, multiple solutions were computed on all profiles. Gireesha et al. [14] compared the content of a simulated study on the impact of thermal stratification on the hydro-magnetic flow and the transfer of heat of dusty material above a vertically stretchy surface in the presence of a uniform source of heat. Both the surface shear stress and the thermal transport were greatly impacted by thermal stratification. Rehman and Malik [15] looked at the flow of an oblique stretchy cylinder-driven Eyring–Powell fluid in a MHD free convective flow separation. The phenomenon of thermal stratification explains flow field analysis. The surface of the cylinder is supposed to have a higher temperature than the surrounding fluid. Koriko et al. [16] looked into the dynamics of the smooth, two-dimensional free heat conduction of micropolar fluid across a vertically porous surface submerged in a thermally stratified material. The heat that best compensates for climatic factors at the wall and in the flow field was chosen.

Mukhopadhyay and Ishak [17] reported an assessment of an axisymmetric flow separation of convective heat transport of viscous and incompressible fluid approaching a stretchy cylinder submerged in a thermally stratified material. Using iterative methods, simulation solutions to these issues were produced. In his evaluation, Besthapu et al. [18] focused on how thermal and solutestratification affected the flow of magnetic properties of nanofluid across an exponential stretched surface. Abbas et al. [19] conducted a theoretical search for the MHD and heat transfer characteristics. By extending a non-rotationg surface under the situation of no slipping, a flow pattern was produced. Yasin et al. [20] presented the quantitative simulation of the sustained mixed convection flow separation above a vertical surface encased in a thermally stratified porous material filled with a nanofluid. By evaluating the consequences of thermal stratification, the investigators Geeta and Moorthy [21] explored an assessment of mass and heat transport features in a fluid medium above a quasi-vertical porous plate. Hayat et al. [22] examined how heat stratification affected the Maxwell fluid flow separation in mixed convective flows. With internal heat generation, Moorthy et al. [23] discussed the impact of differential viscosity and thermal stratification on free convection flow across a non-isothermal vertical channel.

According to the above literature review, the fluctuating mixed convective flow of fluctuating heat and fluctuating current density along a circular heated and nonconducting cylinder has not yet been studied in relation to lowered gravity and stratification. Using concepts from the prior literature review and the works of Chawla [2] and Kay et al. [24], the transient and fluctuating mixed convective heat–current flow mechanism along a circular heated cylinder at four prominentpositions of $30°$, $60°$, $90°$, $180°$ with effects of lowered gravity and stratification was first developed. The circular heated and nonconducting cylinder’s surface was examined numericallyregarding the current heat–transport phenomena. The present research is new in that it involves magnetic-thermal analysis of an electrically charged fluid along the horizontal heated and nonconducting cylinder with reduced gravity and thermal stratification. This is a numerical study of the physical mechanism using the finite-difference scheme. These mechanisms are no longer significant in mechanical and manufacturing procedures because of high temperatures. The surface has magnetization and the liquid is electrically conductive, which is essential to prevent high temperatures throughout the surface. First, we secure the numerical solution for the steady part, and then these results are used to find the phase angle, amplitude of oscillatory skin friction, heat transfer, and magnetic intensity. It is pertinent to mention here that the convective heat transfer is practically associated with oscillatory flow behavior. In the literature, various unsteady models are converted into ordinary equations and just solved for the steady part; however, in current work, the periodic behavior of skin friction, heat transfer, and current density is obtained using oscillating Stokes conditions. Because they satisfy the boundary requirements that were specified, these outcomes are significant and precise.

2. Flow Geometry and Mathematical Formulation

Consider the boundary-layer fluid flow field in two dimensions that arise around a horizontally, nonconducting cylinder. The x-direction along the surface is displayed in Figure 1, together with the y-normal directions towards the surface and the u and v velocities along the $xy$ direction. The temperature of the system is $T$, the exterior flow velocity is $U$, the magnetization is $H_x$ along the base, $H_y$, towards the plane surface. Additionally, the magnetization $H_o$ behaves normally toward the surface of the horizontallynonconducting cylinder. The dimensionless form of the boundary-layer equations is given below:

The relationship between maximum density ${\rho }_m$ and maximum temperature $T_m$ is given below:

$$\frac{\rho -{\rho }_m}{{\rho }_m}=-\gamma {\left(T-T_m\right)}^2$$

(1)

The above equation at the free-stream surface is:

$$T\to T_m \pm \mathsf{\Delta }T,\hspace{1em}\hspace{1em}\hspace{1em}as\hspace{1em}\hspace{1em}\hspace{1em}y\to \pm \infty$$

(2)

under a certain constant $T$. Consider the region $y\ge 0$ according to the boundary requirements to obtain the symmetry in this situation, where $T_{\infty }=T_m+\mathsf{\Delta }T$ is related to ${\rho }_{\infty }$ by Equation (1). The reduced gravity is described below:

$$g^{\prime }=g\frac{({\rho }_m-{\rho }_{\infty })}{{\rho }_{\infty }},\hspace{1em}\hspace{1em}\hspace{1em}{g}^{\prime }=g\gamma \frac{{\rho }_m}{{\rho }_{\infty }}{\left(T_{\infty }-T_m\right)}^2.$$

(3)

Integrating the appropriate non-dimensional elements into the controlling algebraic framework in the context of reduced gravity by following Millsaps and Pohlhausen [1], Chawla [2], and Kay et al. [24] gives:

$$\frac{\partial \overline{u}}{\partial \overline{x}}+\frac{\partial \overline{v}}{\partial \overline{y}}=0$$

(4)

$$\frac{\partial \overline{u}}{\partial \tau }+\overline{u}\frac{\partial \overline{u}}{\partial \overline{x}}+\overline{v}\frac{\partial \overline{u}}{\partial \overline{y}}=\frac{d\overline{U}}{d\tau }+\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}+\xi \left({\overline{h}}_x\frac{\partial {\overline{h}}_x}{\partial \overline{y}}+{\overline{h}}_y\frac{\partial {\overline{h}}_x}{\partial \overline{y}}\right)+R_g\left(2\overline{\theta }-{\overline{\theta }}^2\right)\lambda sin\alpha$$

