Comment on Rasool et al. Spectral Relaxation Methodology for Chemical and Bioconvection Processes for Cross Nanofluid Flowing Around an Oblique Cylinder with a Slanted Magnetic Field Effect. Coatings 2022, 12, 1560

1. First Error

In the present comment, the units presented in the Nomenclature section in [1] are used.

The momentum Equation (2) in [1] is as follows:

$$\begin{array}{l}{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}}+(u{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}+\upsilon {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }r}})u={\mu }_f{\displaystyle \frac{1}{\left(1+{\tilde{\gamma }}^n{\left(\frac{\mathsf{\partial }u}{\mathsf{\partial }r}\right)}^n\right)}}-{\displaystyle \frac{\sigma {\sin }^2(\omega )B_0^2}{{\rho }_f}}u+{\displaystyle \frac{g}{{\rho }_f}}\left({\rho }_f{\beta }_f(1-C_{\infty })(T-T_{\infty })\right)\\ +({\rho }_p-{\rho }_f)(C-C_{\infty })-{\gamma }^{*}({\rho }_{\mathrm{m}}-{\rho }_f)(N-N_{\infty })\cos \mathsf{\Omega }-{\mu }_f{\tilde{\gamma }}^n{\left({\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }r}}\right)}^n{\displaystyle \frac{{\mathsf{\partial }}^{2}u}{\mathsf{\partial }{r}^2}}{\left(1+{\tilde{\gamma }}^n{\left({\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }r}}\right)}^n\right)}^{-2}\end{array}$$

(2)

In Equation (2), there are five errors:

In the Nomenclature section in [1], the units of the power law index n are $N·{\mathrm{m}}^{-2}=\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$. It is well known in mathematics that the term ${\mathsf{\Lambda }}^n$ is correct only when the power n is dimensionless. For example, the term ${\mathsf{\Lambda }}^{\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}}$ is meaningless. Thus, the three terms ${\tilde{\gamma }}^n{\left({\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }r}}\right)}^n$ in Equation (2) are wrong.

The term $\left(1-C_{\infty }\right)$ is wrong because the pure number 1 is dimensionless, whereas the units of concentration $C_{\infty }$ are ${\mathrm{m}}^{-3}$. In physics, it is not permitted to add quantities with different units.

The term $({\rho }_p-{\rho }_f)(C-C_{\infty })$ is wrong because its units are $\mathrm{kg}{\mathrm{m}}^{-6}$ instead of $\mathrm{m}{\mathrm{s}}^{-2}$.

2. Second Error

The energy Equation (3) in [1] is as follows:

$$\frac{\partial T}{\partial t}+\left(u\frac{\partial }{\partial x}+v\frac{\partial }{\partial r}\right)u=\frac{1}{\left(\rho c_p\right)}\frac{1}{r}\left(\frac{\partial }{\partial r}\left(K_f\frac{\partial T}{\partial r}\right)\right)+j\left(D\frac{\partial T}{\partial r}\frac{\partial C}{\partial r}+\frac{\overline{D}}{T_{\infty }}{\left(\frac{\partial T}{\partial r}\right)}^2\right)$$

(3)

In Equation (3), there are two errors:

The units of the term ${\displaystyle \frac{1}{\left(\rho c_p\right)}}{\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }r}}\left(K_f{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }r}}\right)\right)$ are ${\mathrm{m}}^{-1}{\mathrm{s}}^{-1}\mathrm{Kelvin}$ instead of ${\mathrm{s}}^{-1}\mathrm{Kelvin}$.

The units of the term $D{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }r}}{\displaystyle \frac{\mathsf{\partial }C}{\mathsf{\partial }r}}$ are ${\mathrm{m}}^{-3}{\mathrm{s}}^{-1}\mathrm{Kelvin}$, whereas the units of the term ${\displaystyle \frac{\overline{D}}{T_{\infty }}}{\left({\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }r}}\right)}^2$ are ${\mathrm{s}}^{-1}\mathrm{Kelvin}$. In physics, it is not permitted to add quantities with different units.

