Corrigendum to “Rheology of magnetohydrodynamic viscoelastic fluid flow and heat transfer during the blade coating process with blade slip”

Abbas Z, Javed M, Hanif A, Rafiq MY, Khaliq S. Rheology of magnetohydrodynamic viscoelastic fluid flow and heat transfer during the blade coating process with blade slip. Polymers and Polymer Composites, 2024; 32: 1-14. doi:10.1177/09673911241273

This Corrigendum is regarding the comments and suggestions by Prof. Asterios Pantokratoras on the paper “Rheology of magnetohydrodynamic viscoelastic fluid flow and heat transfer during the blade coating process with blade slip” [Polymers and Polymer Composites, 2024; 32: 1-14]”, to whom we like to share our appreciation. In the subsequent paragraphs, first, we list the corrections made in accordance with his comments. Finally, for the sake of completeness, we include his original comments.

(Error ¹).

Typos in Equation (2), the magnetic term should be $\frac{\sigma }{\rho }{B_0}^2\overline{u}$ instead of $\sigma {B_0}^2\overline{u}$ .

(Error ²).

Typos in Equation (3), the magnetic term should be $\frac{\sigma }{\rho }{B_0}^2\overline{v}$ instead of $\sigma {B_0}^2\overline{v}$ .

(Error ³, Error ⁴, Error ⁵, Error ⁶, and Error ⁷).

The correct form of Equation (4) is: $\begin{array}{l}\rho C_p\left(\overline{u}\frac{\partial T}{\partial \overline{x}}+\overline{v}\frac{\partial T}{\partial \overline{y}}\right)=k\left(\frac{{\partial }^{2}T}{\partial {\overline{x}}^2}+\frac{{\partial }^{2}T}{\partial {\overline{y}}^2}\right)+\mu \left(2\frac{{\partial }^2\overline{u}}{\partial {\overline{x}}^2}+2\frac{{\partial }^2\overline{v}}{\partial {\overline{y}}^2}+\frac{{\partial }^2\overline{v}}{\partial {\overline{x}}^2}+\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}+2\frac{\partial \overline{u}}{\partial \overline{y}}\frac{\partial \overline{v}}{\partial \overline{x}}\right)\\ +{\alpha }_1\left(4\left(u\frac{{\partial }^2\overline{u}}{\partial {\overline{x}}^2}+v\frac{{\partial }^2\overline{u}}{\partial \overline{x}\overline{y}}\right)\frac{\partial \overline{u}}{\partial \overline{x}}+u\left(\frac{\partial \overline{v}}{\partial \overline{x}}+\frac{\partial \overline{u}}{\partial \overline{y}}\right)\left(\frac{{\partial }^2\overline{u}}{\partial \overline{x}\overline{y}}+\frac{{\partial }^2\overline{v}}{\partial {\overline{x}}^2}\right)+v\left(\frac{\partial \overline{v}}{\partial \overline{x}}+\frac{\partial \overline{u}}{\partial \overline{y}}\right)\left(\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}-\frac{{\partial }^2\overline{u}}{\partial {\overline{x}}^2}\right)\right)\\ +2{\beta }_3\left(16\frac{{\partial }^4\overline{u}}{\partial {\overline{x}}^4}+8\frac{{\partial }^2\overline{u}}{\partial {\overline{x}}^2}\left(\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}+\frac{{\partial }^2\overline{v}}{\partial {\overline{x}}^2}+2\frac{\partial \overline{u}}{\partial \overline{y}}\frac{\partial \overline{v}}{\partial \overline{x}}\right)+4\frac{{\partial }^3\overline{u}}{\partial {\overline{y}}^3}\frac{\partial \overline{v}}{\partial \overline{x}}+4\frac{{\partial }^3\overline{v}}{\partial {\overline{x}}^3}\frac{\partial \overline{u}}{\partial \overline{y}}+6\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}\frac{{\partial }^2\overline{v}}{\partial {\overline{x}}^2}+\frac{{\partial }^4\overline{u}}{\partial {\overline{y}}^4}+\frac{{\partial }^4\overline{v}}{\partial {\overline{x}}^4}\right)\end{array}$ .

(Error ⁸).

In Equation (5), the slip parameter (dimensional) should be signified by $\overline{\alpha }$ . Also, in Equation (8), a term $\alpha =\frac{\overline{\alpha }}{H_0}$ (where $\alpha$ is a dimensionless slip parameter) should be added to avoid confusion.