(5)

$$\frac{\partial {\overline{h}}_x}{\partial \overline{x}}+\frac{\partial {\overline{h}}_y}{\partial \overline{y}}=0$$

(6)

$$\frac{\partial \overline{h}}{\partial \tau }+\overline{u}\frac{\partial \overline{h}}{\partial \overline{x}}+\overline{v}\frac{\partial \overline{h}}{\partial \overline{y}}-{\overline{h}}_x\frac{\partial \overline{u}}{\partial \overline{x}}-{\overline{h}}_y\frac{\partial \overline{u}}{\partial \overline{y}}=\frac{1}{\gamma }\frac{{\partial }^2\overline{h}}{\partial {\overline{y}}^2}$$

(7)

$$\frac{\partial \overline{\theta }}{\partial \tau }+\overline{u}\frac{\partial \overline{\theta }}{\partial \overline{x}}+\overline{v}\frac{\partial \overline{\theta }}{\partial \overline{y}}+S_t\overline{u}=\frac{1}{P_r}\frac{{\partial }^2\overline{\theta }}{\partial {\overline{y}}^2}$$

(8)

where $\xi$ represents the electromagnetic force factor, $S_t$ is the stratification number,$\lambda$ represents the mix-convection number, $R_g$ represents the reduced gravity parameter, $\gamma$ is the magnetic Prandtl number, $P_r$ represents the Prandtl number, $H_o$ represents the strength of the electromagnetic force across the cylinder, and the non-dimension temperature of fluid is designated by $\theta$ in Equation (9).

$$\overline{\theta }=\frac{T_w - T_{\infty , \overline{x}}}{T_w - T_{\infty , 0}}=1-\frac{T_{\infty ,\overline{x}} - T_{\infty , 0}}{T_w - T_{\infty , 0}}$$

(9)

Since $T_{\infty ,\overline{ x}}$ is a linear function and $\Delta T_0=T_w-T_{\infty ,0}$ ($T_{\infty ,0}$ is constant), the dimensionless temperature can be written as:

$$\overline{\theta }=1-\frac{1}{\Delta T_0}\frac{d{T}_{\infty ,\overline{x}}}{d\overline{x}}\overline{x}=1-S_t\overline{x}$$

(10)

In linear form, the thermal stratification $S_t$ is considered to be constant, but for other variations it may be characterized as a function of $\overline{x}$. Thus, the boundary conditions are:

$$\begin{gathered} \overline{u}=0,\hspace{1em}\overline{v}=0,\hspace{1em}{\overline{h}}_y={\overline{h}}_x=0,\hspace{1em}\overline{\theta }=1-S_t\overline{x}\hspace{1em}at\hspace{1em}\overline{y}=0 \\ \hspace{1em}\hspace{1em}\hspace{1em}\overline{u}\to \overline{U}\left(\tau \right),\hspace{1em}\overline{\theta }\to 0,\hspace{1em}{\overline{h}}_y\to 1\hspace{1em}as\hspace{1em}\overline{y}\to \infty \end{gathered}$$

(11)

Consequently, by taking into consideration the stream’s velocity, $U\left(\tau \right)=1+ϵe^{i\omega \tau }$, using $\left|ϵ\right|<<1$, which is the oscillation module’s frequency index $\omega$ with a lower amplitude element. The quantities $u, v, h_x, h_y$, and $\theta$ for flow rate, electromagnetic flow-rate, and heating rate can be expressed as the summation of steady and transient/unsteady factors:

$$\begin{gathered} \overline{u}=u_s+ϵu_t{e}^{i\omega \tau },\hspace{1em}\overline{v}=v_s+ϵv_t{e}^{i\omega \tau },\hspace{1em}{\overline{h}}_x=h_{xs}+ϵh_{xt}{e}^{i\omega \tau } \\ {\overline{h}}_y=h_{ys}+ϵh_{yt}{e}^{i\omega \tau },\hspace{1em}\hspace{1em}\overline{\theta }={\theta }_s+ϵ{\theta }_t{e}^{i\omega \tau } \end{gathered}$$

(12)

Once more, constant controlling steady and unsteady/transient models can be incorporated separately through order evaluation in terms of $O\left({\epsilon }^0\right)$ and $O\left(\epsilon e^{i\omega \tau }\right)$ by utilizing Equations (4)–(8) with (11) in the following manner:

Steady Part:

$$\frac{\partial u_s}{\partial x}+\frac{\partial v_s}{\partial y}=0$$

(13)

$$u_s\frac{\partial u_s}{\partial x}+v_s\frac{\partial u_s}{\partial y}=\frac{{\partial }^2{u}_s}{\partial y^2}+\xi \left(h_{xs}\frac{\partial h_{xs}}{\partial x}+h_{ys}\frac{\partial h_{xs}}{\partial y}\right)+R_g\left(2{\theta }_s-{\theta }_s{}^2\right)\lambda sin\alpha$$

(14)

$$\frac{\partial h_{xs}}{\partial x}+\frac{\partial h_{ys}}{\partial y}=0$$

(15)

$$u_s\frac{\partial h_s}{\partial x}+v_s\frac{\partial h_s}{\partial y}-h_{xs}\frac{\partial u_s}{\partial x}-h_{ys}\frac{\partial u_s}{\partial y}=\frac{1}{\gamma }\frac{{\partial }^2{h}_s}{\partial y^2}$$

(16)

$$u_s\frac{\partial {\theta }_s}{\partial x}+v_s\frac{\partial {\theta }_s}{\partial y}+S_t{u}_s=\frac{1}{P_r}\frac{{\partial }^2{\theta }_s}{\partial y^2}$$

(17)

with appropriate boundary conditions:

$$\begin{gathered} u_s=v_s=0, h_{ys}=h_{xs}=0,\hspace{1em}{\theta }_s=1-S_{t}x\hspace{1em}at\hspace{1em}y=0 \\ {u}_s\to 1,\hspace{1em}\hspace{1em}{\theta }_s\to 0,\hspace{1em}\hspace{1em}{h}_{ys}\to 1\hspace{1em}as\hspace{1em}\hspace{1em}y\to \infty \end{gathered}$$

(18)

