Tangent hyperbolic ternary hybrid nanofluid flow over a rough-yawed cylinder due to impulsive motion

Abstract

This work aims to examine the flow of tangent hyperbolic (T-H) ternary hybrid nanofluid (THNF) over a rough-yawed cylinder generated by impulsive motion in a mixed convection mechanism with periodic magnetohydrodynamics. The surface roughness is depicted by a high-frequency sine wave with a small amplitude. The governing equations of a system were converted to a dimensionless form by using semi-similar transformations. After linearizing the equations with a Quasilinearization technique, they were discretized with the implicit finite difference method. Including a third component (i.e. MoS₂) of the THNF decays the fluid velocity. Raising magnetic field parameter M from 0 to 2 boosts the $R{e}_x^{1/2}{C}_f$ for THNF by 19% at $\xi =0.55$. The $R{e}_x^{1/2}{C}_f$ increases to a maximum of 288% for the T-H and 101% for the Newtonian THNF as the roughness attribute $\epsilon$ increases from 0.001 to 0.01 for $\xi =0.31$ at $n=15$. The findings were compared to previously published to demonstrate the numerical method's reliability.

KEYWORDS:

• Impulsive motion
• Unsteady combined convection flow
• Periodic magnetohydrodynamics (MHD)
• Ternary hybrid nanofluid
• Tangent-hyperbolic (T-H) fluid
• Yawed cylinder

1. Introduction

There are many different technological and industrial circumstances in which one can observe the consequences of heat transfer phenomena. Water, ethanol, and other common fluids, among others, have a heat transfer coefficient that is on the low end of the scale. The researchers look into the concept of nanofluid and hybrid nanofluid [1–4] for the effective heat transfer. The idea of a ternary hybrid nanofluid is something that has been developed in order to improve the effectiveness of heat transfer. The mixture is called a hybrid nanofluid when multiple nanoparticle species coexist in the same fluid. Specifically, when three different nanoparticles are combined, a ternary hybrid nanofluid (THNF) is produced. Until recently, research on hybrid nanofluids has only involved those with two components. Because of the increased energy transfer rates displayed by THNF [5–7], researchers are currently focusing on understanding their unique characteristics. These fluids interest scientists and engineers because of their potential applications in various fields, including the pharmaceutical and nuclear security sectors, the cooling of electrical devices, and many others. Oke [5] has recently investigated the effect of a revolving surface on a THNF based on ethylene glycol. Gul and Saeed [6] have recently conducted research on THNF flow in a porous medium over a nonlinearly expanding surface. Researchers Nazir et al. [7] looked at using ternary hybrid nanomaterials in a partially ionized hyperbolic tangent material to generate thermal energy.

In response to a sudden change in the velocity field, inviscid flow develops instantly. The flow in the viscous layer close to the wall, on the other hand, grows more slowly but eventually stabilizes into a steady state. Over short time scales, viscous forces and unstable acceleration dominate the flow, whereas viscosity, pressure gradient, and convective acceleration dominate over long-time scales. In contrast to the big-time scale, where the flow is influenced by conditions at the leading edge and the stagnation point, the flow on the small-time scale is largely unaffected by these factors. The Rayleigh equation applies for a short period of time, while the Falkner-Skan equation governs the problem mathematically. Only a few studies have looked at impulsive motion on different geometries like cones, flat plates, and rotating spheres [8–12]. In these problems, to simplify the governing equations while maintaining their essential structure, a semi-similar method was employed to simplify the number of independent variables. Since non-Newtonian fluids possess rheological properties distinct from those of Newtonian fluids, numerous models have been developed to describe them adequately. Some of these models include the Carreau, Williamson, Cross, Sisko, Jeffery, Ellis and tangent hyperbolic (T-H) [13–15] models, amongst others. The last model describes a hyperbolic tangent fluid with pseudo-plastic properties. In this type of fluid, the viscosity of the fluid decreases as the shear rate increases. Blood, paints, whipped cream, and ketchup are all types of substances that can be considered examples of this substance. The diverse array of disciplines, including geophysics, biology, metallurgy, chemistry, petroleum, and many more, that use this rheological model, provides the impetus for the research being conducted on it. This non-Newtonian model is also frequently applied when evaluating the machinery used in the laboratory. In addition to this, the fluid accurately reflects the physical characteristics of blood and makes it possible to conduct a more accurate investigation into the phenomenon of shear thinning. Within the context of these implications, researchers have recently reported several different approaches. Awais et al. [13] experimented with optimizing the entropy of a T-H nanofluid in a flat plate using the concept of activation energy. The flow of T-H nanofluid was studied by Ibrahim and Gizewu [14] using a modified version of Fourier's and Fick's diffusion models. This model was used to study the flow of T-H nanofluid around a stretched surface. Jamshed et al. [15] conducted research on a single-phase thermal T-H hybrid nanofluid in conjunction with an entropy study.

During the manufacturing process, the surface of a material may become rough. As a consequence, it significantly impacts the formation of boundary layer features in mixed convective flow over surfaces with a range of different roughness and geometries. Patil et al. [16–18] investigated how surface roughness affects the boundary layer's mixed convective flow and found that it can delay the layer's point of separation across various geometries.

Magnetohydrodynamics (MHD) [19,20] takes into account the behaviour of a large volume of electrically conducting and flowing liquid. This fluid could be anything from plain water to blood plasma. The effects of magnetohydrodynamics (MHD) on liquid metal were related to developing an electromagnetic pump, which is frequently attributed to Hartman. The peculiarities and general behaviour of MHD flow have been investigated using a wide variety of numerical methods. Researchers have nonetheless devoted a lot of time to studying the periodic magnetic field in order to examine the flow patterns. Utilizing magnetohydrodynamic induction, electromagnetic induction drives supply power to induction models and are thus real-world examples of periodic MHD. In the metal purification industry, high-frequency periodic magnetic fields are used to induce eddy currents in a pool of molten metal, which then react in rapid succession to the magnetic field. Other significant applications include the aerospace industry, particularly missile aerodynamics, the petroleum industry, fluid droplet spray, electromagnetically driven heat sinks, seawater boundary drag modification, and so on. However, there is a dearth of research on periodic MHD combined with convection [21–23].

