On squeezed flow of couple stress nanofluid between two parallel plates Academic research paper on "Mathematics"

results in

PHYSICS

Accepted Manuscript

On squeezed flow of couple stress nanofluid between two parallel plates

Tasawar Hayat, Rai Sajjad, Ahmed Alsaedi, Taseer Muhammad, Rahmat Ellahi

PII: DOI:

Reference:

S2211-3797(16)30574-5 http://dx.doi.Org/10.1016/j.rinp.2016.12.038 RINP 505

To appear in:

Results in Physics

Received Date: Accepted Date:

18 November 2016 23 December 2016

Please cite this article as: Hayat, T., Sajjad, R., Alsaedi, A., Muhammad, T., Ellahi, R., On squeezed flow of couple stress nanofluid between two parallel plates, Results in Physics (2017), doi: http://dx.doi.org/10.1016Zj.rinp. 2016.12.038

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On squeezed flow of couple stress nanofluid between

two parallel plates

Tasawar Hayat1'2, Rai Sajjad3, Ahmed Alsaedi2, Taseer Muhammad1* and Rahmat Ellahi3'4 1 Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan 2Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 3Department of Mathematics and Statistics, Faculty of Basic & Applied Sciences, International Islamic University, Islamabad 44000, Pakistan 4Department of Mechanical Engineering, Bourns Hall A373, University of California,

Riverside, CA 92521, USA * Corresponding author E-mail: taseer_qau@yahoo.com (Taseer Muhammad)

Abstract: The present communication provides an analytical treatment of magnetohydrody-namic (MHD) squeezing flow of couple stress nanomaterial between two parallel surfaces. Constitutive relations of couple stress fluid are used in the problem formulation. Novel features regarding thermophoresis and Brownian motion are taken into consideration. Couple stress fluid is electrically conducted subject to time-dependent applied magnetic field. The governing partial differential system is converted into the set of nonlinear ordinary differential system through appropriate transformations. The resulting nonlinear systems have been computed through the homotopic approach. Behaviors of various sundry parameters on velocity, temperature and concentration fields are studied in detail. Further the skin friction and heat and mass transfer rates are also computed and analyzed.

Keywords: Squeezing flow; Couple stress fluid; Nanoparticles; Magnetohydrodynamics.

1 Introduction

The suspension of nanometer sized particles in a convectional base fluid is called as nanofluid. These nanoparticles are specially made of metals (Al, Cu, Ag) or nonmetals (carbon nan-otubes, graphite) and the base fluid is commonly a conductive fluid such as ethylene glycol, water or oil. Investigations have derived that in a traditional base fluid the suspension of nanopartilces sufficiently variates the heat transfer characteristics and transport property. To obtain prominent thermal conductivity enhancement in the nanofluid, studies have been processed both theoretically and experimentally. Many applications in technology and engi-

neering of nanofluids are nuclear reactor, vehicle cooling, vehicle thermal management, heat exchanger, cooling of electronic devices and many others. Moreover magneto nanofluids are assisting removal of blockage in the arteries, hyperthermia, wound treatment, cancer therapy and magnetic resonance imaging. The pioneer investigation on the enhancement of thermal properties of the base fluid by the suspension of nanoparticles in it was done by Choi [1]. Later the growth of mathematical model of nanofluid which demonstrate the characteristics of Brownian diffusion and thermophoresis is presented by Buongiorno [2]. Mustafa et al. [3] studied the stagnation-point flow of nanoliquid towards a stretched surface. Makinde and Aziz [4] analyzed the boundary-layer flow of nanoliquid past a linearly stretched sheet by considering convective surface condition. Turkyilmazoglu [5] derived the exact analytical solutions for MHD slip flow of nanofluids by taking heat and mass transfer characteristics. Having such in view, few relevant attempts on nanofluid flows can be quoted through the analysis [6 — 25] and various studies therein.

