Simultaneous effects of convective conditions and nanoparticles on peristaltic motion

Journal of Molecular Liquids 193 (2014) 74–82

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Simultaneous effects of convective conditions and nanoparticles on peristaltic motion T. Hayat a,b, Humaira Yasmin a,⁎, B. Ahmad b, B. Chen c a b c

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia No. 5 Yiheyuan Road, Haidan District, Beijing, Peking University, China

a r t i c l e

i n f o

Article history: Received 6 November 2013 Received in revised form 30 November 2013 Accepted 17 December 2013 Available online 31 December 2013 Keywords: Peristalsis Nanoparticle phenomenon Convective condition

a b s t r a c t This article models the peristaltic transport of viscous nanofluid in an asymmetric channel. The channel walls satisfy the convective conditions. Effects of Brownian motion and thermophoresis are taken into account. The relevant flow analysis is first modeled and then computed for the series solutions of temperature and concentration fields. Closed form expression of stream function is constructed. Plots are prepared for a parametric study reflecting the effects of Brownian motion, thermophoresis, Prandtl, Eckert and Biot numbers. It is seen that temperature is an increasing function of Brownian motion, thermophoresis, Eckert and Prandtl numbers. However temperature is found to decrease when Biot number increases. It is also observed that the nanoparticle volume fraction field has opposite results for Brownian motion and thermophoresis parameters. Heat transfer coefficient increases via Biot, Brownian, thermophoresis, Prandtl and Eckert parameters. It is also worth mentioning to point out that the trapping increases for channel width and it decreases when the flow rate is increased. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The peristaltic transport of fluid has achieved special attention in the recent years due to its wide applications in engineering and biomechanics. This process is highly important in many physiological systems and industry such as swallowing food through esophagus, in the vasomotion of small blood vessels such as venules, capillaries and arterioles, in sanitary fluid transport, toxic liquid transport in the nuclear industry etc. The peristaltic flows are due to the waves traveling along the walls having elastic properties. Latham [1] and Shapiro et al. [2] initially dealt the peristaltic flow of viscous fluid. Afterwards extensive research has been conducted for the peristaltic flows. We mention here few recent studies [3–12]. Moreover, the peristaltic transport of fluid in a channel with heat transfer is significant in the hemodialysis and oxygenation processes. The heat transfer analysis in the existing attempts on peristalsis has been mostly addressed through prescribed surface temperature or heat flux. Recently the idea of convective boundary condition has been used for the heat transfer analysis [13–15]. At present, the mechanics of nanofluids has motivated the recent researchers for the enhancement of thermal conductivity of base fluid. Choi [16] introduced the word “nanofluid”. The term “nanofluid” refers to a liquid suspension containing ultrafine particles having diameter less than 50 nm. These particles can be found in the metals such as (Al, Cu), oxides (Al2O2), carbides (SiC), nitrides (SiN) or nonmetals (graphite, carbon ⁎ Corresponding author. Tel.: +92 51 90642172. E-mail address: [email protected] (H. Yasmin). 0167-7322/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.molliq.2013.12.036

nanotubes, nanofibers, nanosheets, droplets). Choi verified that the suspension of solid nanoparticles with typical length scales of 1–50 nm with high thermal conductivity enhances the effective thermal conductivity and the convective heat transfer of the base fluid. Buongiorno [17] proposed the nonhomogeneous equilibrium model and revealed that this massive increase in the thermal conductivity occurs due to the presence of two main effects namely the Brownian diffusion and the thermophoretic diffusion of the nanoparticles. Some representative studies on nanofluids have been carried in the investigations [18–40]. To our knowledge, there is yet scant information on the peristaltic transport of nanofluids. Even a single attempt is not available for peristaltic transport of nanofluid in an asymmetric channel with convective effects. The object of present communication is to present model taking into account such considerations. The Brownian motion and thermophoresis effects are given due attention in the mathematical modeling. Impact of Brownian motion, thermophoresis and Biot number on the temperature, concentration and heat transfer coefficient at the upper wall is analyzed. Trapping phenomena is also discussed for various parameters. 2. Fundamental equations The balances of mass, momentum, temperature and nanoparticle volume fraction are given by [18]:

div V ¼ 0;

