Numerical analysis of nanofluid flow conveying nanoparticles through expanding and contracting gaps between permeable walls

Journal of Molecular Liquids 212 (2015) 785–791

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Numerical analysis of nanofluid flow conveying nanoparticles through expanding and contracting gaps between permeable walls M. Hatami a,⁎, S.A.R. Sahebi b, A. Majidian b, M. Sheikholeslami c, D. Jing d, G. Domairry c a

Department of Mechanical Engineering, Esfarayen University of Technology, North Khorasan, Iran Department of Mechanical Engineering, Islamic Azad University, Sari Branch, Sari, Iran Department of Mechanical Engineering, Babol University of Technology, Babol, Iran d International Research Center for Renewable Energy, State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China b c

a r t i c l e

i n f o

Article history: Received 15 May 2015 Received in revised form 14 October 2015 Accepted 19 October 2015 Available online xxxx Keywords: Least Square Method Nanofluid Non-dimensional wall dilation rate Porous wall

a b s t r a c t In this study, the problem of nanofluid flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions is solved using Least Square Method. The concept of this method is briefly introduced, and it's application for this problem is studied. Then, the results are compared with numerical results and the validity of these methods is shown. Graphical results are presented to investigate the influence of the volume fraction of nanoparticle, non-dimensional wall dilation rate and permeation Reynolds number on the velocity, normal pressure distribution and wall shear stress. The present problem for slowly expanding or contracting walls with weak permeability is a simple model for the transport of biological fluids through contracting or expanding vessels. The results indicate that velocity boundary layer thickness near the walls decreases with increase of Reynolds number and nanoparticle volume friction and it increases as non-dimensional wall dilation rate increases. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Recently, due to the rising demands of modern technology, including chemical production, power station, and microelectronics, there is a need to develop new types of fluids that will be more effective in terms of heat exchange performance. Nanofluids are produced by dispersing the nanometer-scale solid particles into base liquids with low thermal conductivity such as water, ethylene glycol (EG), oils, etc. [1]. The term “nanofluid” was first coined by Choi [2] to describe this new class of fluids. The materials with sizes of nanometers possess unique physical and chemical properties [3]. The presence of the nanoparticles in the fluids noticeably increases the effective thermal conductivity of the fluid and consequently enhances the heat transfer characteristics. Therefore, numerous methods have been taken to improve the thermal conductivity of these fluids by suspending nano/micro-sized particle materials in liquids. Asymmetric laminar flow and heat transfer of nanofluid between contracting rotating disks was investigated by Hatami et al. [4]. Their results indicated that temperature profile becomes more flat near the middle of two disks with the increase of injection but opposite trend is observed with increase of expansion ratio. The problem of laminar nanofluid flow in a semi-porous channel in the presence of transverse magnetic field was investigated analytically by Sheikholeslami et al. [5].Their results showed that velocity boundary ⁎ Corresponding author. E-mail address: [email protected] (M. Hatami).

http://dx.doi.org/10.1016/j.molliq.2015.10.040 0167-7322/© 2015 Elsevier B.V. All rights reserved.

layer thickness decrease with increase of Reynolds number and it increases as Hartmann number increases. Several studies have been published recently on the modeling of natural convection heat transfer in nanofluids such as [6–8]. Studies of fluid transport in biological organisms often concern the flow of a particular fluid inside an expanding or contracting vessel with permeable walls. For a valve vessel exhibiting deformable boundaries, alternating wall contractions produce the effect of a physiological pump. The flow behavior inside the lymphatic exhibits a similar character. In such models, circulation is induced by successive contractions of two thin sheets that cause the downstream convection of the sandwiched fluid. Seepage across permeable walls is clearly important to the mass transfer between blood, air and tissue [9]. Therefore, a substantial amount of research work has been invested in the study of the flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions. Dauenhauer and Majdalani [10] studied the unsteady flow in semi-infinite expanding channels with wall injection. They are characterized by two non-dimensional parameters, the expansion ratio of the wall α and the cross-flow Reynolds number Re. Majdalani and Zhou [11] studied moderate to large injection and suction driven channel flows with expanding or contracting walls. Using perturbations in cross-flow Reynolds number Re, the resulting equation is solved both numerically and analytically. Boutros et al. [12] studied the solution of the Navier–Stokes equations which described the unsteady incompressible laminar flow in a semi-infinite porous circular pipe with