Studying fluid flows around yawed cylinders [24–29] in the design of fluid equipment used in various industrial transport processes has increased interest in this area of study. Yawed-shaped bodies are used in a wide variety of practical contexts, such as in overhead cables, bridge stay cables, chimney stacks, and many more. Engineers working with such systems must comprehend how yaw angle affects the rate of energy transfer via variables such as fluid velocity, temperature, etc. For the first time, Roy [24] looked at the consequences of non-uniform mass transfer on a yawed cylinder. Later, Roy and Saikrishnan [25] used the non-uniform slot suction they had constructed to further investigate yawed cylinders, building on the work of Roy [24]. Thakur et al. [26] analyzed the wake flow effects of both single and multiple cylinders that have been yawed. Revathi et al. [27] conducted a study on forced convective flow through a yawed cylinder, taking temporal variations and non-uniform suction into account (blowing). It is important to know the impact of sudden change in the fluid motion (impulsive motion) and roughness on the fluid flow characteristics.

Based on the literature review, we can confidently assert that no academic has studied the effect of roughness and T-H ternary hybrid nanofluid flow around a yawed cylinder in the context of mixed convection caused by impulsive motion. Here, the surface roughness is depicted by a high-frequency sine wave with a small amplitude. The governing equations of a system were converted to a dimensionless form by using semi-similar transformations. After linearizing the equations with a Quasilinearization technique, they were discretized with the implicit finite difference method. This study aims to investigate the following novel cutting-edge concepts in particular:

• Influence of cylinder's surface roughness on the flow.

• Effects of ternary hybrid nanofluid and impulsive motion.

• Influence of the tangent hyperbolic (T-H) fluid model on the flow characteristics.

• Periodic MHD impact.

• A comparison of various fluids is taken into account.

2. Mathematical analysis

Here, we assume a two-dimensional unsteady laminar incompressible T-H ternary hybrid nanofluid flow across a yawed cylinder with a yaw angle between 15° and 60° in mixed convection mechanism as seen in Figure 1. When $\theta =0^{\circ }$, the cylinder is vertical and when $\theta ={90}^{\circ }$, it is horizontal. Thus, we have studied the yaw angle between 15° and 60° to achieve the effects of a yawed cylinder. Flow in the x-direction is called chordwise flow, and flow in the z-direction is called spanwise flow. Let u, v, and w stand for the x, y, and z components of velocities, respectively. The assumed wall temperature $T_w$ is higher than the fluid's actual temperature of flow. Flow velocities along the chord and spanwise flow directions are represented by $u_{\mathrm{\infty }}$ and $w_{\mathrm{\infty }}$, respectively. Let $\theta$ denote the yaw angle and R the radius of the infinitely yawed cylinder. An impulsive motion caused by the freestream is considered here. A periodic magnetic field of strength B is applied in $z$-direction, which is also responsible for the sudden movement in the flow field. A high-frequency, low-amplitude sine wave represents surface roughness in simulations. To incorporate density variations into the momentum equation, the Boussinesq approximation has been used [16–18,30–32].

Figure 1. Physical model.

For a tangent hyperbolic fluid, we have the following constitutive equation [13,14]: $\overline{\tau }=[{\mu }_{\mathrm{\infty }}+({\mu }_0+{\mu }_{\mathrm{\infty }})\tanh {(\mathrm{\Gamma }\dot{\mathrm{\Omega }})}^s]\dot{\mathrm{\Omega }},$where, $\overline{\tau }$ is the extra stress tensor, ${\mu }_0\mathrm{\&}{\mu }_{\mathrm{\infty }}$ are zero and infinite shear rate viscosity, s is the power law index, $\mathrm{\Gamma }$ is material constant and $\dot{\mathrm{\Omega }}$ is given by: $\dot{\mathrm{\Omega }}=\sqrt{\frac{1}{2}\sum _i\sum _j{\dot{\mathrm{\Omega }}}_{\mathrm{ij}}{\dot{\mathrm{\Omega }}}_{\mathrm{ji}}}=\sqrt{\frac{1}{2}\mathrm{\Pi }},$where, $\mathrm{\Pi }=\frac{1}{2}\mathrm{tr}(\mathrm{\nabla V}+{(\mathrm{\nabla }V)}^T{)}^2$. Here, ${\mu }_{\mathrm{\infty }}=0$ as it is impossible to measure infinite shear rate and invoking the shear thinning effect (i.e. $\mathrm{\Gamma }\dot{\mathrm{\Omega }}<<1$) for the tangent hyperbolic fluid, extra stress tensor becomes, $\begin{array}{rl}\overline{\tau }& ={\mu }_{\mathrm{\infty }}[{(\mathrm{\Gamma }\dot{\mathrm{\Omega }})}^s]\dot{\mathrm{\Omega }}={\mu }_0[{(1+\mathrm{\Gamma }\dot{\mathrm{\Omega }}-1)}^s]\dot{\mathrm{\Omega }}\\ & ={\mu }_{\mathrm{\infty }}[1+s(\mathrm{\Gamma }\dot{\mathrm{\Omega }}-1)]\dot{\mathrm{\Omega }}.\end{array}$With the above assumptions taken into account, once the boundary layer approximations are applied, the governing equations expressed as [13–15,16,21,22, 28,29]: (1) $\begin{array}{rl}& \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0,\end{array}$(1) (2) $\begin{array}{rl}& \begin{array}{rl}& \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\\ & =u_e\frac{d{u}_e}{\mathrm{dx}}+{\nu }_{\mathrm{thnf}}(1-m)\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}\\ & +\sqrt{2}\mathrm{\Gamma }{\nu }_{\mathrm{thnf}}m\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}\\ & +\frac{{\sigma }_{\mathrm{thnf}}{B}^2}{{\rho }_{\mathrm{thnf}}}{\sin }^2(\pi (1-e^{-\tau }))(u_e-u)\\ & +g{\beta }_{\mathrm{thnf}}(T-T_{\mathrm{\infty }})\frac{x}{R}\sin (\theta ),\end{array}\}\end{array}$(2) (3) $\begin{array}{rl}& \begin{array}{rl}& \frac{\mathrm{\partial }w}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }w}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }w}{\mathrm{\partial }y}\\ & ={\nu }_{\mathrm{thnf}}(1-m)\frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{z}^2}+\sqrt{2}\mathrm{\Gamma }{\nu }_{\mathrm{thnf}}m\frac{\mathrm{\partial }w}{\mathrm{\partial }y}\frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{y}^2}\\ & -\frac{{\sigma }_{\mathrm{thnf}}{B}^2}{{\rho }_{\mathrm{thnf}}}{\sin }^2(\pi (1-e^{-\tau }))w\\ & +g{\beta }_{\mathrm{thnf}}(T-T_{\mathrm{\infty }})\frac{x}{R}\cos (\theta ),\end{array}\}\end{array}$(3) (4) $\begin{array}{rl}& \frac{\mathrm{\partial }T}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\\ & =\frac{k_{\mathrm{thnf}}}{{(\rho C_p)}_{\mathrm{thnf}}}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}\\ & +\frac{{\sigma }_{\mathrm{thnf}}{B}^2}{{\rho }_{\mathrm{thnf}}{C}_p}{\sin }^2(\pi (1-e^{-\tau }))(u_e-u{)}^2.\end{array}$(4)