ð1Þ

T. Hayat et al. / Journal of Molecular Liquids 193 (2014) 74–82

dV 1 2 ¼ − ∇P þ ν∇ V; ρf dt

ð2Þ

h 
dT ν
2 ¼ α∇ T þ τ·L þ τ DB ∇C∇T þ DT =T m Þ∇T∇T; c dt p

ð3Þ

 dC 2 2 ¼ DB ∇ C þ DT =T m Þ∇ T: dt

ð4Þ

Here V is the velocity field, T is the fluid temperature, C the nanoparticle concentration, d=dt the material time derivative, v the kinematic viscosity, P the pressure, ρf the density of an incompressible fluid, ρp the density of nanoparticles, α the thermal diffusivity, DB the Brownian motion coefficient, DT the thermophoretic diffusion coefficient, τ = (ρc)p/(ρc)f the ratio of effective heat capacity of the nanoparticle material to heat capacity of the fluid and T m is the mean temperature.

Considering a wave
frame  ðx; yÞ that moves with a velocity c away from the fixed frame X; Y , we write

 x ¼ X−ct; y ¼ Y; uðx; yÞ ¼ U X; Y; t −c; vðx; yÞ ¼ V X; Y; t ;


 T ðx; yÞ ¼ T X; Y; t ; C ðx; yÞ ¼ C X; Y; t ; pðx; yÞ ¼ P X; Y; t ;

Let us consider the peristaltic
motion of viscous nanofluid in an asymmetric channel. We take X; Y as the Cartesian coordinates with X being measured in the direction of wave propagation and Y in the direction normal to the X-axis. Let Y ¼ h1 and Y ¼ h2 be the upper and lower boundaries of the channel respectively (see Fig. 1). The motion is induced by sinusoidal wave trains propagating with a constant speed c along the channel walls. These are defined by    2π
h1 X; t ¼ d1 þ a1 cos X−ct ; upperwall; λ

ð5Þ

 
  2π
h2 X; t ¼ −d2 −a2 cos X−ct þ ϕ ; lower wall; λ

ð6Þ


 in which ðu; vÞ and U; V are the velocity components in the wave and the fixed frames respectively, p and P stand for pressure in the wave and fixed frames. The exchange of heat with ambient at the walls through Newton's cooling law is given by k


 ∂T at y ¼ h1 ; ¼ −h T−T 0 ∂y

k


 ∂T ¼ −h T 1 −T ∂y



ð10Þ

at y ¼ h2 ;

ð11Þ

! ∂u ∂u 1 ∂p ∂2 u ∂2 u u þv þ 2 ; ¼− þν 2 ρ f ∂x ∂x ∂y ∂x ∂y

ð12Þ

! 2 2 ∂v ∂v 1 ∂p ∂ v ∂ v u þv þ 2 ; ¼− þν 2 ρ f ∂y ∂x ∂y ∂x ∂y

ð13Þ

u

u 2

ð9Þ

∂u ∂v þ ¼ 0; ∂x ∂y

where a1 and a2 are the wave amplitudes, λ is the wavelength, d1 + d2 is the width of the asymmetric channel and the phase difference ϕ varies in the range 0 ≤ ϕ ≤ π (ϕ = 0 corresponds to the symmetric channel with waves out of phase and ϕ = π the waves are in phase). Further a1, a2, d1, d2 and ϕ satisfy the condition 2

ð8Þ

where h is the heat transfer coefficient and T0 and T1 are the ambient temperatures at the upper and lower channel walls respectively. The relevant flow equations in wave frame can be expressed as

3. Mathematical modeling




75

2

a1 þ a2 þ 2a1 a2 cosϕ ≤ ðd1 þ d2 Þ :

ð7Þ

! "     # ∂T ∂T ∂2 T ∂2 T ν ∂u 2 ∂u ∂v 2 þ þ þv ¼α þ þ 4 2 2 cp ∂x ∂y ∂x ∂y ∂x ∂x ∂y " ! !2 !2 )# ( D ∂C ∂T ∂C ∂T ∂T ∂T þ þτ DB þ þ T ; ∂x ∂x ∂y ∂y ∂x ∂y Tm

∂C ∂C ∂2 C ∂2 C þ 2 þv ¼ DB 2 ∂x ∂y ∂x ∂y

Fig. 1. Geometry of the problem.