Squeezing flow between two parallel surfaces is investigated by many researchers due to its attraction in engineering and technological processes. Such applications include compression and injection shaping, liquid metal lubrication, food and polymer industries etc. The initial analysis on lubrication approximation has been reported by Stefan [26]. The squeezing flow of power-law fluid between the parallel disks is investigated by Leider and Bird [27]. Effects of suction/injection on the squeezed flow was illustrated by Hamza and MacDonald [28]. Mahmood et al. [29] analyzed the flow and heat transfer characteristics in a squeezing flow. Siddiqui et al. [30] considered the MHD flow of viscous fluid between parallel surfaces. Domairry and Aziz [31] examined the magnetohydrodynamic squeezing flow between two parallel disks under the influence of suction and injection. Hayat et al. [32] reported the magnetohydrodynamic squeezed flow of second grade fluid. Sheikholeslami and Ganji [33] studied the squeezing flow of Cu-water nanofluid between parallel surfaces. Domairry and Hatami [34] investigated the unsteady squeezing flow of Cu-water nanofluid between parallel surfaces by using differential transform method. Recently Hayat et al. [35] investigated the squeezing flow of nanofluid between two parallel plates with time-dependent magnetic field.

The literature survey shows that the couple stress fluid model in the presence of Brownian motion and thermophoresis has not been reasonably discussed. Earlier Stokes [36] gives the simplest generalization of the classical viscous fluid theory that maintains the body couples and couple stresses. The accurate flow behavior of such fluid cannot be analyzed by the clas-

sical viscous fluid theory. Moreover their stress tensor do not satisfy the symmetric property. The main effect of couple stresses introduces a size dependent effect that is not available in the classical viscous theories. Srinivasacharya and Kaladhar [37] considered a non-Darcy porous medium with Soret and Dufour effects in mixed convection flow of couple stress fluid. Alsaedi et al. [38] investigated the peristaltic flow in a uniform porous medium by taking couple stress fluid. Two-dimensional flow over a continuously stretching/shrinking surface in an electrically conducting quiescent couple stress fluid is discussed by Turkyilmazoglu [39]. Hayat et al. [40] examined the magnetic field effects in three-dimensional (3D) flow of couple stress nanofluid over a nonlinear stretched surface subject to convective boundary condition. Recently Ramesh [41] investigated the inclined magnetic field characteristics in peristaltic flow of couple stress material in a porous medium.

Existing information on the topic witnessed that an analytical treatment of magnetohy-drodynamic (MHD) squeezing flow of couple stress nanomaterial between two parallel surfaces with novel features of thermophoresis and Brownian motion has never been reported. Cou-

ple stress fluid is electrically conducted subject to time-dependent applied magnetic field. The lower surface of the channel is porous. The upper impermeable surface moves towards the lower surface with a time-dependent velocity. The governing nonlinear partial differential system is converted into the set of nonlinear ordinary differential equations through the appropriate transformations. The resulting nonlinear system has been solved through ho-motopic approach [42 — 50]. Attributes of pertinent parameters on the velocity, temperature and concentration distributions are observed for suction case. Moreover the skin friction coefficients and heat and mass transfer rates are also computed numerically at the channel boundar

aaries.

2 Mathematical analysis

We consider the time-dependent two-dimensional (2D) magnetohydrodynamic (MHD) squeezing flow of an incompressible couple stress nanomaterial between two parallel surfaces separated by a distance y7^ (1 — 71). The upper surface at y = h (t) = y7^ (1 — 71) is moving towards the lower surface with velocity — i ^ «(1-7*7 the l°wer permeable surface at y = 0 is stretching with velocity j3^ (t < ^ . Couple stress fluid is electrically conducted in the presence of time-dependent magnetic field applied in the y—direction (see Fig.

1). Further the Hall current and electric field effects have been omitted. Induced magnetic field for low magnetic Reynolds number approximation is not taken. Brownian diffusion and thermophoresis characteristics are present. Relevant expressions are

,pD , dTd£ + dTdC_N B y dx dx dy dy ,

d2C d2C\ Dt (d^T d^T\ dx2 + dy2) + Tm Vdx2 + dy2) ' ^ '

Here u and v show the components of velocity along the horizontal and vertical directions respectively, a the electrical conductivity, p* stands for pressure, p^ the density of base liquid, p the dynamic viscosity, n the couple stress viscosity parameter, T the temperature, (pc)p the effective heat capacity of nanoparticles, a k/ (pcstands for thermal diffusivity, k for thermal conductivity, (pc)^ the heat capacity of fluid, Db stands for Brownian diffusion coefficient, Dt the thermophoresis diffusion coefficient, Tm stands for mean temperature and