Initial conditions: (5) $\begin{array}{rl}& u(x,y,t)=u_i(x,y);v(x,y,t)=v_i(x,y);\\ & w(x,y,t)=w_i(x,y);T(x,y,t)=T_i(x,y).\end{array}$(5) Boundary conditions (B. Cs.): (6) $\begin{array}{rl}& \mathrm{At}y=0:u(x,0,t)=u_0\mathrm{\alpha sin}\left(\frac{\nu }{u_e^2}\omega \left(1-e^{-\frac{u_{e}t}{x}}\right)\right);\\ & v(x,0,t)=0;w(x,0,t)=0;T(x,0,t)=T_w;\\ & \mathrm{As}y\to \mathrm{\infty }:u(x,\mathrm{\infty },t)\to u_e=u_{\mathrm{\infty }}x;\\ & w(x,\mathrm{\infty },t)\to w_e=w_{\mathrm{\infty }}\cos (\theta );T(x,\mathrm{\infty },t)=T_{\mathrm{\infty }}.\end{array}\}$(6) Where, $u_{\mathrm{\infty }}=w_{\mathrm{\infty }}\sin (\theta )$.

Semi-similar transformations: $\xi =1-e^{-\tau }$; $\eta ={\left(\frac{u_e}{x{\nu }_f\xi }\right)}^{1/2}y$; $\tau =\frac{u_{e}t}{x}$; $w(x,z,t)=w_e\mathrm{xS}(\xi ,\eta )$; $\psi (x,y,t)=-(x{u}_e{\nu }_f\xi {)}^{1/2}f(\xi ,\eta )$; $G=\frac{T-T_{\mathrm{\infty }}}{T_w-T_{\mathrm{\infty }}}$; $u=\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}$; $v=-\frac{\mathrm{\partial }\psi }{\mathrm{\partial }x}$.

Therefore, we get $u=u_{e}F$ and $v=-(u_{\mathrm{\infty }}\xi {\nu }_f{)}^{1/2}f$.

With these semi-similar transformations, while Eq. (1) is identically satisfied, Eqs. (2)–(4) take the form: (7) $\begin{array}{rl}& \begin{array}{rl}& a_1\left\{(1-m)+\frac{m}{\sqrt{\xi }}\mathrm{We}\cdot F_{\eta }\right\}{F}_{\mathrm{\eta \eta }}\\ & +\left\{\mathrm{\xi f}+\frac{\eta }{2}(1-\xi )\right\}{F}_{\eta }\\ & -\xi F^2+\xi -\xi (1-\xi )F_{\xi }\\ & +a_2{a}_3\mathrm{\xi M}(1-F){\sin }^2(\mathrm{\pi \xi })+a_4\mathrm{Ri\xi sin}(\theta )G=0,\end{array}\}\end{array}$(7) (8) $\begin{array}{rl}& \begin{array}{rl}& a_1\left\{(1-m)+\frac{m}{\sqrt{\xi }}\cot (\theta )\mathrm{We}\cdot S_{\eta }\right\}{S}_{\mathrm{\eta \eta }}\\ & +\left\{\mathrm{\xi f}+\frac{\eta }{2}(1-\xi )\right\}{S}_{\eta }-\mathrm{\xi S}-\xi (1-\xi )S_{\xi }\\ & -a_2{a}_3\mathrm{\xi M}{\sin }^2(\mathrm{\pi \xi })S+a_4\mathrm{Ri\xi sin}(\theta )G=0,\end{array}\}\end{array}$(8) (9) $\begin{array}{rl}& b_1{G}_{\mathrm{\eta \eta }}+\mathrm{Pr}\left\{\mathrm{\xi f}+\frac{\eta }{2}(1-\xi )\right\}{G}_{\eta }-\mathrm{Pr\xi }(1-\xi )G_{\xi }\\ & +b_2\mathrm{Pr\xi M}\cdot \mathrm{Ec}\cdot {\sin }^2(\mathrm{\pi \xi })(1-F{)}^2=0.\end{array}$(9)

The pertinent boundary constraints are: (10) $\begin{array}{l}\mathrm{At}\eta =0:F=\mathrm{\epsilon sin}(\mathrm{n\xi });S=0;G=1,\\ \mathrm{As}\eta \to \mathrm{\infty }:F\to 1;S\to 1;G\to 0.\end{array}\}$(10) Where, $\begin{array}{rl}\mathrm{We}& =\mathrm{\Gamma x}{\left(\frac{u_{\mathrm{\infty }}^3}{{\nu }_f}\right)}^{1/2};a_1=\frac{{\nu }_{\mathrm{thnf}}}{{\nu }_f};a_2=\frac{{\sigma }_{\mathrm{thnf}}}{{\sigma }_f};a_3=\frac{{\rho }_f}{{\rho }_{\mathrm{thnf}}};\\ a_4& =\frac{{\beta }_{\mathrm{thnf}}}{{\beta }_f};M=\frac{{\sigma }_f{B}_0^2}{{\rho }_f{u}_{\mathrm{\infty }}};\mathrm{Ri}=\frac{\mathrm{Gr}}{R{e}^2};\\ \mathrm{Gr}& =\frac{g{\beta }_f(T_w-T_{\mathrm{\infty }})R^3}{{\nu }_f^2};\mathrm{Re}=\frac{u_{\mathrm{\infty }}R}{{\nu }_f};\mathrm{Pr}=\frac{{\mu }_f{C}_{p_f}}{k_f};\\ \mathrm{Ec}& =\frac{u_e^2}{C_{p_f}(T_w-T_{\mathrm{\infty }})};b_1=\frac{k_{\mathrm{thnf}}/k_f}{{(\rho C_p)}_{\mathrm{thnf}}/{(\rho C_p)}_f};\\ b_2& =\frac{{\sigma }_{\mathrm{thnf}}/{\sigma }_f}{{(\rho C_p)}_{\mathrm{thnf}}/{(\rho C_p)}_f}.\end{array}$

Also $f(\xi ,\eta ,\tau )={\int }_0^{\eta }F\cdot \mathrm{d\eta }+f_w$, as we have considered impermeable surface, $f_w=0$.