! þ

DT

∂2 T

Tm

2

∂x

þ

∂2 T 2

∂y

ð14Þ

! :

ð15Þ

76

T. Hayat et al. / Journal of Molecular Liquids 193 (2014) 74–82

Defining dimensionless variables, Reynolds number (Re), Prandtl number (Pr), Eckert number (Ec), Brownian motion parameter (Nb), the thermophoresis parameter (Nt), Schmidt number (Sc) and Biot number (Bi) as x¼

x y u v d2 p d ;y ¼ ;u ¼ ;v ¼ ;p ¼ 1 ;δ ¼ 1 ; λ d1 c c cλμ λ

h1 h ct T−T 0 C−C 0 ; h ¼ 2; t ¼ ; θ ¼ ;σ ¼ ; λ d1 2 d1 T 1 −T 0 C 1 −C 0 2 ρcd1 ν c τDB ðC 1 −C 0 Þ ; ; Nb ¼ ; Pr ¼ ; Ec ¼ Re ¼ ν α μ c f ðT 1 −T 0 Þ τDT ðT 1 −T 0 Þ ν hd Nt ¼ ; Sc ¼ ; Bi ¼ 1 ; DB k T mν h1 ¼

ð16Þ

to wavelength i.e. a/λ = 0.156. It is also declared by Lew et al. [42] that Reynolds number in small intestine was small. Further, the situation of intrauterine fluid flow due to myomaterial contractions is a peristaltic type fluid motion in a cavity. The sagittal cross section of the uterus reveals a narrow channel enclosed by two fairly parallel walls [43]. The 1–3 mm width of this channel is very small compared with its 50 mm length [44], defining an opening angle from cervix to fundus of about 0.04 rad. Analysis of dynamics parameters of the uterus revealed that frequency, wavelength, amplitude and velocity of the fluid–wall interface during a typical contractile wave were found to be 0.01–0.057 Hz, 10–30 mm, 0.05–0.2 mm and 0.5–1.9 mm/s respectively. Therefore adopting low Reynolds number and long wavelength analysis [2] we have 3

the governing Eqs. ((11)–(15)) can be written as δ

∂u ∂v þ ¼ 0; ∂x ∂y

ð17Þ

  2 ∂ ∂ ∂p ∂2 u 2∂ u þ 2; u¼− þδ Re δu þ v 2 ∂x ∂y ∂x ∂x ∂y

ð18Þ

!   2 2 ∂ ∂ ∂p ∂ v 2∂ v ; þ Reδ δu þ v v¼− þδ δ ∂x ∂y ∂y ∂x2 ∂y2

ð19Þ

!   2 2 ∂ ∂ 1 ∂ θ 2∂ θ Re δu þ v δ þ θ¼ Pr ∂x ∂y ∂x2 ∂y2 "  2  2 # ∂u 2 ∂u 2 ∂v þ þδ þ Ec 4δ ∂x ∂y ∂x   ∂σ ∂θ 2 ∂σ ∂θ þ þNb δ ∂x ∂x ∂y ∂y "    2 # 2 ∂θ 2 ∂θ þ ; þNt δ ∂x ∂y

!   2 2 ∂ ∂ ∂2 σ Nt ∂2 θ 2∂ σ 2∂ θ δ þ þ þ : σ ¼δ ReSc δu þ v ∂x ∂y ∂x2 ∂y2 Nb ∂x2 ∂y2