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M. Hatami et al. / Journal of Molecular Liquids 212 (2015) 785–791 Table 1 Thermo physical properties of water and nanoparticles.

Nomenclature c Injection/suction coefficient NM Numerical method Pressure drop in the normal direction Δpn Re Permeation Reynolds number u ,v Velocity components along x, y axes, respectively Injection velocity Vw Greek symbols υ Kinematic viscosity α Non-dimensional wall dilation rate τ Shear stress ρ Fluid density Subscripts ∞ Condition at infinity nf Nanofluid f Base fluid s Nano-solid-particles

Pure water Copper (Cu) Silver (Ag) Alumina (Al2O3)

ρ(kg/m3)

μ(Pa ⋅ s)

997.1 8933 10,500 3970

0.001 – – –

the fluid inflow velocity Vw is independent of position. The equations of continuity and motion for the unsteady flow are given as follows:

injection or suction. Through the pipe wall whose radius varies with time. The resulting fourth-order nonlinear differential equation is then solved using small-parameter perturbations. The objective of the present paper is to study the nanofluid flow in a rectangular domain bounded by two moving porous walls. The reduced ordinary differential equations are solved via Least squares method. The effects of the parameters governing the problem are studied and discussed.

∂u ∂v þ ¼ 0; ∂x ∂y

ð3Þ

" 2 # 2 ∂u ∂u ∂u 1 ∂p μ nf ∂ u ∂ u ; þ þ þ u  þ v  ¼ − ρnf ∂x ρnf ∂x2 ∂y2 ∂t ∂x ∂y

ð4Þ

" 2 # 2 ∂v ∂v ∂v 1 ∂p μ nf ∂ v ∂ v : þ þ þ u  þ v  ¼ − ρnf ∂y ρnf ∂x2 ∂y2 ∂t ∂x ∂y

ð5Þ

In the above equations, u⁎ and v⁎ indicate the velocity components in x and y directions, p⁎ denotes the dimensional pressure, ρnf , μnf and t are the density, dynamic viscosity of nanofluid and time, respectively. The boundary conditions will be: y ¼ aðt Þ :

∂u ¼ 0; v ¼ 0; ∂y  u ¼ 0:

y ¼ 0 : x ¼ 0 :

2. Flow analysis and mathematical formulation

u ¼ 0; v ¼ −V w ¼ −

a• ; c

ð6Þ

•

Consider the laminar, isothermal and incompressible flow in a rectangular domain bounded by two permeable surfaces that enable the fluid to enter or exit during successive expansions or contractions. A schematic diagram of the problem is shown in Fig. 1. The fluid is a water based nanofluid containing Cu. It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. The thermo physical properties of the nanofluid are given in Table 1. The effective density ρnf, the effective dynamic viscosity μnf, the heat capacitance (ρCp)nf and the thermal conductivity knf of the nanofluid are given as: ρnf ¼ ρ f ð1−ϕÞ þ ρs ϕ μ nf ¼

μf ð1−ϕÞ2:5

ð1Þ

Where c ¼ Vaw is the wall presence or injection/suction coefficient, that is a measure of wall permeability. The stream function and mean flow vorticity can be introduced by putting: ∂ψ  ∂ψ  ∂v ∂u ;v ¼  ;ξ ¼  −  ∂y ∂x ∂x" ∂y #    μ nf ∂2 ξ ∂2 ξ ∂ξ  ∂ξ  ∂ξ : þv ¼ þ þu ∂t ∂x ∂y ρnf ∂x2 ∂y2

u ¼

Due to mass conservation, a similar solution can be developed with respect to x⁎ [13]. Starting with:

ð2Þ

Here, ϕ is the solid volume fraction. The walls expand or contract uniformly at a time-dependent ratea•. At the wall, it is assumed that



vx f ðy; t Þ ; a  y  ∂f : y ¼ ; fy ≡ a ∂y ψ ¼

:

ð7Þ

u ¼

vx f a2



y

;

v ¼



−v f ðy; t Þ ; a

ð8Þ

Substitution Eq. (8) into Eq. (7) yields: uy t þ u uy x þ v uy y ¼ vuy y y :

ð9Þ

In order to solve Eq. (9), one uses the chain rule to obtain:  f yyyy

  μ nf       þ α yf yyy þ 3 f yy þ f f yyy − f y f yy −a2 ρnf

!−1 

f yyt ¼ 0;

ð10Þ

With the following boundary conditions: at y ¼ 0 : at y ¼ 1 :





f ¼ 0; f yy ¼ 0; 

•

Fig. 1. Two-dimensional domain with expanding or contracting porous walls.



f ¼ ReA ð1−ϕÞ2:5 ; f y ¼ 0;

ð11Þ

Where αðtÞ ≡ aaυ is the non-dimensional wall dilation rate which is defined positive for expansion and negative for contraction and

M. Hatami et al. / Journal of Molecular Liquids 212 (2015) 785–791

A ¼ ð1−ϕÞ þ ρρs ϕ is a parameter. Furthermore, Re ¼ f

ρ f aV w μf

is the per-

meation Reynolds number of base fluid defined positive for injection and negative for suction through the walls. Eqs. (8), (10) and (11) can be normalized by putting: 



ψ u v f ψ ¼ • ;u ¼ • ;v ¼ ; f ¼ ;  aa a a ReA ð1−ϕÞ2:5

787

~, which is a linIt is considered that u is approximated by a function u ear combination of basic functions chosen from a linearly independent set. That is, ~¼ u≅u

n X

ci φi :

ð22Þ

i¼1

ð12Þ

Now, when substituted into the differential operator, D, the result of the operations generally isn't p(x). Hence an error or residual will exist:

And so: 0

xf xf −f α ;v ¼ ; ;u ¼ ;c ¼ c c c ReA ð1−ϕÞ2:5

ψ¼ IV

f

 000   000  00 0 00 þ α yf þ 3f þ ReA ð1−ϕÞ2:5 f f −f f ¼ 0:

ð13Þ

~ ðxÞÞ−pðxÞ ≠ 0: RðxÞ ¼ Dðu

ð14Þ

The notion in LSM is to force the residual to zero in some average sense over the domain. That is:

ð23Þ

Z The boundary conditions (11) will be: X

f ¼ 0; f ¼ 0 : 0 f ¼ 1; f ¼ 0

y¼0: y¼1:

ð15Þ

The resulting Eq. (14) is the classic Berman's formula [14], with α = 0 (channel with stationary walls). After the flow field is found, the normal pressure gradient can be obtained by substituting the velocity components into Eqs. (3)–(5). Hence it is:   −1  −1  ″ 0 0 f þ f f þ α ReA ð1−ϕÞ2:5 f þ yf ; py ¼ − ReA ð1−ϕÞ2:5 p

p¼

A ρ f V 2w

: ð16Þ

We can determine the normal pressure distribution, if we integrate Eq. (16). Let pc be the centerline pressure, hence: Z

pðyÞ

pc

RðxÞ W i ðxÞ ¼ 0

Z dp ¼

y

0

  −1  −1  ″ 0 0 − ReA ð1−ϕÞ2:5 f þ f f þ α ReA ð1−ϕÞ2:5 f þ yf ;