Oscillating Part:

$$\frac{\partial u_t}{\partial x}+\frac{\partial v_t}{\partial y}=0$$

(19)

$$\begin{array}{c}i\omega \left(u_t-1\right)+u_s\frac{\partial u_t}{\partial x}+u_t\frac{\partial u_s}{\partial x}+v_s\frac{\partial u_t}{\partial y}+v_t\frac{\partial u_s}{\partial y}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{{\partial }^2{u}_t}{\partial y^2}+\xi \left(h_{xs}\frac{\partial h_{xt}}{\partial x}+h_{xt}\frac{\partial h_{xs}}{\partial x}+h_{ys}\frac{\partial h_{xt}}{\partial y}+h_{yt}\frac{\partial h_{xs}}{\partial y}\right)+R_g\left(2{\theta }_t-2{\theta }_s{\theta }_t\right)\lambda sin\alpha \end{array}$$

(20)

$$\frac{\partial h_{xt}}{\partial x}+\frac{\partial h_{yt}}{\partial y}=0$$

(21)

$$i\omega h_{xt}+u_s\frac{\partial h_t}{\partial x}+u_t\frac{\partial h_s}{\partial x}+v_s\frac{\partial h_t}{\partial y}+v_t\frac{\partial h_s}{\partial y}-h_{xs}\frac{\partial u_t}{\partial x}-h_{xt}\frac{\partial u_s}{\partial x}-h_{ys}\frac{\partial u_t}{\partial y}-h_{yt}\frac{\partial u_s}{\partial y}=\frac{1}{\gamma }\frac{{\partial }^2{h}_{xt}}{\partial y^2}$$

(22)

$$-\omega {\theta }_2+u_s\frac{\partial {\theta }_1}{\partial x}+u_1\frac{\partial {\theta }_s}{\partial x}+v_s\frac{\partial {\theta }_1}{\partial y}+v_1\frac{\partial {\theta }_s}{\partial y}+S_t{u}_1=\frac{1}{P_r}\frac{{\partial }^2{\theta }_1}{\partial y^2}$$

(23)

with appropriate boundary conditions;

$$\begin{gathered} u_t=v_t=0,\hspace{1em}{h}_{yt}=h_{xt}=0,\hspace{1em}{\theta }_1=1-S_{t}x\hspace{1em}at\hspace{1em}y=0 \\ {u}_t\to 1,\hspace{1em}\hspace{1em}{\theta }_t\to 0,\hspace{1em}\hspace{1em}{h}_{yt}\to 1\hspace{1em}\hspace{1em}as\hspace{1em}\hspace{1em}y\to \infty \end{gathered}$$

(24)

Dividing the fluctuating component into imaginary and real components as demonstrated in Equation (19) through (24) and taking into account the fluctuating Stokes’ parameters outlined in Equation (25) gives:

$$u_t=u_1+i{u}_2,\hspace{1em}{v}_t=v_1+i{v}_2,\hspace{1em}{\theta }_t={\theta }_1+i{\theta }_2,\hspace{1em}{h}_{xt}=h_{x1}+i{h}_{x2},\hspace{1em}{h}_{yt}=h_{y1}+i{h}_{y2}$$

(25)

Using Equation (25) in Equations (19)–(24), the imaginary and real components are:

For real components:

$$\frac{\partial u_1}{\partial x}+\frac{\partial v_1}{\partial y}=0$$

(26)

$$\begin{array}{c}-\omega u_2+u_s\frac{\partial u_1}{\partial x}+u_1\frac{\partial u_s}{\partial x}+v_s\frac{\partial u_1}{\partial y}+v_1\frac{\partial u_s}{\partial y}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{{\partial }^2{u}_1}{\partial y^2}+\xi \left(h_{xs}\frac{\partial h_{x1}}{\partial x}+h_{x1}\frac{\partial h_{xs}}{\partial x}+h_{ys}\frac{\partial h_{x1}}{\partial y}+h_{y1}\frac{\partial h_{xs}}{\partial y}\right)+R_g\left(2{\theta }_1-2{\theta }_s{\theta }_1\right)\lambda {\theta }_{1}sin\alpha \end{array}$$

(27)

$$\frac{\partial h_{x1}}{\partial x}+\frac{\partial h_{y1}}{\partial y}=0$$

(28)

$$-\omega h_2+u_s\frac{\partial h_1}{\partial x}+u_1\frac{\partial h_s}{\partial x}+v_s\frac{\partial h_1}{\partial y}+v_1\frac{\partial h_s}{\partial y}-h_{xs}\frac{\partial u_1}{\partial x}-h_{x1}\frac{\partial u_s}{\partial x}-h_{ys}\frac{\partial u_1}{\partial y}-h_{y1}\frac{\partial u_s}{\partial y}=\frac{1}{\gamma }\frac{{\partial }^2{h}_1}{\partial y^2}$$

(29)

$$-\omega {\theta }_2+u_s\frac{\partial {\theta }_1}{\partial x}+u_1\frac{\partial {\theta }_s}{\partial x}+v_s\frac{\partial {\theta }_1}{\partial y}+v_1\frac{\partial {\theta }_s}{\partial y}+S_t{u}_1=\frac{1}{P_r}\frac{{\partial }^2{\theta }_1}{\partial y^2}$$

(30)

along with boundary conditions:

$$\begin{gathered} u_1=v_1=0,\hspace{1em}\hspace{1em}{h}_{y1}=h_{x1}=0,\hspace{1em}{\theta }_1=1-S_{t}x\hspace{1em}at\hspace{1em}y=0 \\ {u}_1\to 1,\hspace{1em}\hspace{1em}{\theta }_1\to 0,\hspace{1em}\hspace{1em}{h}_{y1}\to 1\hspace{1em}\hspace{1em}as\hspace{1em}\hspace{1em}y\to \infty \end{gathered}$$

(31)

For imaginary components:

$$\frac{\partial u_2}{\partial x}+\frac{\partial v_2}{\partial y}=0$$

(32)

$$\begin{array}{c}\omega \left(u_1-U_o\right)+u_s\frac{\partial u_2}{\partial x}+u_2\frac{\partial u_s}{\partial x}+v_s\frac{\partial u_2}{\partial y}+v_2\frac{\partial u_s}{\partial y}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{{\partial }^2{u}_2}{\partial y^2}+\xi \left(h_{xs}\frac{\partial h_{x2}}{\partial x}+h_{x2}\frac{\partial h_{xs}}{\partial x}+h_{ys}\frac{\partial h_{x2}}{\partial y}+h_{y2}\frac{\partial h_{xs}}{\partial y}\right)+R_g\left(2{\theta }_2-2{\theta }_s{\theta }_2\right)\lambda {\theta }_{2}sin\alpha \end{array}$$