∂p ∂ ψ ; ¼ ∂x ∂y3

ð23Þ

∂p ¼ 0; ∂y

ð24Þ

!2    2 2 2 1∂ θ ∂ ψ ∂σ ∂θ ∂θ þ Ec þ Nb ¼ 0; þ Nt 2 2 Pr ∂y ∂y ∂y ∂y ∂y

ð25Þ

2

2

∂ σ Nt ∂ θ þ ¼ 0; ∂y2 Nb ∂y2

ð26Þ

where continuity equation is identically satisfied and the above equations indicate that p ≠ p(y). Defining Θ and F as the dimensionless time-mean flows in the laboratory and wave frames respectively we have

ð20Þ

Θ ¼ F þ 1 þ d;

Z F¼

ð21Þ

h1 h2

∂ψ dy: ∂y

ð27Þ

ð28Þ

Keeping in view the physical constraints of the problem, the boundary conditions in the fixed frame may be expressed mathematically as follows U ¼ 0; at Y ¼ h1 and Y ¼ h2 ;

ð29Þ

Defining the velocity components u and v in terms of the stream function ψ by

C ¼ C 0 at Y ¼ h1 ;

ð30Þ

u ¼ ψy ; v ¼ −δψx :

C ¼ C 1 at Y ¼ h2 :

ð31Þ

ð22Þ

In the limit Re → 0, the inertialess flow corresponds to Poiseuillelike longitudinal velocity profile. The pressure gradient depends upon x and t only in laboratory frame. It does not depend on y. Such features can be expected because there is no streamline curvature to produce transverse pressure gradient when δ = 0. The assumptions of long wavelength and small Reynolds number give δ = 0 and Re = 0. It should be pointed out that the theory of long wavelength and zero Reynolds number remains applicable for the case of chyme transport in male small intestine [41]. In this case c = 2 cm/min, a = 1.25 cm and λ = 8.01 cm. Here half width of intestine is small in comparison

Writing the above conditions through Eq. (8), the dimensionless boundary conditions are ψ ¼ −1; at y ¼ h1 and y ¼ h2 ;

ð32Þ

ψ ¼ F=2; σ ¼ 0 at y ¼ h1 ;

ð33Þ

ψ ¼ −F=2; σ ¼ 1 at y ¼ h2 :

ð34Þ

T. Hayat et al. / Journal of Molecular Liquids 193 (2014) 74–82

77

Fig. 4. Temperature profile θ(y) for Nb and Nt.

Fig. 2. Temperature profile θ(y) for Ec.

4. Solution procedure Also Eqs. (9) and (10) give ∂θ þ Biθ ¼ 0 ∂y

The computations of Eqs. (25) and (26) are made through homotopy perturbation method (HPM). For that we write ð35Þ

at y ¼ h1 ;

∂θ þ Bið1−θÞ ¼ 0 at y ¼ h2 : ∂y

ð36Þ

h i 2 H ðq; θÞ ¼ ð1−qÞ LðθÞ þ Ecψyy −Lðθ0 Þ "    2 # ∂σ ∂θ ∂θ þ q LðθÞ þ Nb þ Nt ; ∂y ∂y ∂y "

The dimensionless forms of hi's (i = 1,2) are h1 ðxÞ ¼ 1 þ acosð2πxÞ; h2 ðxÞ ¼ −d−bcosð2πx þ ϕÞ;

H ðq; σ Þ ¼ ð1−qÞ½Lðσ Þ−Lðσ 0 Þ þ q Lðσ Þ þ

Nt ∂2 θ Nb ∂y2

ð39Þ

!# ;

ð40Þ

ð37Þ where we have taken L ¼ ∂y∂ as the linear operator and ψ(y) is computed by using Eq. (23) and boundary conditions (Eqs. (32)–(34)). Exact solution for ψ is 2

2

where a = a1/d1, b = a2/d1, d = d2/d1 and ϕ satisfy the following relation 2

2

ψðyÞ ¼ −

2

a þ b þ 2abcosϕ≤ ð1 þ dÞ :

" # ðh1 þ h2 −2yÞ  −2ðh1 −h2 Þðh1 −yÞðh2 −yÞ : 2 2 2 þF h1 −4h1 h2 þ h2 þ 2ðh1 þ h2 Þy−2y 2ðh1 −h2 Þ3 ð41Þ

The pressure rise per wavelength (ΔPλ) is Zλ ΔP λ ¼

dp dx: dx

We define initial guesses as

ð38Þ

0

Fig. 3. Temperature profile θ(y) for Pr.

