Comment on Ghali et al. Mathematical Entropy Analysis of Natural Convection of MWCNT—Fe₃O₄/Water Hybrid Nanofluid with Parallel Magnetic Field via Galerkin Finite Element Process. Symmetry 2022, 14, 2312

Abstract

A serious error and many typographical errors exist in the above paper.

• Serious Error

The horizontal and vertical momentum Equations (2) and (3) in [1] are as follows

$$u{\displaystyle \frac{\partial u}{\partial x}}+\upsilon {\displaystyle \frac{\partial u}{\partial y}}=-{\displaystyle \frac{1}{{\rho }_{hnf}}}{\displaystyle \frac{\partial P}{\partial x}}+{\nu }_{hnf}\left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}\right)$$

(1)

$$u{\displaystyle \frac{\partial \upsilon }{\partial x}}+\upsilon {\displaystyle \frac{\partial \upsilon }{\partial y}}=-{\displaystyle \frac{1}{{\rho }_{hnf}}}{\displaystyle \frac{\partial P}{\partial y}}+{\nu }_{hnf}\left({\displaystyle \frac{{\partial }^2\upsilon }{\partial x^2}}+{\displaystyle \frac{{\partial }^2\upsilon }{\partial y^2}}\right)+{\beta }_{hnf}g(T-T_{avg})+{\displaystyle \frac{{\sigma }_{hnf}}{{\rho }_{hnf}}}{B}_0^2\upsilon$$

(2)

where $u,\upsilon$ are the horizontal and vertical fluid velocities and $B_0$ is the magnetic flux density. In Figure 1a in [1] it is shown that the magnetic field is parallel to the horizontal x-axis and perpendicular to the vertical y-axis. This is also mentioned in the title in [1]. In order to check the correctness of the magnetic force in Equations (1) and (2) we present the following momentum equations from reference [2]. The horizontal and vertical momentum equations in [2] are as follows

$$\rho u{\displaystyle \frac{\partial u}{\partial x}}=-{\displaystyle \frac{\partial p}{\partial x}}+{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{4}{3}}\mu {\displaystyle \frac{\partial u}{\partial x}}\right)-\sigma (E+uB\sin {\chi }_b-\upsilon Bcos{\chi }_b)B\sin {\chi }_b$$

(3)

$$\rho u{\displaystyle \frac{\partial \upsilon }{\partial x}}=-{\displaystyle \frac{\partial p}{\partial y}}+{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{4}{3}}\mu {\displaystyle \frac{\partial \upsilon }{\partial x}}\right)+\sigma (E+uB\sin {\chi }_b-\upsilon Bcos{\chi }_b)B\cos {\chi }_b$$

(4)

In Equations (3) and (4) the magnetic field is inclined and ${\chi }_b$ is the angle between the magnetic field and horizontal axis $x$ (Figure 1). If the magnetic field in Equations (3) and (4) becomes horizontal, as happens in Equations (1) and (2), the angle ${\chi }_b$ is zero, the $\sin {\chi }_b$ is zero and the $\cos {\chi }_b$ is equal to 1. The electric field $E$ in [1] is zero. Then, the magnetic term in Equation (3) is $-\sigma (uB\sin {\chi }_b-\upsilon Bcos{\chi }_b)B\sin 0=0$ and the magnetic term in Equation (4) is $+\sigma (uB\sin {\chi }_b-\upsilon Bcos{\chi }_b)B\cos {\chi }_b=+\sigma (uB\sin 0-\upsilon Bcos0)B\cos 0=-\sigma (\upsilon B)B=-\sigma B^2\upsilon$. It is clear that the magnetic term in the vertical momentum in Equation (4) is $-\sigma B^2\upsilon$ whereas the magnetic term in the vertical momentum in Equation (2) is $+{\displaystyle \frac{{\sigma }_{hnf}}{{\rho }_{hnf}}}{B}_0^2\upsilon$; that is, it is positive instead of negative.

Another argument that the magnetic term in Equation (2) must be negative is included in the NACA technical report 3971 ([3], Equation (47)). The Equation (47) in NACA [3] is as follows

$$u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}=\nu {\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}-{\displaystyle \frac{\sigma B_0^2}{\rho }}(u-u_{\infty })$$

(5)

where the velocity $u$ is horizontal and the magnetic field acts transversely to $u$. In the above Equation (5), $u_{\infty }$ is the free stream velocity which is zero in [1].