Coefficient of skin friction (in chordwise direction): (11) $\begin{array}{rl}{C}_f=\frac{{\tau }_w}{{\rho }_f{u}_e^2}& =\frac{{\mu }_{\mathrm{thnf}}{\left\{(1-m)\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+m\mathrm{\Gamma }{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2\right\}}_{y=0}}{{\rho }_f{u}_e^2}\\ \Rightarrow R{e}_x^{1/2}{C}_f& =\frac{{\mu }_{\mathrm{thnf}}}{{\mu }_f}\left\{\frac{(1-m)}{\sqrt{\xi }}+\frac{m}{\xi }\mathrm{We}\cdot F_{\eta }(\xi ,0)\right\}\\ & \times F_{\eta }(\xi ,0),\end{array}$(11) Coefficient of skin friction (in spanwise direction): (12) $\begin{array}{rl}{\overline{C}}_f=\frac{{\tau }_{w_y}}{{\rho }_f{u}_e^2}& =\frac{{\mu }_{\mathrm{thnf}}{\left\{(1-m)\frac{\mathrm{\partial }w}{\mathrm{\partial }y}+m\mathrm{\Gamma }{\left(\frac{\mathrm{\partial }w}{\mathrm{\partial }y}\right)}^2\right\}}_{y=0}}{{\rho }_f{u}_e^2}\\ \Rightarrow R{e}_x^{1/2}{\overline{C}}_f& =\frac{{\mu }_{\mathrm{thnf}}}{{\mu }_f}\cot (\theta )\\ & \times \left\{\frac{(1-m)}{\sqrt{\xi }}+\frac{m}{\xi }\cot (\theta )\mathrm{We}\cdot S_{\eta }(\xi ,0)\right\}\\ & \times S_{\eta }(\xi ,0),\end{array}$(12) Heat transfer rate: (13) $\begin{array}{rl}\mathrm{Nu}& =-\frac{k_{\mathrm{thnf}}x{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)}_{y=0}}{k_f(T_w-T_{\mathrm{\infty }})}\\ \Rightarrow R{e}_x^{-1/2}\mathrm{Nu}& =-\frac{k_{\mathrm{thnf}}}{k_f}\frac{1}{\sqrt{\xi }}{G}_{\eta }(\xi ,0,\tau ).\end{array}$(13)

3. Numerical method

The following are the set of linear equations obtained by employing the Quasilineazation technique [28,33, 34–36] to the dimensionless Eqs. (7)–(9). (14) $\begin{array}{rl}& F_{\mathrm{\eta \eta }}^{i+1}+A_1^i{F}_{\eta }^{i+1}+A_2^i{F}_{\xi }^{i+1}+A_3^i{F}^{i+1}+A_4^i{G}^{i+1}=A_5^i,\end{array}$(14) (15) $\begin{array}{rl}& S_{\mathrm{\eta \eta }}^{i+1}+B_1^i{S}_{\eta }^{i+1}+B_2^i{S}_{\xi }^{i+1}+B_3^i{S}^{i+1}+B_4^i{G}^{i+1}=B_5^i,\end{array}$(15) (16) $\begin{array}{rl}& G_{\mathrm{\eta \eta }}^{i+1}+C_1^i{G}_{\eta }^{i+1}+C_2^i{G}_{\xi }^{i+1}+C_3^i{F}^{i+1}=C_4^i.\end{array}$(16)

The equivalent boundary conditions are: (17) $\begin{array}{ll}\text{At }\eta =0& :F^{i+1}(\xi ,0)=\mathrm{\epsilon sin}(\mathrm{n\xi }),\\ S^{i+1}(\xi ,0)=0,& G^{i+1}(\xi ,0)=1,\\ \text{At }\eta ={\eta }_{\mathrm{\infty }}& :F^{i+1}(\xi ,0)=1,\\ S^{i+1}(\xi ,0)=1,& G^{i+1}(\xi ,0)=0.\end{array}\}$(17) The implicit finite difference technique [37–39] is applied to the obtained set of linear equations (14)–(16) for discretization. After that, Varga's matrix inverse technique [40] is implemented to solve the resulting finite difference equations with the matrix of coefficients in block tridiagonal form. Here, the step lengths $\mathrm{\Delta }\xi \mathrm{and}\mathrm{\Delta \eta }$ are taken to be 0.001. With a precision of 10⁻⁴, the findings iteratively converge to the desired values, i.e.: $\begin{array}{rl}& \mathrm{Max}\{|(F_{\eta }{)}_w^{i+1}-(F_{\eta }{)}_w^i|,|(S_{\eta }{)}_w^{i+1}-(S_{\eta }{)}_w^i|,\\ & |(G_{\eta }{)}_w^{i+1}-(G_{\eta }{)}_w^i|\}\le {10}^{-4}.\end{array}$

