Comment on the paper “rheology of magnetohydrodynamic viscoelastic fluid flow and heat transfer during the blade coating process with blade slip,polymers and polymer composites,vol. 32,1–14,2024”

First error

In equation (2) in 1 the magnetic term $-\sigma B_0^2\overline{u}$ is wrong because its units are $kg{m}^{-2}\hspace{0.5em}{\sec }^{-2}$ instead of $m\hspace{0.5em}{\sec }^{-2}$ . The correct magnetic term is $-\frac{\sigma B_0^2}{\rho }\overline{u}$ .

Second error

In Figure 1 in 1 it is clearly shown that the velocity $\overline{u}$ is horizontal and the velocity $\overline{v}$ is vertical and the fluid moves in the horizontal direction. It is also written that the magnetic field is applied in the normal direction of the flow. This means that the magnetic field acts perpendicular to $\overline{u}$ velocity and parallel to velocity $\overline{v}$ . It is well known in magnetohydrodynamics (MHD) that the Lorentz force, which retards the flow, appears when the magnetic field acts perpendicular to flow (equation (5.32, page 151) in 2). This means that the magnetic term $-\frac{\sigma B_0^2}{\rho }\overline{v}$ in equation (3) in 1 must be zero. However this error has no effect on the results because the two-dimensional flow has been changed into one-dimensional and the equation (3) was not used.

Third error

In the two-dimensiomal energy equation (4) appear the following typographical errors:

The term $\left(\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}+6\frac{\partial \overline{u}}{\partial \overline{y}}\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}\right)\left(\frac{\partial \overline{u}}{\partial \overline{x}}\right)$ is wrong because the units of $\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}$ are $m^{-1}\hspace{0.5em}{\sec }^{-1}$ whereas the units of the term $6\frac{\partial \overline{u}}{\partial \overline{y}}\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}$ are $m^{-1}\hspace{0.5em}{\sec }^{-2}$ . In Physics it is not allowed to add quantities with different units.

The term $\left(\frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}\right)\left(\frac{\partial \overline{u}}{\partial \overline{x}}\right)$ is wrong because its units are $m^{-1}\hspace{0.5em}{\sec }^{-2}$ instead of $kg{m}^{-1}\hspace{0.5em}{\sec }^{-3}$ .

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 $m^{-1}\hspace{0.5em}{\sec }^{-3}$ instead of $kg{m}^{-1}\hspace{0.5em}{\sec }^{-3}$ .

The term $\frac{{\alpha }_1}{\rho }\left(\frac{\partial \overline{v}}{\partial \overline{y}}13\frac{{\partial }^2\overline{v}}{\partial {\overline{y}}^2}+\overline{v}\frac{{\partial }^3\overline{v}}{\partial {\overline{y}}^3}+2\frac{{\alpha }_2}{\rho }\frac{\partial \overline{v}}{\partial \overline{y}}\frac{{\partial }^2\overline{v}}{\partial {\overline{y}}^2}\right)$ is wrong because the units of $\overline{v}\frac{{\partial }^3\overline{v}}{\partial {\overline{y}}^3}$ are $m^{-1}\hspace{0.5em}{\sec }^{-2}$ 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 $m\hspace{0.5em}{\sec }^{-2}$ .

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 $m\hspace{0.5em}{\sec }^{-1}$ instead of $kg{m}^{-1}\hspace{0.5em}{\sec }^{-3}$ .

Between equations (8) and (9) in 1 it is written that the flow is considered as one-dimensional between two parallel plates. In the one-dimensional flow the vertical velocity $\overline{v}$ is zero and the temperature T is independent of $\overline{x}$ which means that $\frac{\partial T}{\partial \overline{x}}=0$ . Taking into account these facts all the above typographical errors disappear from the energy equation (4) and its final form is as follows

$\frac{d\overline{u}}{d\overline{y}}\left(\mu \frac{d\overline{u}}{d\overline{y}}+2\left({\beta }_2+{\beta }_3\right){\left(\frac{d\overline{u}}{d\overline{y}}\right)}^3\right)+k\frac{d^{2}T}{d{\overline{y}}^2}=0$

(1)

With changing the flow from two-dimensional to one-dimensional the partial differential equation (4) has been changed into ordinary differential equation (1). The above dimensional energy equation (1) is absent from. ¹ However the above equation (1) is important because, from this equation, was derived the dimensionless equation (11) in 1.

Fourth 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 $meters\left(length\right)$ .This dimensional parameter appears in the dimensionless equation (12) in 1 and in the figures. However in a dimensionless equation all terms must be dimensionless and for that reason the equation (12) in 1 is wrong. The parameter $\alpha$ in the dimensionless equation (12) in 1 must be replaced by a new dimensionless parameter $\lambda =\frac{\alpha }{H_0}$ .

Fifth error

In figures 11, 12, 13 and 14 in 1 appears the vertical velocity $v$ . However the flow is horizontal and $v$ must be replaced by $u$ .

Sixth 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.