Biðh2 −yÞ−1 3EcPr ð F þ h1 −h2 Þ2 þ θ0 ðyÞ ¼ 1 þ 4ðh1 −h2# Þ3 " Biðh1 −h2 Þ þ 2 4 8 þ Biðh1 −h2 Þ ðh1 −h2 −2yÞ −  ; Bi ðh1 −h2 Þ3

Fig. 5. Temperature profile θ(y) for Bi.

ð42Þ

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T. Hayat et al. / Journal of Molecular Liquids 193 (2014) 74–82

Fig. 6. Nanoparticle volume fraction ϕ(y) for Ec.

σ 0 ðyÞ ¼

Fig. 8. Nanoparticle volume fraction ϕ(y) for Nb.

h1 −y : h1 −h2

ð43Þ

Let us write 2

θðy; qÞ ¼ θ0 þ qθ1 þ q θ2 þ …;

ð44Þ

2

σ ðy; qÞ ¼ σ 0 þ qσ 1 þ q σ 2 þ …:

ð45Þ

The solutions of temperature and nanoparticle phenomena (for q = 1) are constructed as 8

7

6

5

4

3

2

θðyÞ ¼ A1 y þ A2 y þ A3 y þ A4 y þ A5 y þ A6 y þ A7 y þ A8 y þ A9 ; 4

3

2

σ ðyÞ ¼ B1 y þ B2 y þ B3 y þ B4 y þ B5 ;

ð46Þ ð47Þ

where the Ai's (i = 1–9) and Bj's (j = 1–5) can be obtained through simple algebraic computations. 5. Results and discussion In this section, the effects of various emerging parameters on the temperature θ, nanoparticle volume fraction ϕ and heat transfer coefficient at the upper wall are discussed in detail. Figs. 2 and 3 show the effects of Eckert number Ec and Prandtl number Pr on temperature profile θ respectively. These Figs. depict that the temperature θ increases with an increase in Eckert and Prandtl numbers due to the strong viscous

Fig. 7. Nanoparticle volume fraction ϕ(y) for Pr.

dissipation effects. Also Ec = 0 corresponds to the situation in which viscous dissipation effects are absent. Effect of Brownian motion parameter Nb and thermophoresis parameter Nt on temperature profile θ is illustrated in Fig. 4. It is observed from this Fig. that the temperature θ increases when Nb and Nt are increased. As the Brownian motion and thermophoretic effects strengthen this corresponds to the effective movement of nanoparticles from the wall to the fluid which results in the significant increase in temperature θ. The temperature profile θ decreases with an increase in the Biot number Bi because of the nonuniform temperature distribution in the fluid (see Fig. 5). Effects of Eckert number (Ec) and Prandtl number (Pr) on nanoparticle volume fraction field ϕ are seen in the Figs. 6 and 7. The nanoparticle volume fraction field ϕ is found to decrease when the effects of Eckert number and Prandtl number intensify. However the nanoparticle volume fraction field ϕ increases with an increase in Brownian motion parameter Nb. On the other hand ϕ decreases when we intensify the thermophoresis parameter Nt (see Figs. 8 and 9). The behaviors of parameters on the heat transfer coefficient at the upper wall have been observed in Figs. 10–13. The heat transfer coefficient is denoted by Z(x) = (h1)xθy(h1) which actually defines the rate of heat transfer or heat flux at the upper wall. The heat transfer coefficient Z(x) increases when the values of Biot number Bi are set to be increased (see Fig. 10). According to Fig. 11 the heat transfer coefficient Z(x) increases by increasing the values of Brownian motion parameter Nb and thermophoresis variable Nt. As a consequence the nanoparticle enhances the heat transfer rate. Further it is noted that by increasing the values of Prandtl number Pr, the heat transfer coefficient Z(x) is increased (see Fig. 12). It is important

Fig. 9. Nanoparticle volume fraction ϕ(y) for Nt.

