School of Engineering, Democritus University of Thrace s1

Comment on the paper "MHD mixed convective heat transfer over a permeable stretching wedge with thermal radiation and ohmic heating, by Xiaohong Su, Liancun Zheng, Xinxin Zhang and Junhong Zhang, Chemical Engineering Science, Vol. 78, 2012, pp. 1-8."

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

In the above paper a new analytical method named DTM-BF is proposed and applied to study the MHD mixed convective flow and heat transfer of an incompressible, viscous and electrically conducting fluid over a stretching permeable wedge in the presence of thermal radiation and ohmic heating. The boundary layer governing equations of PDES are transformed into highly non linear coupled ODES and the approximate solutions are derived by the new proposed analytical technique—the differential transformation and basis functions method (DTM-BF) on unbounded domains. The effects of the velocity ratio parameter, magnetic field parameter, the wedge angle parameter, the mixed convection parameter, suction/injection parameter, thermal radiation parameter, Eckert number and Prandtl number are presented. Moreover, the reliability and effectiveness of the DTM-BF have been verified by comparing analytical results with numerical solutions. Some new characteristics of the mixed convective flow and heat transfer over a wedge are investigated and discussed. The partial differential equations of this problem, using the same symbols by Su et al. (2012) are

u v   0 x y (1) 2

u u dU  2u  B 2 u  v  U   g (T  T )sin  (u U ) x y dx y 2 0  2  (2)

T T dU  2T q c (u  v )  u(U  B 2U )  a  r  B 2u 2 p x y dx y 2 y (3) where u and v are the velocity components in the x and y- directions, U is the free stream velocity, υ is the fluid kinematic viscosity, g is the gravity acceleration, β0 is the thermal expansion coefficient, T is the fluid temperature, Ω is the wedge angle, σ is the electrical conductivity of the fluid, B is the strength of the magnetic field, ρ is the fluid density, cp is the specific heat, a is the fluid thermal conductivity and qr is the radiation heat flux. The energy equation (3) takes the following form

T T 1 dU 1 a  2T q B 2u 2 u  v   uU  B 2uU   r  2 (4) x y c p dx c p c p y c p y c p

1 Let us calculate the units of the term B 2uU . The units of specific heat c p are

1 2 2 c p  Kelvin sec m

The units of electrical conductivity are (Davidson, 2001, page 417)

   Ohm 1 (resis tan ce1 )m 1 (length 1 )

The unit of the magnetic field is Tesla. The definition of Tesla is

T   kg(mass)A1 (ampere 1 )sec2 (time 2 )

Eliminating ampere from the units we have

B  Ohm1/ 2m 1 sec 1/ 2 kg1/ 2 3

The units of velocity are

u  msec1

1 Therefore the units of the term B 2uU are c p

 1   B 2uU   m 3 (length 3 )kg(mass)sec 1 (time 1 )Kelvin(temperature) c p 

The units of all other terms in equation (4) are sec 1 Kelvin and this means 1 that the term B 2uU is wrong (in a dimensional equation all terms must c p have the same units). This means that the equation (3) is wrong and the presented results also wrong.

References

1. Xiaohong Su, Liancun Zheng, Xinxin Zhang and Junhong Zhang. MHD mixed convective heat transfer over a permeable stretching wedge with thermal radiation and ohmic heating, Chemical Engineering Science, Vol. 78, 2012, pp. 1-8.

2. Davidson, P.A. (2001). An Introduction to Magnetohydrodynamics, 1st ed., Cambridge University Press, Cambridge.