A Model for Electro-osmotic Flow of Pseudoplastic Nanofluids …
195
Rδ
∂ψ
∂ y
∂ 2 ψ
∂ x∂ y
−
∂ψ
∂ x
∂ 2 ψ
∂ y 2
= −
∂ p
∂ x
+ δ
∂ S xx
∂ x
+
∂ S xy
∂ y
+ U hs
δ
2 ∂ 2
∂ x 2 +
∂ 2
∂ y 2
+ Grθ + Brσ,
(23)
−Rδ 3
∂ 2 ψ
∂t∂ x
+
∂ψ
∂ y
∂ 2 ψ
∂ x 2 −
∂ψ
∂ x
∂ 2 ψ
∂ y∂ x
= −
∂ p
∂ y
+ δ 2 ∂ S xy
∂ x
+ δ
∂ S yy
∂ y
,
(24)
Rδ
∂ψ
∂ y
∂θ
∂ x
−
∂ψ
∂ x
∂θ
∂ y
=
1
Pr
δ
2 ∂
2
θ
∂ x 2 +
∂
2
θ
∂ y 2
+ Rn
∂
2
θ
∂ y 2
+ N b
δ
2 ∂σ
∂ x
∂θ
∂ x
+
∂σ
∂ y
∂θ
∂ y
+ N t
δ
2
∂θ
∂ x
2
+
∂θ
∂ y
2
+ βθ,
(25)
R δ
∂ψ
∂ y
∂σ
∂ x
−
∂ψ
∂ x
∂σ
∂ y
=
δ
2 ∂
2
σ
∂ x 2 +
∂
2
σ
∂ y 2
+
N t
N b
δ
2 ∂
2
θ
∂ x 2 +
∂
2
θ
∂ y 2
,
(26)
in which
2δ
∂
2
ψ
∂ x∂ y
= S xx + λ 1
δ
∂ψ
∂ y
∂ S xx
∂ x
+ δ
∂ψ
∂ x
∂ S xx
∂ y
− 2δ
∂
2
ψ
∂ x∂ y
S xx − 2
∂
2
ψ
∂ y 2 S xy
+
1
2
(λ 1 − μ 1 )
4δS xx
∂
2
ψ
∂ x∂ y
+ 2S xy
∂
2
ψ
∂ y 2 − δ
2 ∂
2
ψ
∂ x 2
,
(27)
−2δ
∂
2
ψ
∂ y∂ x
= S yy + λ 1 δ
∂ψ
∂ y
∂ S yy
∂ x
−
∂ψ
∂ x
∂ S yy
∂ y
+ 2
∂
2
ψ
∂ y∂ x
S yy + 2δ
∂
2
ψ
∂ x 2 S xy
+
1
2
(λ 1 − μ 1 )
−4δS yy
∂
2
ψ
∂ y∂ x
+ 2S xy
∂
2
ψ
∂ y 2 − δ
2 ∂
2
ψ
∂ x 2
, (28)
∂
2
ψ
∂ y 2 − δ
2 ∂
2
ψ
∂ x 2 = S xy + λ 1
δ
∂ψ
∂ y
∂ S xy
∂ x
− δ
∂ψ
∂ x
∂ S xy
∂ y
+ δ
2 ∂
2
ψ
∂ x 2 S xx −
∂
2
ψ
∂ y 2 S yy
+
1
2
(λ 1 − μ 1 )
S xx + S yy
∂
2
ψ
∂ y 2 − δ
2 ∂
2
ψ
∂ x 2
,
(29)
where δ, R, U hs, Gr, Br, Pr, Rn, N t, N b, β, and ξ are the wave number, Reynolds
number, Helmholtz–Smoluchowski velocity, local temperature Grashof number,
local nanoparticle Grashof number, Prandtl number, thermal radiation, thermophoresis parameter, Brownian motion parameter, Joule heating parameter, pseuoplastic
fluid parameter, and continuity equation is automatically satisfied.
Under the assumptions of long wavelength and low-Reynolds number and
neglecting the terms of order δ and higher, Eqs. (23–29) become
∂ p
∂ x
=
∂ S xy
∂ y
+ U hs
∂
2
∂ y 2 + Grθ + Brσ,
(30)