T. Hayat et al. / Journal of Molecular Liquids 193 (2014) 74–82

Fig. 10. Heat transfer coefficient Z(x) for Bi.

to note that in all these Figs. the viscous dissipation effects are significantly large (Ec = 1.0) which enhances the heat transfer rate. Fig. 13 shows the effects of Eckert number Ec on heat transfer coefficient Z(x). This Fig. depicts that the heat transfer coefficient Z(x) is much larger for higher values of the Eckert number (Ec N 1) when compared with the case when viscous dissipation is negligible (Ec = 0.01). Fig. 14 is plotted to see the effects of channel width d on trapping. For this Fig. d = 0.6 (panels (a) and (b)), d = 0.7 (panels (c) and (d)), d = 1.3 (panels (e) and (f)) and the other parameters are a = 0.5, b = 0.7 and Θ = 1.2. It is observed from this Fig. that the size of the trapping bolus decreases by increasing the value of channel width d. The left panel shows the trapping for phase difference ϕ = 0 and right panel for ϕ = π/2. A comparative study of these panels reveals that the trapping bolus is symmetric with respect to the channel for ϕ = 0. On the other hand for ϕ = π/2 the trapping bolus shifts towards left because of the asymmetry of channel. Fig. 15 depicts the effects of flow rate Θ on trapping. This Fig. shows that the trapping bolus increases in size when we increase the values of flow rate Θ. In this Fig. Θ = 0.9 (panels (a) and (b)), Θ = 1.2 (panels (c) and (d)), Θ = 1.8 (panels (e) and (f)) and other parameters are a = 0.5, b = 0.7 and d = 1.2. Also the trapping bolus shifts towards left in an asymmetric case (right panel) when compared with the symmetric case (left panel). 6. Main points This attempt addresses the effects of nanoparticles and convective conditions on peristaltic motion in an asymmetric channel. Viscous

Fig. 11. Heat transfer coefficient Z(x) for Nb and Nt.

79

Fig. 12. Heat transfer coefficient Z(x) for Pr.

dissipation effects are also considered. The conducted study leads to the following observations: • The temperature increases while nanoparticle volume fraction decreases when there is an increase in Eckert number and Prandtl number. • By increasing the value of Brownian motion and thermophoresis parameter, temperature profile increases. • Temperature is a decreasing function of Biot number. • Nanoparticle volume fraction field increases with an increase in Brownian motion parameter and it decreases by increasing thermophoresis parameter. • Heat transfer coefficient increases when there is an increase in the Biot, Brownian motion, thermophoresis, Prandtl and Eckert parameters. • Trapping decreases with an increase in channel width while it increases by increasing the flow rate.

Acknowledgments We are grateful to the reviewers for their useful suggestions. This paper was funded by the Deanship of Scientific Research (DSR) at the King Abdulaziz University (KAU), under grant no. (25-130/1433 HiCi). The authors, therefore, acknowledge technical and financial support of KAU.

Fig. 13. Heat transfer coefficient Z(x) for Ec.

80

T. Hayat et al. / Journal of Molecular Liquids 193 (2014) 74–82

Fig. 14. Streamlines for different values of d (panels (a), (c) and (e) for ϕ = 0 and panels (b), (d), and (f) for ϕ = π/2.

T. Hayat et al. / Journal of Molecular Liquids 193 (2014) 74–82

Fig. 15. Streamlines for different values of Θ (panels (a), (c) and (e) for ϕ = 0 and panels (b), (d) and (f) for ϕ = π/2).

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