C for concentration. The subjected boundary conditions are

u = U0 = aX , v = - V° , T = T0, C = C0 at y = 0, (6)

1 — 71 1 — 71

u = 0, v = Vh = f = -y^Jj^y= T„ + C = Co + at j, = ft (t) ,

where Co and T0 are the concentration and temperature distributions at the lower stretching porous surface, a stands for stretching rate of the lower surface, Vo > 0 indicates the suction and Vo < 0 shows the blowing/injection velocity. Introducing the suitable transformations

« = Uof (C), « = (0, C = 4),

t = To + ^ e (C), c = Co + ^ 0 (C) •

Eliminating pressure gradient from Eqs. (2) and (3) and employing Eq. (8) in the Eqs. (2) - (7) we get

. „-

/™ - y (Cf"' + 3f') ~ f7" + //"' - KT - M2f" = 0, (9)

e" + pr ^fd' - (26 + co') + Nhey + Nte'^ = o, 4>" + Le Pr (- (20 + C^')) + ^e" = 0, (11)

/ (0) = /' (o) = i, e (o) = o, 0 (o) = o, 1 / (1) = f, f (1) = 0, d (1) = 1, 0 (1) = 1, J

where K stands for couple stress parameter, M for magnetic parameter, Sq for squeezing parameter, Nt for thermophoresis parameter, Pr for Prandtl number, N^ for Brownian motion parameter, Le for Lewis number and S for suction/blowing parameter. These parameters are defined by

C _ 7 K — na M2 — Pr — ^ Q — V0

°<1 ~ a ' — ulpf (1-7i) ' — pf a ' rL ~ a ' ° ~ ah(t) '

N _ (,pc)vDBCo N _ (,Pc)vDtTq _

JVb ~ (pc)f v(l-yt) ' ~ (pc)f vTm(l-yt) ' ~ DB ■

Skin friction coefficients at lower and upper surfaces are given by

Cf 1 = = if" (0) - Kfiv (0)) , (14)

Cf2 = Tw » = 2h(t) = (Rex)-1/2 (/" (1) - Kf™ (1)) , (15)

Local Nusselt numbers at lower and upper surfaces are given by

Nuxl =

NuTo =

(Tw - T0) dy

= - (Rexf2 9' (0),

(Tw - T0) dy

= - (Rex)1/2 9' (1),

y=h(t)

Local Sherwood numbers at lower and upper surfaces are given by

Shxi —

ShTo —

(Cw - C0) dy

= - (Rex)1/2 $ (0),

(Cw - C0) dy

= - (Re.)1/2 (1),

(18) (19)

where Rex = Uqx/u represents the local Reynolds number.

3 Homotopic solutions

The appropriate initial approximations an

,ry linear operators are given by

/o(C) = C - 2C2 + C3 + s (1 - 3(2 + 2C3) + C2 (1 - C),

C <MC) = C

fiv, Ce = 9", ^ =

The above operators satisfy

cf (c? + c2*c + esc2 + QC3) = o,

(20) (21)

ce (c*5 + Qc) = 0, c* (c; + esc) = o,

where C* (i = 1 — 8) elucidate the arbitrary constants. Deformation problems at zeroth-order

(i - p) cf [f{c,p) - f0 (C)] = phfMf [/(C,P)], (23)

(1 - p) C0 [9 (C,p) - 90 (C)] = pheNe [f{Ç,p)№,p)M,P)] , (24)

(1 - p) C^ [0 (C,p) - 00 (c )] = PhM^ [f((,p)M,P)M,P)] , (25)

f(0,p) = s, f' (0,p) = i, e (o,p) = o, 0 (o,p) = o, (26)

fd,p) = f, f (l,p) = 0, 9 (l,p) = l, 0 a,p) = l, (27)

Kr W S* d2f\ df d* f - d2f

Uf = â? - T №+3~dë) - -dc^ë + W ""M~ïë' (28)