The coefficients of equations (14)–(16) are: $\begin{array}{rl}{A}_1^i& =\sqrt{\xi }\left\{\mathrm{\xi f}+\frac{\eta }{2}(1-\xi )\right\}\mathrm{dff}-\xi \sqrt{\xi }\{F^2-1\\ & +(1-\xi )F_{\xi }-a_2{a}_{3}M{\sin }^2(\mathrm{\pi \xi })(1-F)\\ & -a_4\mathrm{Ri\xi sin}(\theta )G\}\mathrm{df};A_2^i=-\frac{\xi \sqrt{\xi }}{Q_1}(1-\xi );\\ A_3^i& =-\frac{\xi \sqrt{\xi }}{Q_1}\{2F+a_2{a}_{3}M{\sin }^2(\mathrm{\pi \xi })\};\\ A_4^i& =\frac{a_4}{Q_1}\mathrm{Ri\xi }\sqrt{\xi }\sin (\theta );\\ A_5^i& =A_1^i{F}_{\eta }-\frac{\sqrt{\xi }}{Q_1}\{\left[\mathrm{\xi f}+\frac{\eta }{2}(1-\xi )\right]F_{\eta }+\xi (1+F^2)\\ & +a_2{a}_3\mathrm{\xi M}{\sin }^2(\mathrm{\pi \xi })\};\\ B_1^i& =\sqrt{\xi }\left\{\mathrm{\xi f}+\frac{\eta }{2}(1-\xi )\right\}\mathrm{dss}-\sqrt{\xi }\{\mathrm{\xi S}+\xi (1-\xi )S_{\xi }\\ & +a_2{a}_3\mathrm{\xi M}\cdot S\cdot {\sin }^2(\mathrm{\pi \xi })-a_4\mathrm{Ri\xi sin}(\theta )G\}\mathrm{ds};\\ B_2^i& =-\frac{\xi \sqrt{\xi }}{Q_2}(1-\xi );\\ B_3^i& =-\frac{\xi \sqrt{\xi }}{Q_2}\{1+a_2{a}_{3}M{\sin }^2(\mathrm{\pi \xi })\};\\ B_4^i& =a_4\mathrm{Ri}\frac{\xi \sqrt{\xi }}{Q_2}\sin (\theta );\\ B_5^i& =B_1^i{S}_{\eta }-\frac{\sqrt{\xi }}{Q_2}\left\{\mathrm{\xi f}+\frac{\eta }{2}(1-\xi )\right\}{S}_{\eta };\\ C_1^i& =\frac{\mathrm{Pr}}{b_1}\left\{\mathrm{\xi f}+\frac{\eta }{2}(1-\xi )\right\};C_2^i=-\frac{\mathrm{Pr}}{b_1}\xi (1-\xi );\\ C_3^i& =-\frac{2\mathrm{Pr}}{b_1}{b}_2\mathrm{\xi EcM}{\sin }^2(\mathrm{\pi \xi })(1-F);\\ C_4^i& =-\frac{\mathrm{Pr}}{b_1}{b}_2\mathrm{\xi EcM}{\sin }^2(\mathrm{\pi \xi })(1-F^2).\end{array}$

Where, $\begin{array}{rl}{Q}_1& =a_1\left\{\sqrt{\xi }(1-m)+\mathrm{mWe}{F}_{\eta }\right\};\\ Q_2& =a_1\left\{\sqrt{\xi }(1-m)+\mathrm{mWe}\cdot \cot (\theta )F_{\eta }\right\};\\ \mathrm{df}& =\frac{-a_1\mathrm{mWe}}{Q_1^2};\mathrm{dff}=\frac{a_1\sqrt{\xi }(1-m)}{Q_1^2};\\ \mathrm{ds}& =\frac{-a_1\mathrm{mWecot}(\theta )}{Q_2^2};\mathrm{dss}=\frac{a_1\sqrt{\xi }(1-m)}{Q_2^2}.\end{array}$

3.1. Results comparison

For the authentication of the numerical approach, the current outcomes are compared with the results attained by Roy and Anilkumar [41] and Takhar et al. [42] for the Nusselt number $R{e}^{-1/2}\mathrm{Nu}$ in Table 1 when Pr = 7.0, $\theta =0^{\circ }$ and $\mathrm{We}=\mathrm{Ec}={\beta }_t=M=0$. The comparison demonstrates a definitive agreement between the current findings and those obtained in [41] and [42].

Table 1. Comparison of heat transfer rate.

		Heat transfer rate ($R{e}^{-1/2}\mathrm{Nu}$)
$\xi$	Ri	Roy and Anilkumar [41]	Takhar et al. [42]	Our results
0	0	0.5854	0.5854	0.5853
0	1	0.8220	0.8219	0.8219
0	2	0.9304	0.9302	0.9303
1	0	0.8666	0.8666	0.8666
1	1	1.0621	1.0617	1.0620
1	2	1.1688	1.1685	1.1686

3.2. Grid discretization

The grid discretization of a numerical method is defined by the number of grids used (or the grid size). The accuracy and computing time of the solution will grow as the number of grids is increased until a particular ideal number of grids is reached. The grid discretization is shown in Figure 2 with $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ mesh sizes in the x and y directions. The grid is denoted by the ordered pair (i, j), and the total number of such pairings is called the grid number.

Figure 2. Grid discretization.

4. Results and discussion

In this section, various graphs on profiles and gradients were used to illustrate the typical effects of a number of variables, including the Weissenberg number $(\mathrm{We})$, periodic magnetic field $(M)$, nanoparticle volume fraction $({\phi }_i)$, Richardson number $(\mathrm{Ri})$, yaw angle $(\theta )$, roughness parameter $(\epsilon )$, frequency $(n)$, and power-law index $(m)$. Unless and until they were explicitly stated, some parameters, such as, $m=0.3$, $\mathrm{Pr}=7.0$, $\mathrm{Ec}=0.1$ were assigned constant values throughout the numerical computation. In addition, for the current study, we considered a three-component or ternary hybrid nanofluid (THNF) with Ag (Silver), TiO₂ (Titanium dioxide), and MoS₂ (Molybdenum disulphide). Table 2 shows the expressions for the various properties of these nanoparticles, and Table 3 shows the thermal properties.

Table 2. Expressions for ternary hybrid nanofluid characteristics [6].