The correct negative term is shown also in Equation ((5.32), page 151) in [4] as follows

$$-{\displaystyle \frac{\partial p}{\partial x}}+\rho \nu {\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}-\sigma B^{2}u=0$$

(6)

The wrong positive term $+{\displaystyle \frac{{\sigma }_{hnf}}{{\rho }_{hnf}}}{B}_0^2\upsilon$ in Equation (2) appears also in Equation (19) in [1] and as $+{\displaystyle \frac{{\sigma }_f}{{\sigma }_{hnf}}}{\displaystyle \frac{{\rho }_f}{{\rho }_{hnf}}}{\displaystyle \frac{\mathrm{Pr}{H}_a^2}{\epsilon \sqrt{Ra}}}V$ in the transformed Equation (28) in [1] where Ha is the dimensionless Hartmann number which represents the magnetic field.

• Typographical Errors

• In the nomenclature it is written that U,V are dimensional velocity components. However, U,V are dimensionless.
• In the nomenclature it is written that the units of dynamic viscosity $\mu$ are $W·m^{-1}·K^{-1}=kg·m·{\mathrm{s}}^{-3}·K^{-1}$. However, the units of $\mu$ are $kg·m^{-1}·{\mathrm{s}}^{-1}$.
• In the nomenclature it is written that the units of electrical conductivity $\sigma$ are $\Omega ·M$. However, the units of $\sigma$ are ${\Omega }^{-1}·m^{-1}={(Ohm)}^{-1}{(length)}^{-1}$.
• In the nomenclature it is written that $\lambda$ is the length of the baffle. However, in Figure 1 in [1] no baffle with length $\lambda$ appears.
• In the dimensional Equations (2), (3), (18) and (19) in [1] the pressure P is dimensional, whereas in Equation (21) in [1], the pressure P (same symbol) is dimensionless.
• In Equation (21) in [1] appears an unknown parameter $g_y$.
• Above Equation (29) in [1] it is written that “The following formula is the energy equation for normal convection inside a porous region, with σ the thermal capacity ratio”. However, no parameter σ exists in Equation (29) in [1]. In addition, σ is the electrical conductivity.
• Between Equations (33) and (34) in [1] are presented the boundary conditions concerning the dimensional velocities and temperatures. The corresponding boundary conditions for dimensionless quantities do not exist in [1].
• In the dimensionless Figures 3, 8, 12, 15 and 17, for the isotherms, the symbol T (Kelvin) is presented. The correct term is $\theta (\dim ensionless)$.
• The caption of Figure 5 says. “Variation of $N{u}_{avg}$ with Ra for different Da at Ha = 0, $\varphi =0.02$, and $\epsilon =0.4$”. However, in the figure there is a different $\varphi$, not a different Da.
• The caption of Figure 6 says. “Variation of $N{u}_{avg}$ with Ra for different $\varphi$ at Da = 0.01, Ha = 0, and $\epsilon =0.4$”. However, in the figure there is a different Da, not a different $\varphi$.
• In the caption of Figure 9 $\epsilon =0$ and on the figure $\epsilon =0.4$.
• In the caption of Figure 10 $\epsilon =0$ and on the figure $\epsilon =0.4$.
• In the caption of Figure 11 it is written that Ha = 0 whereas the Ha in the horizontal axis varies between 0 and 100.
• In Figure 28 the $\varphi$ varies between 0.02 and 0.08, whereas in the caption it is written that $\varphi =0.02$.
• In the problem description in [1] it is written that “NF is power-law non-Newtonian”. However, no NF power-law non-Newtonian exists in [1].
• The entropy generation is presented in Equation (36) in [1]. Above Equation (36) is written “Non-Dimensional Entropy Generation” and exactly below it is written “The dimensional local entropy produced is represented by equation (36)”. In Equation (36) $k_m,T_{avg},{\mu }_{nf},K,{\sigma }_{nf},B$ are dimensional and $U,V,X,Y,\theta$ are dimensionless. Therefore, Equation (36) in [1] is neither dimensional nor dimensionless.