(Error ⁹).

The temperature in equation (17) depends on Br (Brinkman number), $\alpha$ (Slip parameter), k (normalized coating thickness) and m (magnetic parameter). We have shown some effects in the form of graphs as the total number of figures already reached 18.

Original Comments/Suggestions

School of Engineering, Democritus University of Thrace, 67100 Xanthi – Greece email:[e-mail].

Some errors exist in the above paper.

First error

In equation (2) in [1] the magnetic term $\sigma B_0^2\overline{u}$ is wrong because its units are $kg{m}^{-2}\hspace{0.17em}{\sec }^{-2}$ instead of $m\hspace{0.17em}{\sec }^{-2}$ .

Second error

In equation (3) in [1] the magnetic term $\sigma B_0^2\overline{u}$ is wrong because its units are $kg{m}^{-2}\hspace{0.17em}{\sec }^{-2}$ instead of $m\hspace{0.17em}{\sec }^{-2}$ .

Third error

In equation (4) in [1] appears the term $\left(\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}+6\frac{\partial \overline{u}}{\partial \overline{y}}\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}\right)\left(\frac{\partial \overline{u}}{\partial \overline{x}}\right)$ . The units of $\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}$ are $m^{-1}\hspace{0.17em}{\sec }^{-1}$

whereas the units of the term $6\frac{\partial \overline{u}}{\partial \overline{y}}\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}$ are $m^{-1}\hspace{0.17em}{\sec }^{-2}$ . In Physics it is not allowed to add quantities with different units.

Fourth error

In equation (4) in [1] the term $\left(\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}\right)\left(\frac{\partial \overline{u}}{\partial \overline{y}}\right)$ is wrong because its units are $m^{-1}\hspace{0.17em}{\sec }^{-2}$ instead of $kg{m}^{-1}\hspace{0.17em}{\sec }^{-3}$ .

Fifth error

In equation (4) in [1] the term $\left(6\frac{\partial \overline{u}}{\partial \overline{y}}\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}\right)\left(\frac{\partial \overline{u}}{\partial \overline{x}}\right)$ is wrong because its units are $m^{-1}\hspace{0.17em}{\sec }^{-3}$ instead of $kg{m}^{-1}\hspace{0.17em}{\sec }^{-3}$ .

Sixth error

In equation (4) in [1] appears the term $\frac{{\alpha }_1}{\rho }\left(\frac{\partial \overline{v}}{\partial \overline{y}}13\frac{{\partial }^2\overline{v}}{\partial {\overline{y}}^2}+\overline{v}\frac{{\partial }^3\overline{v}}{\partial {\overline{y}}^3}+2\frac{{\alpha }_2}{\rho }\frac{\partial \overline{v}}{\partial \overline{y}}\frac{{\partial }^2\overline{v}}{\partial {\overline{y}}^2}\right)$ . The units of $\overline{v}\frac{{\partial }^3\overline{v}}{\partial {\overline{y}}^3}$ are $m^{-1}\hspace{0.17em}{\sec }^{-2}$ whereas the units of the term $2\frac{{\alpha }_2}{\rho }\frac{\partial \overline{v}}{\partial \overline{y}}\frac{{\partial }^2\overline{v}}{\partial {\overline{y}}^2}$ are $m\hspace{0.17em}{\sec }^{-2}$ . In Physics it is not allowed to add quantities with different units.

Seventh error

In equation (4) in [1] the term $\frac{{\beta }_3}{\rho }\left(40\frac{\partial \overline{v}}{\partial \overline{y}}\frac{{\partial }^2\overline{v}}{\partial {\overline{y}}^2}\right)$ is wrong because its units are $m\hspace{0.17em}{\sec }^{-2}$ instead of $kg{m}^{-1}\hspace{0.17em}{\sec }^{-3}$ .

Eighth error

In a Physics equation all terms must have the same units and from equation (5) in [1] it is found that the units of parameter $\alpha$ are meters(length). This dimensional parameter appears in the dimensionless equation (12) in [1]. However in a dimensionless equation all terms must be dimensionless and for that reason the equation (12) in [1] is wrong.

Ninth error

In the dimensionless equation (17) in [1] the dimensionless temperature $\theta$ depends on the Brinkman number Br. However, in the temperature results the Brinkman number is absent.