(33)

$$\frac{\partial h_{x2}}{\partial x}+\frac{\partial h_{y2}}{\partial y}=0$$

(34)

$$\omega h_1+u_s\frac{\partial h_2}{\partial x}+u_2\frac{\partial h_s}{\partial x}+v_s\frac{\partial h_2}{\partial y}+v_2\frac{\partial h_s}{\partial y}-h_{xs}\frac{\partial u_2}{\partial x}-h_{x2}\frac{\partial u_s}{\partial x}-h_{ys}\frac{\partial u_2}{\partial y}-h_{y2}\frac{\partial u_s}{\partial y}=\frac{1}{\gamma }\frac{{\partial }^2{h}_2}{\partial y^2}$$

(35)

$$\omega {\theta }_1+u_s\frac{\partial {\theta }_2}{\partial x}+u_2\frac{\partial {\theta }_s}{\partial x}+v_s\frac{\partial {\theta }_2}{\partial y}+v_2\frac{\partial {\theta }_s}{\partial y}+S_t{u}_2=\frac{1}{P_r}\frac{{\partial }^2{\theta }_2}{\partial y^2}$$

(36)

along with boundary conditions:

$$\begin{gathered} u_2=v_2=0,\hspace{1em}{h}_{y2}=h_{x2}=0,\hspace{1em}{\theta }_2=0\hspace{1em}at\hspace{1em}y=0 \\ {u}_2\to 0,\hspace{1em}\hspace{1em}{\theta }_2\to 0,\hspace{1em}\hspace{1em}{h}_{y2}\to 1\hspace{1em}\hspace{1em}as\hspace{1em}\hspace{1em}y\to \infty \end{gathered}$$

(37)

3. Computational Scheme and Solution Methodology

The compacting fundamental steady, real, and imaginary variables stated above are quantitatively determined by utilizing the finite-difference algorithm. To acquire the primitive form of the associated partial differential equations (PDEs) for further iteration, the primitive-variable substitution is applied for the smooth form. In order to achieve this, the following expression of (38) is implemented for the steady section, transforming it into a structure that incorporates all independent and dependent variables as follows:

$$\begin{gathered} u_s\left(x,y\right)=U_s\left(X,Y\right), v_s\left(x,y\right)=x^{\frac{-1}{2}}{V}_s\left(X,Y\right),h_{ys}\left(x,y\right)=x^{\frac{-1}{2}}{\phi }_{ys}\left(X,Y\right), \\ {h}_{xs}\left(x,y\right)={\phi }_{xs}\left(X,Y\right),\hspace{1em}\hspace{1em}{\theta }_s\left(x,y\right)={\theta }_s\left(X,Y\right),\hspace{1em}\hspace{1em}Y=x^{\frac{-1}{2}}y,\hspace{1em}\hspace{1em}X=x \end{gathered}$$

(38)

Steady Equations:

$$X\frac{\partial U_s}{\partial X}-\frac{Y}{2}\frac{\partial U_s}{\partial Y}+\frac{\partial V_s}{\partial Y}=0$$

(39)

$$X{U}_s\frac{\partial U_s}{\partial X}+\left[V_s-\frac{Y}{2}{U}_s\right]\frac{\partial U_s}{\partial Y}=\frac{{\partial }^2{U}_s}{\partial Y^2}+\xi \left[X{\phi }_{xs}\frac{\partial {\phi }_{xs}}{\partial Y}+ \left({\phi }_{ys}-\frac{Y}{2}{\phi }_{xs}\right)\frac{\partial {\phi }_{xs}}{\partial Y}\right]+R_g\left(2{\theta }_s-{\theta }_s{}^2\right)\lambda sin\alpha$$

(40)

$$X\frac{\partial {\phi }_{xs}}{\partial X}-\frac{Y}{2}\frac{\partial \phi x_s}{\partial Y}+\frac{\partial {\phi }_{ys}}{\partial Y}=0$$

(41)

$$X{U}_s\frac{\partial {\phi }_s}{\partial X}+\left[V_s-\frac{Y}{2}{U}_s\right]\frac{\partial {\phi }_s}{\partial Y}-X{\phi }_{xs}\frac{\partial U_s}{\partial Y}-\left({\phi }_{ys}-\frac{Y}{2}{\phi }_{xs}\right)\frac{\partial U_s}{\partial Y}=\frac{1}{\gamma }\frac{{\partial }^2{\phi }_s}{\partial Y^2}$$

(42)

$$X{U}_s\frac{\partial {\theta }_s}{\partial X}+\left[V_s-\frac{Y}{2}{U}_s\right]\frac{\partial {\theta }_s}{\partial Y}+S_{t}X{U}_s=\frac{1}{P_r}\frac{{\partial }^2{\theta }_s}{\partial Y^2}$$

(43)

with boundary conditions as:

$$\begin{gathered} U_s=V_s=0,\hspace{1em}{\phi }_{ys}={\phi }_{xs}=0,\hspace{1em}{\theta }_s=1-S_{t}X\hspace{1em}at\hspace{1em}Y=0 \\ {U}_s\to 1,\hspace{1em}{\theta }_s\to 0,\hspace{1em}{\phi }_{ys}\to 1\hspace{1em}as\hspace{1em}Y\to \infty \end{gathered}$$

(44)

Real Equations:

$$X\frac{\partial U_1}{\partial X}-\frac{Y}{2}\frac{\partial U_1}{\partial Y}+\frac{\partial V_1}{\partial Y}=0$$