ð17Þ

Then using ff′ = ( f2)′/2 and (f + yf′) = (yf)′, the resulting normal pressure drop will be: " # 2  −1  −1  −1 f 0 0 Δpn ¼ ReA ð1−ϕÞ2:5 f ð0Þ− ReA ð1−ϕÞ2:5 f þ þ α ReA ð1−ϕÞ2:5 yf : 2

ð24Þ

Where the number of weight functions Wi is exactly equal the ~ . The result is a set of n algebraic number of unknown constants ci in u equations for the unknown constants ci. If the continuous summation of all the squared residuals is minimized, the rationale behind the LSM's name can be seen. In other words, a minimum of



τ ¼ μ nf vx þ uy



ρnf v2 x f ¼ a3

″

:

ð19Þ 

Introducing the non-dimensional shear stress τ ¼ ρ τV 2 w, we have: nf

xf

″



ReA ð1−ϕÞ2:5

w

:

ð20Þ

3. Least Square Method (LSM) and application on the problem There existed an approximation technique for solving differential equations called the Least Square Method (LSM). Suppose a differential operator D is acted on a function u to produce a function p: DðuðxÞÞ ¼ pðxÞ:

ð21Þ

R2 ðxÞdx:

RðxÞRðxÞdx ¼ X

ð25Þ

X

In order to achieve a minimum of this scalar function, the derivatives of S with respect to all the unknown parameters must be zero. That is, Z ∂S ∂R ¼ 2 RðxÞ dx ¼0: ∂ci ∂ci

ð26Þ

X

Comparing with Eq. (24), the weight functions are seen to be Wi ¼ 2

∂R : ∂ci

ð27Þ

However, the “2” coefficient can be dropped, since it cancels out in the equation. Therefore the weight functions for the Least Squares Method are just the derivatives of the residual with respect to the unknown constants Wi ¼

Another important quantity is the shear stress. The shear stress can be determined from Newton's law for viscosity:

Z

Z S¼

ð18Þ

τ¼

i ¼ 1; 2; :::; n

00

∂R : ∂ci

ð28Þ

Many advantages of LSM compared to other analytical and numerical methods make it more valuable and motivate researchers to use it for solving heat transfer problems. Some of these advantages are listed below [15–20]: a) It solves the equations directly and no simplifications are needed. For example it solves power nonlinear terms without expanding or using Taylor expansion against Differential Transformation Method (DTM). b) It does not need to any perturbation, linearization or small parameter versus Homotopy Perturbation Method (HPM) and Parameter Perturbation Method (PPM). c) It is simple and powerful compared to numerical methods and reaches to final results faster than numerical procedures while its results are acceptable and have excellent agreement with numerical outcomes, furthermore its accuracy can be increased by increasing the statements of the trial functions. d) It does not need to determine the auxiliary parameter and auxiliary function versus Homotopy Analysis Method (HAM).

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M. Hatami et al. / Journal of Molecular Liquids 212 (2015) 785–791

Fig. 2. Comparison between LSM and NUM for Cu–water nanofluid with ϕ = 0.04 when I: α = Re = 5, II: α = 1, Re = 4, III: α = −0.5, Re = −2 and IV: α = −1, Re = 1.

Because trial function must satisfy the boundary conditions in Eq. (15), so it will be considered as,

f ðyÞ ¼

    1 y3 y4 y5 þ c2 y− þ c3 y− þ ::: 3y−y3 þ c1 y− 3 4 5 2  ynþ2 þ cn y− : nþ2

nanofluid with ϕ = 0.04, Re = −1 and α = 1, LSM result in 0 ≤ y≤ 1 is: f ðyÞ ¼

ð29Þ

By introducing this equation to Eq. (14), residual function will be found and via substituting the residual function into Eq. (24), a set of equation with three equations will be appeared. It is necessary to inform that because one of the boundaries (f(1) = 1) is not satisfied completely in the above trial function, it should be considered as a separate equation to obtain the unknown coefficients. In this paper, three unknown terms from Eq. (29) is considered and by solving the system of equations, coefficients c1–c3 will be determined. For instance, for Cu–water

   1 y3 y4 −0:8782479462 y− 3y−y3 þ 1:132194849 y− 3 4 2  y5 : ð30Þ −0:1201382581 y− 5

As the same manner for Re = 4 and α = 1 it is f ðyÞ ¼

 1 3y−y3 2

  y3 y4 þ 1:729163089 y− −2:029539681 y− 4  3 5 y : þ 0:4617242105 y− 5

ð31Þ

Comparing these equations with fourth-order Runge–Kutta numerical method show the accuracy of applied method.