Dynamic viscosity	${\mu }_{\mathrm{thnf}}={\displaystyle \frac{{\mu }_f}{{(1-{\phi }_1)}^{2.5}{(1-{\phi }_2)}^{2.5}{(1-{\phi }_3)}^{2.5}}}$
Thermal conductivity	$k_{\mathrm{thnf}}={\displaystyle \frac{k_3+2k_{\mathrm{hnf}}-(s-1){\phi }_3(k_{\mathrm{hnf}}-k_3)}{k_3+2k_{\mathrm{hnf}}+{\phi }_2(k_{\mathrm{hnf}}-k_3)}}\ast k_{\mathrm{hnf}},$
	where $k_{\mathrm{hnf}}={\displaystyle \frac{k_2+2k_{\mathrm{nf}}-(s-1){\phi }_1(k_{\mathrm{nf}}-k_2)}{k_2+2k_{\mathrm{nf}}+{\phi }_1(k_{\mathrm{nf}}-k_2)}}\ast k_{\mathrm{nf}},$
	$k_{\mathrm{nf}}={\displaystyle \frac{k_1+2k_f-(s-1){\phi }_1(k_f-k_1)}{k_1+2k_f+{\phi }_1(k_f-k_1)}}\ast k_f$
Density	${\rho }_{\mathrm{thnf}}=(1-{\phi }_1)[(1-{\phi }_2)\{(1-{\phi }_3)+{\phi }_3{\rho }_3\}+{\phi }_2{\rho }_2]+{\phi }_1{\rho }_1$
Thermal expansion coefficient	$(\rho C_p{)}_{\mathrm{thnf}}=(1-{\phi }_1)[(1-{\phi }_2)\{(1-{\phi }_3)+{\phi }_3(\mathrm{\rho \beta }{)}_3\}+{\phi }_2(\mathrm{\rho \beta }{)}_2]+{\phi }_1(\mathrm{\rho \beta }{)}_1$
Heat capacitance	$(\rho C_p{)}_{\mathrm{thnf}}=(1-{\phi }_1)[(1-{\phi }_2)\{(1-{\phi }_3)+{\phi }_3(\rho C_p{)}_3\}+{\phi }_2(\rho C_p{)}_2]+{\phi }_1(\rho C_p{)}_1$
Electrical conductivity	${\sigma }_{\mathrm{thnf}}={\displaystyle \frac{{\sigma }_3+2{\sigma }_{\mathrm{hnf}}-2{\phi }_3({\sigma }_{\mathrm{hnf}}-{\sigma }_3)}{{\sigma }_3+2{\sigma }_{\mathrm{hnf}}+{\phi }_2({\sigma }_{\mathrm{hnf}}-{\sigma }_3)}}\ast {\sigma }_{\mathrm{hnf}}$,
	where ${\sigma }_{\mathrm{hnf}}={\displaystyle \frac{{\sigma }_2+2{\sigma }_{\mathrm{nf}}-2{\phi }_1({\sigma }_{\mathrm{nf}}-{\sigma }_2)}{{\sigma }_2+2{\sigma }_{\mathrm{nf}}+{\phi }_1({\sigma }_{\mathrm{nf}}-{\sigma }_2)}}\ast {\sigma }_{\mathrm{nf}}$,
	${\sigma }_{\mathrm{nf}}={\displaystyle \frac{{\sigma }_1+2{\sigma }_f-2{\phi }_1({\sigma }_f-{\sigma }_1)}{{\sigma }_1+2{\sigma }_f+{\phi }_1({\sigma }_f-{\sigma }_1)}}\ast {\sigma }_f$

Table 3. Thermal characteristics of the nanoparticles [43–45].

Properties	Ag	TiO2	MoS2	Water
$C_p(\text{J/kgK})$	235	686.2	717	4179
$k(\text{W/mK})$	429	8.953	5000	0.613
$\sigma (\mathrm{\Omega m}{)}^{-1}$	2.3$\times$10^6	3.5$\times$10^6	6.3$\times$10^7	5.5$\times$10^−6
$\beta (\text{1/K})\times {10}^{-5}$	1.89	0.9	0.284	21
$\rho (\text{kg/}{\text{m}}^3)$	10,500	4250	1800	997.1

4.1. Effects of $\mathrm{Ri},\mathrm{We},M,{\phi }_3$

Figures 3–8 present the impact of $\mathrm{Ri},\mathrm{We},M,{\phi }_3$ on velocity profile $(F(\xi ,\eta ))$, and corresponding gradients in the chordwise $(R{e}_x^{1/2}{C}_f)$ and spanwise $(R{e}_x^{1/2}{\overline{C}}_f)$ directions. The roughness of the surface is responsible for the sinusoidal oscillations in the chordwise friction coefficient. Here, $\mathrm{We}=0$ characterizes the Newtonian fluid, whereas $\mathrm{We}\ne 0$ characterizes the tangent-hyperbolic (T-H) fluid (non-Newtonian). Compared to Newtonian THNF, the fluid velocity slows, and the chord and spanwise friction coefficients accelerate impulsively (i.e. $\mathrm{We}$ enhanced from 0 to 2). This has relied on the viscosity of the T-H THNF, which is more viscous than the Newtonian one. Notably, the highest percentage enhancement, approximately 241% and 148% in the chord and spanwise friction coefficients, are observed at $\xi =0.23$ and $\xi =0.01$ respectively when $\mathrm{We}$ enhanced from 0 to 1. Also, since the $\mathrm{Ri}$ serves as a favourable pressure gradient, one can observe the increasing friction coefficients through Figures 4 and 5. As a result of the flow's impulsive nature, smaller $\xi$ values reveal substantial fluctuations across all gradients.

Figure 3. Variation in chordwise velocity $F(\xi ,\eta )$ for varying $\mathrm{We}$ and ${\phi }_3$ at $M=0.1$, ${\phi }_1={\phi }_2=0.02$, $\theta ={30}^{\circ }$, $n=10$, $\epsilon =0.01$.

Figure 4. Variation in chordwise friction coefficient $R{e}_x^{1/2}{C}_f$ for varying $\mathrm{Ri}$ and $\mathrm{We}$ at $M=0.1$, ${\phi }_1={\phi }_2={\phi }_3=0.02$, $\theta ={30}^{\circ }$, $\epsilon =0.01$.

Figure 5. Variation in spanwise friction coefficient $R{e}_x^{1/2}{\overline{C}}_f$ for varying $\mathrm{Ri}$ and $\mathrm{We}$ at $M=0.1$, ${\phi }_1={\phi }_2={\phi }_3=0.02$, $\theta ={30}^{\circ }$, $\epsilon =0.01$.

Figure 6. Variation in chordwise velocity $F(\xi ,\eta )$ for varying $M$ and ${\phi }_3$ at $\mathrm{We}=1.0$, ${\phi }_1={\phi }_2=0.02$, $\theta ={30}^{\circ }$, $n=10$, $\epsilon =0.01$.

Figure 7. Variation in chordwise friction coefficient $R{e}_x^{1/2}{C}_f$ for varying $M$ and ${\phi }_3$ at $\mathrm{Ri}=10$, ${\phi }_1={\phi }_2=0.02$, $\mathrm{We}=1.0$, $\theta ={30}^{\circ }$, $\epsilon =0.01$.

Figure 8. Variation in spanwise friction coefficient $R{e}_x^{1/2}{\overline{C}}_f$ for varying $M$ and ${\phi }_3$ at $\mathrm{Ri}=10$, ${\phi }_1={\phi }_2=0.02$, $\mathrm{We}=1.0$, $\theta ={30}^{\circ }$, $\epsilon =0.01$.