(45)

$$\begin{array}{c}X\left[U_s\frac{\partial U_1}{\partial X}+U_1\frac{\partial U_s}{\partial X}\right]+\left[V_s-\frac{Y}{2}{U}_s\right]\frac{\partial U_1}{\partial Y}+\left[V_1-\frac{Y}{2}{U}_1\right]\frac{\partial U_s}{\partial Y}-\omega X(U_2+1)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{{\partial }^2{U}_1}{\partial Y^2}+\xi \left[X\left({\phi }_{xs}\frac{\partial {\phi }_{x1}}{\partial x}+{\phi }_{x1}\frac{\partial {\phi }_{xs}}{\partial x}\right)+\left({\phi }_{ys}-\frac{Y}{2}{\phi }_{xs}\right)\frac{\partial {\phi }_{x1}}{\partial Y}+\left({\phi }_{y1}-\frac{Y}{2}{\phi }_{x1}\right)\frac{\partial {\phi }_{xs}}{\partial Y}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+R_g\left(2{\theta }_1-2{\theta }_s{\theta }_1\right)\lambda sin\alpha \end{array}$$

(46)

$$X\frac{\partial {\phi }_{x1}}{\partial X}-\frac{Y}{2}\frac{\partial {\phi }_{x1}}{\partial Y}+\frac{\partial {\phi }_{y1}}{\partial Y}=0$$

(47)

$$\begin{array}{c}X\left[U_s\frac{\partial {\phi }_1}{\partial X}+U_1\frac{\partial {\phi }_s}{\partial X}\right]+\left[V_s-\frac{Y}{2}{U}_s\right]\frac{\partial {\phi }_1}{\partial Y}+\left[V_1-\frac{Y}{2}{U}_1\right]\frac{\partial {\phi }_s}{\partial Y}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\omega X{\phi }_2\left[X\left({\phi }_{xs}\frac{\partial U_1}{\partial x}+{\phi }_{x1}\frac{\partial U_s}{\partial x}\right)+\left({\phi }_{ys}-\frac{Y}{2}{\phi }_{xs}\right)\frac{\partial U_1}{\partial Y}+\left({\phi }_{y1}-\frac{Y}{2}{\phi }_{x1}\right)\frac{\partial U_s}{\partial Y}\right]=\frac{1}{\gamma }\frac{{\partial }^2{\phi }_1}{\partial Y^2}\end{array}$$

(48)

$$X\left[U_s\frac{\partial {\theta }_1}{\partial X}+U_1\frac{\partial {\theta }_s}{\partial X}\right]+\left[V_s-\frac{Y}{2}{U}_s\right]\frac{\partial {\theta }_1}{\partial Y}+\left[V_1-\frac{Y}{2}{U}_1\right]\frac{\partial {\theta }_s}{\partial Y}-\omega X{\theta }_2+S_{t}X{U}_1=\frac{1}{P_r}\frac{{\partial }^2{\theta }_1}{\partial Y^2}$$

(49)

with boundary conditions as:

$$\begin{gathered} U_1=V_1=0,\hspace{1em}{\phi }_{y1}={\phi }_{x1}=0,\hspace{1em}{\theta }_1=1-S_{t}X\hspace{1em}at\hspace{1em}Y=0 \\ {U}_1\to 1,\hspace{1em}{\theta }_1\to 0,\hspace{1em}{\phi }_{Y1}\to 1\hspace{1em}as\hspace{1em}Y\to \infty \end{gathered}$$

(50)

Imaginary Equations:

$$X\frac{\partial U_2}{\partial X}-\frac{Y}{2}\frac{\partial U_2}{\partial Y}+\frac{\partial V_2}{\partial Y}=0$$

(51)

$$\begin{array}{c}X\left[U_s\frac{\partial U_2}{\partial X}+U_2\frac{\partial U_s}{\partial X}\right]+\left[V_s-\frac{Y}{2}{U}_s\right]\frac{\partial U_2}{\partial Y}+\left[V_2-\frac{Y}{2}{U}_2\right]\frac{\partial U_s}{\partial Y}+\omega X(U_1-1)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{{\partial }^2{U}_2}{\partial Y^2}+\xi \left[X\left({\phi }_{xs}\frac{\partial {\phi }_{x2}}{\partial x}+{\phi }_{x2}\frac{\partial {\phi }_{xs}}{\partial x}\right)+\left({\phi }_{ys}-\frac{Y}{2}{\phi }_{xs}\right)\frac{\partial {\phi }_{x2}}{\partial Y}+\left({\phi }_{y2}-\frac{Y}{2}{\phi }_{x2}\right)\frac{\partial {\phi }_{xs}}{\partial Y}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+R_g\left(2{\theta }_2-2{\theta }_s{\theta }_2\right)\lambda sin\alpha \end{array}$$

(52)

$$X\frac{\partial {\phi }_{x1}}{\partial X}-\frac{Y}{2}\frac{\partial {\phi }_{x1}}{\partial Y}+\frac{\partial {\phi }_{y1}}{\partial Y}=0$$

(53)

$$\begin{array}{c}X\left[U_s\frac{\partial {\phi }_2}{\partial X}+U_2\frac{\partial {\phi }_s}{\partial X}\right]+\left[V_s-\frac{Y}{2}{U}_s\right]\frac{\partial {\phi }_2}{\partial Y}+\left[V_2-\frac{Y}{2}{U}_2\right]\frac{\partial {\phi }_s}{\partial Y}+\omega X{\phi }_1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left[X\left({\phi }_{xs}\frac{\partial U_2}{\partial x}+{\phi }_{x2}\frac{\partial U_s}{\partial x}\right)+\left({\phi }_{ys}-\frac{Y}{2}{\phi }_{xs}\right)\frac{\partial U_2}{\partial Y}+\left({\phi }_{y2}-\frac{Y}{2}{\phi }_{x2}\right)\frac{\partial U_s}{\partial Y}\right]=\frac{1}{\gamma }\frac{{\partial }^2{\phi }_2}{\partial Y^2}\end{array}$$

(54)

$$X\left[U_s\frac{\partial {\theta }_2}{\partial X}+U_2\frac{\partial {\theta }_s}{\partial X}\right]+\left[V_s-\frac{Y}{2}{U}_s\right]\frac{\partial {\theta }_2}{\partial Y}+\left[V_2-\frac{Y}{2}{U}_2\right]\frac{\partial {\theta }_s}{\partial Y}+\omega X{\theta }_1+S_{t}X{U}_2=\frac{1}{P_r}\frac{{\partial }^2{\theta }_2}{\partial Y^2}$$

