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Discussion on the paper " Fractional boundary layer flow and heat transfer over a stretching sheet with variable thickness, Lin Liu, Liancun Zheng, Yanping Chen, Fawang Liu, Journal of Heat Transfer, 2018, 140(9)"
Asterios Pantokratoras School of Engineering, Democritus University of Thrace, 67100 Xanthi – Greece e-mail:
[email protected]
Abstract The present comment concerns some doubtful results included in the above paper. In the above paper the momentum equation and the energy equation are as follows (equations 6 and 9 in [1]) 1u u u u 1u B( x) 2 u v 1 1 (1 1 ) u t x y ( y) 1 y
(1)
1T T T T 1T u v f 2 1 (1 2 ) t x y ( y ) 1 y
(2)
where is the fluid kinematic viscosity and f is the thermal diffusion coefficient. From equations (1) and (2) it is found that the units of kinematic viscosity and thermal diffusion coefficient are [ ] m 1 (length 1 ) sec 1 (time 1 )
[a f ] m 1 (length 1 ) sec 1 (time 1 )
in order that all terms in each equation have the same units ( m sec 2 in (1) and Kelivin sec 1 in (2))
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The Prandtl number is defined as Pr
f
and its units are m 1 sec 1 [Pr] 1 m (length ) 1 m sec
This means that the Prandtl number is dimensionless and correct only for cases . However, in figure 4 in [1] temperature profiles are presented for 0.9 and 0.8,0.9,1.0 . Therefore the results for 0.8 and 1 are wrong. The authors introduced the following dimensionless parameters marked with superscript * (equations 14, 15 and 16 in [1]) t (a f / ) ( 1) /( ) / f t * y (a f / )1 /( ) y * u b f (a f / ) ( 1) /( ) u * v (a f / ) /( ) f v * u0 b1n f (a f / ) ( 1) /( ) u0 * A (a f / )1 /( ) b ( n1) / 2 A *
It was mentioned previously that the Prandtl number is correct only when . However, for , none of the above parameters can be defined because the exponents with denominator cannot be defined. The question is how the profile with 0.9 in figure 4 was calculated. The dimensionless magnetic parameter is defined as follows
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B0 2 b n f M
f
( 1) /( )
(3)
For the magnetic field the following equation is valid B( x) B0 x b
( n 1 / 2 )
(4) B(x) 2 2 term B0
From equation (1) it is found that the units of the term
are
from equation (4) it is found that the units of the
are
The units of b are m n2 sec 2 .
m.
Therefore the units of the term
and
m12n sec 1 .
B0 2 b n f
are
Thus there are two errors in the definition of the magnetic
parameter M . The first is that the
sec 1
term f
( 1) /( )
cannot be defined for
which implies the correct Prandtl number. The second error is that
the term
B0 2 b n f
is dimensional, not dimensionless.
Taking into account all the above the results presented in [1] are inaccurate.
References [1]. Lin Liu, Liancun Zheng, Yanping Chen, Fawang Liu, Fractional boundary layer flow and heat transfer over a stretching sheet with variable thickness, Journal of Heat Transfer, 2018, Vol. 140/091701.
