A Model for Electro-osmotic Flow of Pseudoplastic Nanofluids …
193
where κ = d 2 ez
2n 0
ε k B T
represents the ratio of the characteristic transverse length to
the Debye length; this indicates the penetration of the zeta potential at the surface
into the bulk fluid.
For the problem under consideration, the velocity is defined as
V = [u(x, y, t), v(x, y, t), 0]
(13)
To simplify the analysis, we lead the following changes in the position between
fixed and wave frames
X = x − ct, Y = y, P(X, Y ) = p(x, y, t), U = u − c and V = v.
(14)
In the presence of electric field, thermal radiation and Joule heating, the equation
for the mass and momentum conservation for pseudoplastic nanofluid Eqs. (2)–(10)
are written in terms of the leading transformation yield as follows:
∂U
∂ X
+
∂ V
∂Y
= 0,
(15)
ρ f
U
∂U
∂ X
+ V
∂ V
∂Y
= −
∂ P
∂ X
+
∂ S X X
∂ X
+
∂ S XY
∂Y
+ ρ e E x
+ (1 − C 0 )ρ f gβ t (T − T 0 ) −
ρ p − ρ f
g(C − C 0 ), (16)
ρ f
U
∂ V
∂ X
+ V
∂ V
∂Y
= −
∂ P
∂Y
+
∂ S XY
∂ X
+
∂ S Y Y
∂Y
,
(17)
ρc
f
U
∂ T
∂ X
+ V
∂ T
∂Y
= α m
∂ 2 T
∂ X 2 +
∂ 2 T
∂Y 2
−
∂q r
∂Y
+
ρc
p
D B
∂C
∂ X
∂ T
∂ X
+
∂C
∂Y
∂ T
∂Y
+
ρc
p
D T
T m
∂ T
∂ X
2
+
∂ T
∂Y
2
+ Q 0 (T − T 0 ),
(18)
U
∂C
∂ X
+ V
∂C
∂Y
= D B
∂
2 C
∂ X 2 +
∂
2 C
∂Y 2
+
D T
T m
∂
2 T
∂ X 2 +
∂
2 T
∂Y 2
,
(19)
in which
2μ
∂U
∂ X
= S X X + λ 1
U
∂ S X X
∂ X
+ V
∂ S X X
∂Y
− 2
∂U
∂ X
S X X − 2
∂U
∂Y
S XY
+
1
2
(λ 1 − μ 1 )
4S X X
∂U
∂ X
+ 2S XY
∂U
∂Y
+
∂ V
∂ X
,
2μ
∂ V
∂Y
= S Y Y + λ 1
U
∂ S Y Y
∂ X
+ v
∂ S Y Y
∂Y
− 2
∂ V
∂Y
S Y Y − 2
∂ V
∂ X
S XY
+
1
2
(λ 1 − μ 1 )
4S Y Y
∂ V
∂Y
+ 2S XY
∂U
∂Y
+
∂ V
∂ X
,