(55)

with boundary conditions as:

$$\begin{gathered} U_2=V_2=0,\hspace{1em}{\phi }_{y2}={\phi }_{x2}=0,\hspace{1em}{\theta }_2=0\hspace{1em}at\hspace{1em}Y=0 \\ {U}_2\to 0,\hspace{1em}{\theta }_2\to 0,\hspace{1em}{\phi }_{y2}\to 0\hspace{1em}as\hspace{1em}Y\to \infty \end{gathered}$$

(56)

The numerical results of transformed expressions are obtained with the implicit finite-difference method. The central difference along the y-axis of the discretized system of steady and unsteady expressions is applied and the backward difference is applied for the x-axis of these equations. The discretized procedure for first- and second-order derivative terms involved in the model is given as:

$$\frac{\partial U}{\partial X}=\frac{U_{\left(i, j\right)}-U_{\left(i, j-1\right)}}{\Delta X},\hspace{1em}\hspace{1em}\hspace{1em}\frac{\partial U}{\partial Y}=\frac{U_{\left(i+1, j\right)}-U_{\left(i-1, j\right)}}{2\Delta Y},\hspace{1em}\hspace{1em}\hspace{1em}\frac{{\partial }^{2}U}{\partial Y^2}=\frac{U_{\left(i+1, j\right)}-2U_{\left(i, j\right)}+U_{\left(i-1, j\right)}}{\Delta Y^2}.$$

(57)

Using the above equations in boundary layer equations, the system of a tri-diagonal matrix is obtained. This tri-diagonal matrix is solved using the Gaussian elimination technique. The current model is unsteady, laminar, and viscous, but not turbulent. The magnetic field effects exactly at the surface are zero but maximum magnetic field effects are present far from the surface which yields a nonconducting mechanism. The magnetic field intensity $H_o$ is used far from the surface which yields an electrically conducting fluid.

Utilizing the finite-difference strategy, Equations (39)–(56) are incorporated in their basic form. The quantitative outputs of modified algebraic expressions with $U, V, \theta$, and $\phi$ uncertain factors can be evaluated in tri-diagonal matrix notation by implementing the Gaussian-elimination strategy for such unknown factors. The obtained results of fluctuating skin friction, fluctuating thermal performance, and fluctuating current density at multiple places on a horizontal, shaped, nonconducting cylinder are displayed in Formula (58) in which $A_s$, $A_t$, and $A_m$ are amplitudes and ${\alpha }_s$, ${\alpha }_t$, and ${\alpha }_m$ are phase angles that are utilized to evaluate the transient rate of skinfriction, fluctuating thermal resistance, and current density.

$${\tau }_w={\left(\frac{\partial U}{\partial Y}\right)}_{y=0}+\epsilon \left|A_s\right|Cos\left(\omega t+{\alpha }_s\right), q_w={\left(\frac{\partial \theta }{\partial Y}\right)}_{y=0}+\epsilon \left|A_t\right|Cos\left(\omega t+{\alpha }_t\right),$$

(58)

$$j_w={\left(\frac{\partial \phi }{\partial Y}\right)}_{y=0}+\epsilon \left|A_m\right|Cos\left(\omega t+{\alpha }_m\right)$$

where:

$$\begin{gathered} A_s={\left(u_1{}^2+u_2{}^2\right)}^{\frac{1}{2}},\hspace{1em}{A}_t={\left({\theta }_1{}^2+{\theta }_2{}^2\right)}^{\frac{1}{2}},\hspace{1em}{A}_m={\left({\phi }_{x1}{}^2+{\phi }_{x2}{}^2\right)}^{\frac{1}{2}}, \\ {\alpha }_s={\tan }^{-1}\left(\raisebox{1ex}{$u_2$}\!\left/ \!\raisebox{-1ex}{$u_1$}\right.\right),\hspace{1em}\hspace{1em}{\alpha }_t={\tan }^{-1}\left(\raisebox{1ex}{${\theta }_2$}\!\left/ \!\raisebox{-1ex}{${\theta }_1$}\right.\right),\hspace{1em}\hspace{1em}{\alpha }_m={\tan }^{-1}\left(\raisebox{1ex}{${\phi }_{x2}$}\!\left/ \!\raisebox{-1ex}{${\phi }_{x1}$}\right.\right) \end{gathered}$$

4. Results and Discussion

Various physical issues caused by high temperatures have had minimal impacts on recent developments and systems. The effect of thermal stratification, magneto-hydrodynamics, and reduced gravity on mixed convection oscillatory/transient conductive fluid over a nonconducting horizontal circular cylinder has been discussed computationally. By means of primitive component substitution and a uniform grid discretization, the flow pattern described in terms of the associated partial differential equations is reduced to a convenient structure using the finite-difference methodology. The computational quantities for each parameter were computed in the FORTRAN program with a smoothing algorithm using primitive properties. The numerical results are plotted in a computational domain as a rectangle frame of lines indicating $x_{min}=0.0, x_{max}=70.0, y_{min}=0.0$, and $y_{max}=50.0$, where $y_{max}$ corresponds to $y\to \infty$, which lies very far from the boundary layers. To obtain an effective and reasonable statistical solution for steady state quantities of flow rate, temperature, and magnetization profile, a grid independence assessment has been suggested. The convergence requirement is used to obtain precise computational outcomes for the flow simulation:

$$max\left|U_s\left(i,j\right)\right|+max\left|V_s\left(i,j\right)\right|+max\left|{\theta }_s\left(i,j\right)\right|+max\left|{\phi }_{ys}\left(i,j\right)\right|+max\left|{\phi }_{xs}\left(i,j\right)\right|\le ϵ$$

The algorithm begins at $x=0$ and proceeds implicitly downward. Here, weselected $\Delta x=0.05$ and $\Delta y=0.01$ quantities of 0.05 and 0.01 for such $i$ and $j$ grid points, respectively, with a tolerance of $ϵ=$ 0.00001 for the convergence of the quantitative findings. In Figure 2b, $X$ means unknown quantities $U,V,\theta ,\phi$ exactly at the surface and $Y$ means far from the surface of the magnetized circular cylinder.