Further, the inclusion of a third component (i.e. MoS₂) of the THNF decays the $F(\xi ,\eta )$ as seen in Figures 3 and 6. That is, the non-zero value of ${\phi }_3$ causes such variations due to the addition of third component of the THNF makes the fluid thicker than that of the 2-components hybrid nanofluid (HNF). Following Figures 7 and 8, the non-zero value of ${\phi }_3$ upsurges both the chord and spanwise friction coefficient due to the collision of the nanoparticles caused by the impulsive motion of the fluid. Furthermore, it is clear from Figures 7 and 8 that Lorentz force is generated by periodic MHD and augments $F(\xi ,\eta )$ and $R{e}_x^{1/2}{C}_f$ for the growing values of M from 0 to 2. Exactly opposite nature of the results can be seen in the case of $R{e}_x^{1/2}{\overline{C}}_f$. In particular, a change in the M from 0 to 2 improves the $R{e}_x^{1/2}{C}_f$ 19% and decays the $R{e}_x^{1/2}{\overline{C}}_f$ 50% approximately at $\xi =0.55$ in the THNF.

4.2. Effect of surface roughness

The variations of $R{e}_x^{1/2}{C}_f$ for various values of rough surface parameters $(\epsilon )$ and frequency $(n)$ are graphed in Figure 9(a–c) for both T-H and Newtonian THNF cases. In Figure 9(a), the rough surface with $\epsilon =0.001$ varying encounters levels of skin friction with more rapid changes as frequency n increases. This impact is accentuated by T-H THNF (We = 1), as shown in Figure 9(a). As can be seen in Figure 9(b), the roughness amplitude $\epsilon =0.005$ has a more pronounced impact on $R{e}_x^{1/2}{C}_f$, with more significant amplitude fluctuations, than the roughness amplitude $\epsilon =0.001$. In Figure 9(c), a substantial amplitude of skin-friction variations is observed for the roughness parameter $\epsilon =0.01$ compared to smaller values of $k(\text{W/mK})$ due to the T-H fluid feature (We = 1) and the flow's impulsive nature. In addition, it is owing to the T-H fluid's shear-thinning behaviour and the fluid being trapped in the deeper cavities generated due to the more excellent surface asperities formed on the rough surface. In particular, the $R{e}_x^{1/2}{C}_f$ increases to a maximum of 288% for the T-H and 101% for the Newtonian THNF as the $\epsilon$ increases from 0.001 to 0.01 at $\xi =0.31$ and frequency $n=15$.

Figure 9. (a-c) Effect of surface roughness $(\epsilon \mathrm{and}n)$ on the chordwise surface friction coefficient for different $\mathrm{We}$.

4.3. Effects of $\text{m and }\theta$

The impacts of the power-law index $(m)$ and yaw angle $(\theta )$ over $F(\xi ,\eta )$, $R{e}_x^{1/2}{C}_f$ and $R{e}_x^{1/2}{\overline{C}}_f$ are portrayed in Figures 10–12. The fluid velocity and skin friction coefficient are observed to increase in the chordwise direction when the yaw angle $\theta$ rises. In contrast, the opposite pattern holds true for $R{e}_x^{1/2}{\overline{C}}_f$. In other words, as the cylinder's yaw angle (the degree to which it is tilted) grows, the flow starts impulsively, and so the fluid pressure inside it rises, increasing its chordwise velocity and friction coefficient. It is noticed that at $\xi =0.86$, $m=0.5$ the chord and spanwise friction coefficients enhance and decay by approximately 38% and 69%, respectively, when $\theta$ improving from 15° to 60°. Further, the effect of the power-law index $m$ can be seen through Figures 10–12. The chordwise velocity distribution and the friction coefficients along both directions augment the higher values $m$. Evidence for this can be seen in the fact that as the power-law index m increases, the fluid's characteristic shifts from shear-thinning to shear-thickening. Also, it is clear that upsurging $m$ from 0.2 to 0.5 enhance the chord and spanwise friction coefficients by approximately 56% and 69%, respectively at $\xi =0.86$.

Figure 10. Variation in chordwise velocity $F(\xi ,\eta )$ for varying $\theta$ and $m$ at $M=0.1$, ${\phi }_1={\phi }_2={\phi }_3=0.02$, $\mathrm{We}=1.0$, $n=10$, $\epsilon =0.01$.

Figure 11. Variation in chordwise drag coefficient for varying $\theta$ and $m$ at $M=0.1$, ${\phi }_1={\phi }_2={\phi }_3=0.02$, $\mathrm{We}=1.0$, $n=10$, $\epsilon =0.01$.

Figure 12. Variation in spanwise drag coefficient for varying $\theta$ and $m$ at $M=0.1$, ${\phi }_1={\phi }_2={\phi }_3=0.02$, $\mathrm{We}=1.0$, $n=10$, $\epsilon =0.01$.

4.4. Influence of $M$ and different fluids

Figures 13–15 discuss the significance of periodic MHD and T-H ternary hybrid nanofluid for the purpose of very efficient heat transfer. Here, ${\phi }_i=0,i=1,2,3$, ${\phi }_1=0.02,{\phi }_2={\phi }_3=0$, ${\phi }_1=0.02,{\phi }_2=0.02,{\phi }_3=0$, ${\phi }_1=0.02,{\phi }_2=0.02,{\phi }_3=0.02$, respectively represent the T-H base fluid, Ag T-H nanofluid, Ag-TiO₂ two component T-H HNF and Ag-TiO₂-MoS₂ T-H THNF The temperature distribution $G(\xi ,\eta )$, energy transfer strength $R{e}_x^{-1/2}\mathrm{Nu}$ and chordwise friction coefficient $R{e}_x^{1/2}{C}_f$ escalate by adding different component's nanoparticles to the base fluid. In other words, especially the $R{e}_x^{-1/2}\mathrm{Nu}$ is significantly less for the T-H base fluid and is large for the T-H THNF. Also, the $R{e}_x^{-1/2}\mathrm{Nu}$ for the two components, HNF vary in between the T-H base fluid and T-H ternary HNF. In particular to say that, the $R{e}_x^{-1/2}\mathrm{Nu}$ found to upsurge approximately by 15% for the T-H THNF than that of only T-H base fluid. Further, the period MHD significance is noticed in Figures 13–15. The non-zero values of M $(M=1)$ indicate the presence of the periodic MHD while $M=0$ representing the absence of periodic MHD. Due to the presence of Lorentz force generated by the periodic MHD, the $G(\xi ,\eta )$ and $R{e}_x^{1/2}{C}_f$ are found to be escalated while the $R{e}_x^{-1/2}\mathrm{Nu}$ de-escalated.

