Comment on Khan et al. Impact of Thermal Radiation on Magnetohydrodynamic Unsteady Thin Film Flow of Sisko Fluid over a Stretching Surface. Processes 2019, 7, 369

Abstract

Many errors exist in the above paper.

1. Brief Description

In the above paper, the most important equation is the momentum Equation (17) describing the flow along the stretching surface. In a physics equation, all terms must have the same units and this is not the case in this equation. Thus, the velocity results, produced by this equation, are not correct. For the same reason, the skin friction equation is not correct and thus the skin friction results, as presented in the paper, are not correct.

2. First Error

Equation (14) in [1] is as follows

$$U(x,t)={\displaystyle \frac{cx}{1-bt}}$$

(1)

In a physics equation, all terms must have the same units. Equation (1) in units is

$$U(m{\sec }^{-1})={\displaystyle \frac{c({\sec }^{-1})x(m)}{1-b({\sec }^{-1})t(\sec )}}$$

$$m{\sec }^{-1}=m{\sec }^{-1}$$

It is clear that the units of b and c are sec⁻¹.

In the basic Equation (17) in [1], the units of the term ${\displaystyle \frac{b}{\rho }}{\displaystyle \frac{\partial }{\partial y}}{\left(-u_y\right)}^n$ are:

$${\displaystyle \frac{b({\sec }^{-1})}{\rho ({\mathit{kgm}}^{-3})}}{\displaystyle \frac{\partial }{\partial y(m)}}{\left(-u_y({\sec }^{-1})\right)}^n={\mathit{kg}}^{-1}{m}^2{\sec }^{-n-1}$$

However, the correct units of each term in Equation (17) are $m{\sec }^{-2}$. In a physics equation, all terms must have the same units and for that reason Equation (17) in [1] is wrong.

3. Second Error

Equation (32) in [1] is as follows

$$S_{xy}={\mu }_0\left(a+b{\left|u_y\right|}^n{u}_y\right)y=0$$

(2)

There is a typographical error and the correct form of Equation (2) is

$$S_{xy}={\mu }_0\left(a+b{\left|u_y\right|}^n{u}_y\right)y$$

(3)

In Equation (3) there are two essential errors. The parameter ${\mu }_0$ is unknown. In addition, the units of $a$ are ${\mathit{kgm}}^{-1}{\sec }^{-1}$ whereas the units of $b{\left|u_y\right|}^n{u}_y$ are ${\sec }^{-2-n}$ and in physics it is not permitted to add quantities with different units.

4. Third Error

The dimensionless stretching parameter is defined between Equations (30) and (31) in [1] as $\xi ={\displaystyle \frac{b}{\rho \nu }}{\left({\left({\displaystyle \frac{cx}{(1-bt)\sqrt{\nu }}}\right)}^{3/2}\right)}^{n-1}$. Its units are:

$$\xi ={\displaystyle \frac{b({\sec }^{-1})}{\rho ({\mathit{kgm}}^{-3})\nu (m^2{\sec }^{-1})}}{\left({\left({\displaystyle \frac{c({\sec }^{-1})x(m)}{(1-b({\sec }^{-1})t(\sec ))\sqrt{\nu (m^2{\sec }^{-1})}}}\right)}^{3/2}\right)}^{n-1}={\mathit{kg}}^{-1}m{\sec }^{{\displaystyle \frac{-3(n-1)}{4}}}$$

The stretching parameter $\xi$ is wrong because it is dimensional with units ${\mathit{kg}}^{-1}m{\sec }^{{\displaystyle \frac{-3(n-1)}{4}}}$.

5. Fourth Error

Between Equations (33) and (34) in [1] is defined the Nusselt number as $Nu={\displaystyle \frac{\delta Q_w}{k(T-T_{\delta })}}$. However, the parameter $\delta$ does not exist in [1].

6. Fifth Error

In Equation (29) in [1], the temperature boundary constrain is ${\theta }^{\prime }(\beta )={\displaystyle \frac{\partial \theta }{\partial \eta }}(\beta )=0$. However, in Figure 4 in [1], this condition is violated because ${\theta }^{\prime }(0.9)={\displaystyle \frac{\partial \theta }{\partial \eta }}(0.9)\ne 0$.

7. Sixth Error

In the basic Equation (17) in [1], the units of the term $\sigma B^{2}u$ are:

$$\sigma (m^{-1}{\mathsf{\Omega }}^{-1})B^2{(tesla={\mathit{kg}}^{1/2}{m}^{-1}{\sec }^{-1/2}{\mathsf{\Omega }}^{1/2})}^{2}u(m{\sec }^{-1})={\mathit{kgm}}^{-2}{\sec }^{-2}$$

The term $\sigma B^{2}u$ is wrong because its units are ${\mathit{kgm}}^{-2}{\sec }^{-2}$ instead of $m{\sec }^{-2}$. The correct term must be ${\displaystyle \frac{\sigma B^{2}u}{\rho }}$.

8. Seventh Error

In Equation (23) in [1], the units of the term ${\displaystyle \frac{1}{\rho C_p}}\left({\displaystyle \frac{16T_{\infty }^{*}{\sigma }^{*}{T}_{yy}^4}{3k^{*}}}\right)$ are:

$${\displaystyle \frac{1}{\rho ({\mathit{kgm}}^{-3})C_p(m^2{\sec }^{-2}{K}^{-1})}}\left({\displaystyle \frac{16T_{\infty }^{*}(K){\sigma }^{*}(\mathit{kg}{\sec }^{-3}{K}^{-4})T_{yy}^4(K^4{m}^{-2})}{3k^{*}(m^{-1})}}\right)=K^2{\sec }^{-1}$$

The term is wrong because its units are ${\mathit{Kelvin}}^2{\sec }^{-1}$ instead of $\mathit{Kelvin}{\sec }^{-1}$. The correct term is

$${\displaystyle \frac{1}{\rho C_p}}\left({\displaystyle \frac{16T_{\infty }^{*}{\sigma }^{*}{T}_{yy}^3}{3k^{*}}}\right)$$

9. Eighth Error

In Equation (33) in [1], the exponent m appears, but this exponent does not exist in the paper. This exponent must be n.

10. Ninth Error

Between Equations (29) and (30) in [1], a dimensionless film appears with the thickness $\beta$. The units of the dimensionless unsteadiness parameter $St={\displaystyle \frac{\beta }{c}}$ are:

$$St={\displaystyle \frac{\beta (\dim ensionless)}{c({\sec }^{-1})}}=\sec$$

The parameter $St$ is wrong because it is dimensional with units $\sec$. The correct form is $St={\displaystyle \frac{b}{c}}$.

11. Conclusions

The last four errors are typographical, whereas the other errors are serious errors.