4.1. The Fluid’s Velocity $\mathit{u}$, the Temperature $\mathit{\theta }$, and the Magnetic Field $\mathit{\varphi }$ Plots to Evaluate the Validity of Findings from Described Conditions

Figure 3a–c shows the velocity of the fluid, temperature distribution, and magnetic plot across four distinct levels of the reduced gravity coefficient at places equal to $\alpha =\pi /6$ and $\pi /3$ around the nonconducting cylinder. The maximal response for the temperature distribution and magnetism feature is depicted at $\alpha =\pi /6$ as $R_g$ grows, but the high peak behavior for the fluid’s non-dimensional motion is illustrated at $\alpha =\pi /6$ as $R_g$ declines. This was expected because the enhancement in the gravity element increases the fluid motion along a thermally and horizontally nonconducting cylinder. It can be noticed that the decline in gravity $R_g$ increases the temperature of fluid, which may overshoot the fluid velocity in Figure 3a, as expected. It is also mentioned that the dimensionless velocity of the fluid increases as gravity factor $R_g$ decreases. The fluid becomes lighter and flows faster as gravity decreases, which may lead to enhancement of the velocity field and shows a good change at each position of the nonconducting cylinder. The graphical representations of velocity $U$, temperature $\theta$, and magnetic plot $\varphi$ are provided in Figure 4a–c for distinct quantities of stratification $S_t$ in the context of reduced gravity and magneto-hydrodynamics at two positions of the non-magnetized surface. Since an increase in $S_t$ means a decrease in surface temperature or an increase in free-stream temperature, it is seen that the fluid velocity and magnetic field decrease for lower $S_t=0.1$ at each position; however, the fluid temperature increases sharply, attains a maximum, and then decreases to its asymptotic value as $S_t$ increases at the $\alpha =\pi /6$ position in the presence of buoyancy forces. The fluid acceleration increased as the magnitude of buoyancy increased, and as a result, the fluid’s velocity inside flow domain increased massively. Additionally, it is discovered that, for $R_g$ = 1.0, the thickness of the velocity and thermodynamic boundary layer grows. The magnetization properties are demonstrated away from the outer layer and are completely absent there, and interesting phenomena develop due to the nonconducting system. The problem is significant because liquids with low Prandtl coefficients display greater temperatures and their velocities diminish as a result of Lorentz pressure. It is also noted that the obtained numerical results are in good agreement and satisfy their boundary condition in Equation (15). These results show that the increase in the magnitude of the thermal stratification parameter $S_t$ enhances the temperature difference, which increases the motion of the fluid, and as a result, a decrease in the thermal and magnetic boundary layer is noted.

4.2. Phase Angle of Heat Transfer ${\mathit{\alpha }}_{\mathit{t}}$, Skin Friction ${\mathit{\alpha }}_{\mathit{s}}$ and Current Density ${\mathit{\alpha }}_{\mathit{m}}$

The solid lines in these figures represent the numerical results at the $\alpha =\pi /6$ station, whereas the dashed lines in these figures represent the simulation solutions at the $\alpha =\pi$ location. The phase angle of friction factor ${\alpha }_s$, heat flux ${\alpha }_t$, and current density ${\alpha }_m$ for various values of $\lambda$ = 0.5, 1.5, and 3.5 versus two stations $\alpha =\pi /6$ and $\pi$ with some fixed values are shown in Figure 5a–c. These data show that for a smaller value of $\lambda$ = 0.5 at both locations, the phase angle rises with maximal lag. Figure 5a shows the maximum phase angle of skin friction ${\alpha }_s$ with reasonable variations at both locations. It was speculated that buoyant momentum would lead to a real change in the rate of acceleration of gravitation for fluids with varying density differences. The phase angle of current density ${\alpha }_m$ and heat flux ${\alpha }_t$ with prominent change is examined for each value of mixed convective/buoyancy parameter at both $\alpha =\pi /6$ and $\pi$ locations, as shown in Figure 5b,c. Logically, phase angle is the term used to describe a specific time interval within a cycle that is measured from an arbitrary zero and expressed as an angle. Increasing $\xi$ enhances the Lorentz forces, which opposes the flow and generates the current in fluid layers. Thus, Lorentz forces increase the friction between the layers of fluid, which yields an increment in temperature distribution. In addition, one of the most crucial aspects of a periodic wave is the phase angle. Figure 6a–c present the components of ${\alpha }_s$, ${\alpha }_t$, and ${\alpha }_m$ in terms of phase angle for different values of stratification number $S_t$ = 0.1, 0.2, and 0.3 at two locations $\pi ∕6$ and $\pi$ of the nonconducting circular surface. The phase angles of ${\alpha }_s$ and ${\alpha }_t$ decrease for the high stratification number $S_t$ = 0.3 against $\pi ∕6$, but the phase angle of current density is noted as being a maximum for larger $S_t$ = 0.3 at both locations, as shown in the below figures. In convection, buoyancy forces increase the pressure gradient and temperature due to using a heated fluid such as water for Pr $=7.0$, which was expected physically. The reasonable behavior of the phase angle for ${\alpha }_s$ and ${\alpha }_t$ is noted at the π location for each value of stratification in the presence of the magnetic field. Physically, this was predicted because the magnetic field intensity, which strongly affects the steady and fluctuating aspects of frictional force, is applied normally to the surface.

4.3. Amplitude Shapes of Oscillating Heat Transfer ${\mathbf{\tau }}_{\mathit{t}}$, Skin Friction ${\mathbf{\tau }}_{\mathit{s}}$, and Current Density ${\mathbf{\tau }}_{\mathit{m}}$