3. Third Error

The concentration Equation (4) in [1] is as follows:

$$\frac{\partial C}{\partial t}+\left(u\frac{\partial }{\partial x}+v\frac{\partial }{\partial r}\right)C=\frac{D}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C}{\partial r}\right)+\frac{\overline{D}}{T_{\infty }}\frac{1}{r}\left(r\frac{\partial T}{\partial r}\right)-K_r^2\left(C-C_{\infty }\right){\left(\frac{\partial T}{\partial r}\right)}^m{e}^{-\frac{E_0}{K^{*}T}}$$

(4)

In Equation (4), there are two errors:

The units of the term ${\displaystyle \frac{\overline{D}}{T_{\infty }}}{\displaystyle \frac{1}{r}}\left(r{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }r}}\right)$ are $\mathrm{m}{\mathrm{s}}^{-1}$ instead of ${\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$.

The units of the term $K_r^2(C-C_{\infty }){\left({\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }r}}\right)}^{\mathrm{m}}{e}^{-{\displaystyle \frac{E_0}{K^{*}T}}}$ are ${\mathrm{m}}^{-\mathrm{m}-3}{\mathrm{s}}^{-2}{\mathrm{Kelvin}}^{\mathrm{m}}$ instead of ${\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$.

4. Fourth Error

In Equation (6) in [1], we have the followings:

$$D{\displaystyle \frac{\mathsf{\partial }C}{\mathsf{\partial }r}}+{\displaystyle \frac{\overline{D}}{T_{\infty }}}{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }r}}=0$$

The units of the term $D{\displaystyle \frac{\mathsf{\partial }C}{\mathsf{\partial }r}}$ are ${\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, whereas the unit of the term ${\displaystyle \frac{\overline{D}}{T_{\infty }}}{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }r}}$ is $\mathrm{m}{\mathrm{s}}^{-1}$, and Equation (6) is wrong.

5. Fifth Error

The dimensionless Weissenberg number $We={\displaystyle \frac{a^3{x}^2{r}^2{\tilde{\gamma }}^2}{{(1-\xi t)}^3{R}^2{\nu }_f}}$ is wrong because it is dimensional, with units ${\mathrm{kg}}^2{\mathrm{m}}^{-2}{\mathrm{s}}^{-6}$.

6. Sixth Error

The dimensionless parameter $R_i={\displaystyle \frac{g{\beta }_f(1-C_{\infty })(T_w-T_{\infty })x}{u_w^2}}$ is wrong because the term $(1-C_{\infty })$ is wrong.

7. Seventh Error

The dimensionless parameter $N_r={\displaystyle \frac{({\rho }_p-{\rho }_f)C_{\infty }}{{\beta }_f{\rho }_f(1-C_{\infty })(T_w-T_{\infty })}}$ is wrong because the term $(1-C_{\infty })$ is wrong.

8. Eighth Error

The dimensionless parameter $R_b={\displaystyle \frac{{\gamma }^{*}({\rho }_p-{\rho }_f)}{{\beta }_f{\rho }_f(1-C_{\infty })(T_w-T_{\infty })}}$ is wrong because the term $(1-C_{\infty })$ is wrong.

9. Ninth Error

The dimensionless parameter $N_b={\displaystyle \frac{jD{C}_{\infty }}{{\nu }_f}}$ is wrong because it is dimensional, with the unit ${\mathrm{m}}^{-3}$.

10. Tenth Error

The equation $\gamma ={\left({\displaystyle \frac{(1-\xi t){\nu }_f}{a{R}^2}}\right)}^{1/2}$ is wrong because the unit of the LHS is ${\mathrm{m}}^{-1}$(Nomenclature), whereas the RHS is dimensionless.

11. Eleventh Error

The equation $A={\displaystyle \frac{\xi }{a}}$ is wrong because the unit of the LHS is $\mathrm{m}{\mathrm{s}}^{-1}$(Nomenclature), whereas the RHS is dimensionless.

12. Twelfth Error

The equation ${\tau }_w={\mu }_f{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }r}}{\left(1+\tilde{\gamma }{\left({\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }r}}\right)}^2\right)}^{{\displaystyle \frac{n-1}{2}}}$ is wrong because the pure number 1 is dimensionless, whereas the units of the term $\tilde{\gamma }{\left({\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }r}}\right)}^2$ are $\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-4}$.

13. Thirteenth Error

In the dimensionless Equation (12) in [1], the equation $\rho =1$ is wrong because $\rho$ is dimensional.

14. Fourteenth Error

The parameter $\epsilon$ in Figures 10 and 16 in [1] is unknown.

15. Fifteenth Error

In Table 2 in [1], the parameters $N,\mathsf{\Gamma }$ are unknown.