Figure 13. Variation in fluid temperature for different fluids considered in the presence and absence of periodic magnetic field.

Figure 14. Variation in heat transfer rate for different fluids considered in the presence and absence of periodic magnetic field.

Figure 15. Variation in chordwise drag coefficient for different fluids considered in the presence and absence of periodic magnetic field.

Figure 16. Fluid temperature $G(\xi ,\eta )$ outcome for different shapes of nanoparticles.

Figure 17. Heat transfer outcome for different shapes of nanoparticles.

4.5. Nanoparticle's shape factor effect

Nanoparticle heat conduction critically depends on their shapes. Figures 16 and 17 throw light on how the shape factors of nanoparticles affect the fluid temperature and $G(\xi ,\eta )$ and energy transport strength $(R{e}^{-1/2}\mathrm{Nu})$. Here, we discuss nanoparticles of various forms, including spherical (s = 3.0), hexahedral (s = 3.7221), tetrahedral (s = 4.0613), column (s = 6.3698), and lamina (s = 16.1576) [36]. The shape factor s can be determined by knowing the sphericity of the nanoparticle's shape. The sphericity and shape factors are inversely proportional to one another, given by $s=\frac{3}{\mathrm{\Omega }}$, where $\mathrm{\Omega }$ is the sphericity. It has been examined from Figures 16 and 17 that the emerging values of s boost the $G(\xi ,\eta )$ and $R{e}^{-1/2}\mathrm{Nu}$ i.e. changing the shapes of nanoparticles from spherical to other different shapes enhances the $G(\xi ,\eta )$ and $R{e}^{-1/2}\mathrm{Nu}$. Concisely at $\xi =0.5$, the $R{e}^{-1/2}\mathrm{Nu}$ boosted approximately by 41%, when the shapes of the nanoparticles were changed from spherical to the lamina. Further, different shapes of the nanoparticles are given in Figure 18.

Figure 18. Shape factors of various nanoparticles [46]. (a) Sphere (s = 3.0); (b) Hexahedron (s = 3.7221); (c) Tetrahedron (s = 4.0613); (d) Column (s = 6.3698); (e) Lamina (s = 16.1576).

5. Conclusions

This study discussed the main topic of the tangent hyperbolic THNF flow across a rough-yawed cylinder caused by impulsive motion in a mixed convection mechanism with periodic MHD. The following are the key insights from this study:

• The fluid velocity decelerates, and chord and spanwise friction coefficients impulsively accelerate for the T-H THNF compared to Newtonian THNF (i.e. $\mathrm{We}$ enhances from 0 to 2).

• Notably, the highest percentage enhancement, approximately 241% and 148% in the chord and spanwise friction coefficients are observed at $\xi =0.23$ and $\xi =0.01$ respectively when $\mathrm{We}$ enhanced from 0 to 1.

• Including the third component (i.e. MoS₂) of the THNF decays the $F(\xi ,\eta )$.

• The $R{e}_x^{1/2}{C}_f$ augments for the presence of periodic magnetic field (i.e. non-zero M) in comparison to the case of absence of the magnetic field (M = 0).

• A change in the M from 0 to 2 improves the $R{e}_x^{1/2}{C}_f$ 19% and decays the $R{e}_x^{1/2}{\overline{C}}_f$ 50% approximately at $\xi =0.55$ in the THNF.

• The $R{e}_x^{1/2}{C}_f$ increases to a maximum of 288% for the T-H and 101% for the Newtonian THNF as the $\epsilon$ increases from 0.001 to 0.01 at $\xi =0.31$ and frequency $n=15$.

• The fluid velocity and skin friction coefficient are observed to increase in the chordwise direction when the yaw angle $\theta$ rises.

• It is noticed that at $\xi =0.86$, $m=0.5$ the chord and spanwise friction coefficients enhance and decay by approximately 38% and 69%, respectively, when $\theta$ improves from 15° to 45°.

• The chordwise velocity distribution and the friction coefficients along both directions augment the emerging higher values of $m$.

• The $R{e}_x^{-1/2}\mathrm{Nu}$ found to upsurge approximately by 15% for the T-H ternary THNF than that of only T-H base fluid.

• The emerging values of s boost the $G(\xi ,\eta )$ and $R{e}^{-1/2}\mathrm{Nu}$, i.e. changing the shapes of nanoparticles from spherical to other different shapes enhances the $G(\xi ,\eta )$ and $R{e}^{-1/2}\mathrm{Nu}$.

Nomenclature

$\mathrm{Pr}$	=	Prandtl number
$\mathrm{Ri}$	=	Richardson number
$R{e}_x$	=	local Reynolds number
$k$	=	thermal conductivity
$m$	=	power-law index
$n$	=	non-dimensional frequency
$\mathrm{We}$	=	Weissenberg number
$F,S$	=	non-dimensional velocities in chord and spanwise directions
$g$	=	acceleration due to gravity
$M$	=	periodic MHD parameter
$\mathrm{Ec}$	=	Eckert number
$G$	=	non-dimensional temperature
$T$	=	fluid temperature
$t$	=	time
$s$	=	nanoparticle's shape factor
$f$	=	non-dimensional stream function
$u_e\mathrm{and}{w}_e$	=	free stream velocities in $x\mathrm{and}z$ directions

Greek symbols

$\beta$	=	thermal expansion coefficient
${\phi }_1,{\phi }_2,{\phi }_3$	=	volume fractions of nanoparticles
$\psi$	=	stream function
$\mathrm{\Gamma }$	=	time constant
$\theta$	=	yaw angle
$\xi ,\eta$	=	transformed variables
$\epsilon$	=	roughness parameter
$\tau$	=	non-dimensional time
$\omega$	=	frequency
$\mu$	=	dynamic viscosity
$\nu$	=	kinematic viscosity
$\sigma$	=	electrical conductivity
$\rho$	=	density

Abbreviations

T-H	=	tangent hyperbolic
HNF	=	hybrid nanofluid
THNF	=	ternary hybrid nanofluid

Subscripts

$\xi ,\eta$	=	partial derivatives with respect to $\xi ,\eta$
$w,\mathrm{\infty }$	=	at the wall and ambient fluid, respectively
f	=	properties of the T-H fluid
nf	=	properties of T-H nanofluid
hnf	=	properties of T-H two-components hybrid nanofluid
$\mathrm{thnf}$	=	properties of T-H ternary hybrid nanofluid
i	=	initial condition

Disclosure statement

No potential conflict of interest was reported by the author(s).