This section outlines the consequences of physically appearing variables on the amplitude of transitory skin friction, fluctuating thermal resistance, and fluctuating current density around a horizontally nonconducting cylinder. Plotting of transitory skin friction, oscillatory heat exchange rate, and current density for more appropriate material quantities at $\alpha =\pi /6$ and $\pi /2$ places is achieved using the derived velocities graph, temperature difference, and electromagnetic profiles. Figure 7a–c are drafted to display the consequences of the reduced gravity parameter $R_g$ on the inconsistent computational results. The transitory skin friction and heat transfer rate show a prominent magnitude of oscillation in Figure 7a,b at both $\alpha =\pi /6$ and $\pi /2$ positions, but current density displays a high magnitude of oscillation, as expected, in Figure 7c. However, skin friction improves and small variations in thermal and electrical density are detected due to the substantial buoyant quantity $\lambda$, which serves as a differential pressure and dominates over the resistance. As the gravity component rises, the fluid’s temperature rises and heat transfer rate increases, which lead to enhancement in magnetic oscillation. Logically, this might be because the driving potential of the fluid flow along a heated, nonconducting horizontal cylinder is affected by a rise in the lowered gravitational component. Figure 8a,b in the aforementioned maps indicate a comparable and reducing tendency with the amplitude of fluctuation under the action of the magnetic Prandtl coefficient $\gamma$ in the skin friction and heat transfer rate. This outcome is anticipated since a rise in Pr leads to an increase in the temperature-dependent density differences, which improves the impact of buoyancy. As Pr rises, the thickness of the thermal boundary layer diminishes. However, good variation in the intensity of fluctuation is noted in the transient current–density rate for each value of $\gamma$, which is shown in Figure 8c in the presence of buoyancy forces. Due to the fact that a rise in buoyancy force behaves as a driving factor, the fluid speed is enhanced until it reaches a significant intensity in the current density graph. The maximum amplitude of fluctuation in the current density is evaluated for high values of $\gamma$ for the Prandtl number of water. This is scientifically possible since an increment in Pr associated with a reduction in the fluid’s thermal properties results in shear stresses between the viscous layers that are less significant. As a result, the prominent amplitude of oscillations for each parameter is evaluated at both angles. Logically, it was anticipated because bigger quantities of $\lambda$ are associated with higher buoyant pressures, which cause a rise in the acceleration of fluid motion. However, accurate electromagnetic properties near the outer layer are zero for every parameter due to nonconducting processes, which are significantly seen away from the surface.

Table 1. The numerical outcomes of heat transfer across the nonconducting horizontally cylinder for some choices of $\lambda =0.0, 1.0, 10.0$ for fixed Pr $=7.0$ at the leading edge.

$\mathit{\lambda }$	Chamkha et al. [25]	Saleem and Nadeem [26]	Malik et al. [27]	Present Results
0.0	1.4110	1.4111	1.4110	1.4103
1.0	1.5662	1.5661	1.5660	1.5654
10.0	2.3580	2.3581	2.3581	2.3579

presents the comparison of numerical outcomes of heat transfer with previous existing research of Chamkha et al. [25], Saleem and Nadeem [26], and Malik et al. [27] for some favorable choices of mixed convective parameter $\lambda$ for Pr $=10.0$ under the influence of an aligned magnetic field across the horizontally nonconducting cylinder. It is inferred that magneto properties for each $\lambda$ are responsible for the notable movement of heat. Additionally, the present heat transmission conclusions are consistent with the earlier findings. From

Table 2. The numerical outcomes across the nonconducting horizontally cylinder for some choices of $\xi$ = 0.1, 0.3, 0.5, 0.7 for skin friction ${\mathit{\tau }}_{\mathit{s}}$, heat rate ${\mathit{\tau }}_{\mathit{t}}$, and current density ${\mathit{\tau }}_{\mathit{m}}$.

$\mathit{\xi }$	${\mathit{\tau }}_{\mathit{s}}$	${\mathit{\tau }}_{\mathit{t}}$	${\mathit{\tau }}_{\mathit{m}}$
0.1	4.3565762	0.251481	0.340288
0.3	5.851118	0.281927	0.285282
0.5	7.019164	0.305644	0.251765
0.7	7.850729	0.321940	0.230962

, the highest skin friction is seen at the larger $\xi =0.7$, while the smallest value of skin friction is investigated at the smaller $\xi =0.1$. It is deduced that current density is enhanced at the higher $\xi =0.7$ and the lower current density is depicted at the smaller $\xi =0.1$. The maximum heat transfer is achieved at the smaller $\xi =0.1$, while the minimum value of the heat rate is noted at the larger $\xi =0.7$ with Prandtl number $Pr=10.0$. This finding was anticipated since an increase in the magnetization element is accompanied by an increase in the force of friction, which restricts motion and consequently leads to a decline in the fluid movement. The electromagnetic fluid, on the other hand, exhibits resistance to the Lorentz pressure as the pressure grows by increasing the resistance across its layers.

5. Concluding Remarks

Different physical issues affect recent advances and manufacturing sectors as a result of high temperature. An essential procedure in current innovations, aligned magnetism behaves as a material for coating that absorbs energy and prevents high temperatures. The main objective of the present research is to use magnetic attraction away from the surface to address this issue. The effect of temperature-dependent reduced gravity and thermal stratification on phase angle and amplitude of mixed convection oscillatory/transient conductive fluid over a non-magnetized horizontal circular cylinder is discussed computationally. By means of primitive component substitution and a uniform grid discretization, the flow pattern described in terms of the associated partial differential equations is reduced to a convenient structure using the finite-difference methodology. First, the physical features of velocity distribution, magnetic distribution, and temperature distribution are evaluated. The gradient quantities, such as the phase angle, rates of fluctuating current density, fluctuating heat transfer, and fluctuating skin friction for diverse controlling elements, are described. The FORTRAN application was executed to derive the quantitative findings, which are then displayed in diagrams and table formats. The main findings are given below:

• The thermal boundary layer thickness and magnetic field increases at each angle as reduced gravity increases. It is also mentioned that the dimensionless velocity of the fluid increases as gravity factor $R_g$ decreases.
• It is noted that the fluid velocity and magnetic field decrease as thermal stratification increases at each position but the temperature distribution increases sharply in the presence of buoyancy forces.
• The phase angle of current density and heat transfer increases as the buoyancy force increases. In convection, buoyancy forces increase the pressure gradient and temperature increases due to the heated fluid (water) for Pr $=7.0$.
• The minimum enhancement in the phase angle is noted for increasing the values of thermal stratification but the current density of the fluid is increased at both angles.
• The periodic heat transfer rate and current density increase with a prominent magnitude of oscillation in the presence of reduced gravity.
• The oscillating amplitude of current density increases with prominent variations as the magnetic Prandtl number increases. This was expected because the magnetic diffusion at the surface of the horizontal circular cylinder is decreased for a higher magnetic Prandtl number.