Fig. 3. Effect of nanoparticles volume fraction on f(y) and f′(y) profiles for Cu–water nanofluid.

M. Hatami et al. / Journal of Molecular Liquids 212 (2015) 785–791

789

Fig. 4. Effect of Reynolds number on velocity profiles for Cu–water nanofluid with ϕ = 0.04.

4. Results and discussion The objective of the present study was to apply LSM to obtain an explicit solution of laminar, isothermal, incompressible nanofluid flow in a rectangular domain bounded by two moving porous walls, which enable the nanofluid to enter or exit during successive expansions or contractions (Fig. 1). Different kinds of nanoparticle (Cu, Ag, Al2O3) considered with water as the base fluid. There is acceptable agreement between the numerical solution obtained by four-order Rung-kutte method and LSM (as shown in Fig. 2). After this validity, results are given for the velocity profile, normal pressure distribution and wall shear stress for various values of volume fraction of nanoparticle, permeation Reynolds number and non-dimensional wall dilation rate. Fig. 3 shows the effect of nanoparticles volume fraction on f(y) and f′(y) profiles. As nanoparticles volume fraction increases horizontal velocity (f′) increases while vertical velocity decreases. Fig. 4 shows the effect of Reynolds number on velocity profiles for Cu–water nanofluid.

The Reynolds number is defined as the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions. As Reynolds number increases maximum values of f′ at centerline decrease while velocity boundary layer thickness near the upper and lower plates decreases. Also it can be seen that f decreases with increase of Re. Effect of non-dimensional wall dilation rate on velocity profiles is shown in Fig. 5. Effect of α on velocity profile is in contrast with Reynolds number. It means that velocity boundary layer thickness near the walls increases with increase of non-dimensional wall dilation rate. Effect of nanoparticles material on the profiles is depicted in Fig. 6. It can be said that the shear stress changes at the lower and upper plates by using different types of nanofluid. This figure shows that selecting silver as nanoparticle leads to higher horizontal velocity near the walls. The wall shear stress and pressure distribution in the normal direction for various permeation Reynolds numbers over a range of non-dimensional wall dilation rates, are plotted in Fig. 7.

Fig. 5. Effect of α number on velocity profiles for Cu–water nanofluid with ϕ = 0.04.

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M. Hatami et al. / Journal of Molecular Liquids 212 (2015) 785–791

Fig. 6. Effect of nanoparticles material on the profiles for water based nanofluid.

Fig. 7. (a) Shear stress and (b) pressure drop for Cu–water nanofluid with ϕ = 0.04.

The absolute pressure change in the normal direction is lowest near the central portion. Furthermore, by increasing non-dimensional wall dilation rates the absolute value of pressure distribution in the normal direction increases. When Reynolds number is positive, by increasing non-dimensional wall dilation rates the absolute value of shear stress decreases while opposite trend is observed for negative Reynolds number.

5. Conclusion In this paper, LSM is used to solve the problem of nanofluid flow in a rectangular domain bounded by two moving porous walls. It can be found that LSM has high accuracy. Effects of volume fraction of nanoparticle, permeation Reynolds number and non-dimensional wall dilation rate on flow are examined. The results indicate that velocity boundary layer thickness near the walls decreases with increase of Reynolds number and nanoparticle volume friction and it increases as non-

dimensional wall dilation rate increases. Maximum velocity near the walls was obtained when silver is used as nanoparticle.

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