Me [f((,p)A(,P),^(,P)} = +Pr

/| ~ f (20 + Cf

^ [f{C,p)№,p)M,p)] =

' + ^ (If

a?+ LePrlfdt -Tl20+cTt)) + ^^•

>rs and

Here p G [0,1] stands for embedding parameter, Mf, Me and for nonlinear operators hf, he and h$ for non-zero auxiliary parameters. Setting p = 0 and p = 1 one obtains

/(C, o) = /o(C), f({, 1) = /(C), (31)

0(C, o) = 0o(C), 0(C, i) = 6>(C), o) = 0O(C), i) = <KC)•

:c )•

) vary fi

When p changes from 0 to 1 then f(C,p), G&p) and 0(C,p) vary from initial approximations /o(C)> ^o (C) and (C) to desired ultimate solutions f((), 9(() and )• The following expressions are derived via Taylor's series expansion:

f(c,p) = /o(c) + e uc)pm, uc) = a'{i:p>

■' ml opm

0(C,P) = 00(C) + E MC)i>m, MC) = ^

= ^o(c) + E ¿»(c<uc) = ^^r^

gardin;

The convergence regarding Eqs. (34) — (36) is strongly based upon the suitable choices of hf, he and Choosing suitable values of hf, he and h$ so that Eqs. (34) — (36) converge at p =1 then

/ (c ) = /0(c ) + E )> (37)

0{C) = 00(C) + E 0m(C),

<KC ) = ^o(C ) + E <UC).

The mth-order deformation problems are defined as follows:

£/ [frn (C) - Xmfm-l (C)] = hf),

[0m (c) - xmom-1 (c)] = ^ wc),

|>m (C) - XmK-1 (C)] = h-R^iC), (42)

fU 0) = fL (0) = 0m (0) = (0) = 0, (43)

fm( 1) = fi(1) = ern (1) = (1) = 0, (44)

m<) = /.S-f (cc-i+3/:-.)+£ (/»->-*/r - /:-■-*/:)-^/ÍS-M2/^, (45)

^T(C ) = C-1 + /—i-*^ - Pry(20-1 + CC-1 )

m—1 m—1

+pr ^ +Pr ^ , (46)

k=0 fc=0

1 '" Pr 2 V— -v ■ Nh

) = C-i + be P^ /m-1-fc& - Le Pr -f (20m_1 + CC-i) + ^C-i, (47) k=0 b

, 0, m < 1,

Xm = < (48)

1, m > 1.

The mth-order problems have the following solutions:

fm (C) = fm (C) + c? + esc + esc2 + Q*C3, (49)

0m (C) = C (C) + ct + CSC, (50)

(C) = C (C) + c; + CSC, (51)

in which f*n {r¡), 6*m {r¡) and 4>*m {rj) represent the special solutions.

4 Convergence analysis

No doubt the homotopic solutions (37) — (39) contain the nonzero auxiliary parameters hf, h$ and Such auxiliary parameters play a key role to adjust and control the convergence of obtained homotopic solutions. To get the acceptable values of such parameters, we have drawn the ft—curves at 7th order of deformations. Figs. 2 and 3 clearly indicate that the convergence zone lies inside the ranges —1.5 < hf < —0.4, —1.4 < H$ < —0.3 and —1.3 < h^ < —0.5 for lower surface case (C = 0) and —1.7 < hf < —0.4, —1.3 < hg < —0.3 and — 1.2 < H$ < —0.4 for upper surface case (C = 1). Table 1 shows that the 15th order of deformations is enough for convergent homotopic solutions in lower surface case whereas the

10th order of deformations is sufficient for convergent homotopic solutions in upper surface case (see Table 2).

Fig. 2. The ft—curves for / (£) ,6 (£) and 0 (£) at lower surface.

S=0.1, M = 0.1, Sq = l£ = Pr=1.0, N, = 0.2, Nb = 0.5, K= 0.02

i^ie. 3. T

2 I / /

1.5 I / J

1 \ * S* \ / / ----- \

~ 0.5 \ ' ' \! / s \ \ \

S3 o / /'

' / i /' —/"CD \

-05 —- »'m ........ \\

-1 ; ' 1 x ............!..

-2 -1.5

hf, h, %

Fig. 3. The H—curves for f (() ,9 (() and 0 (() at upper surface. Table 1: HAM solutions convergence at lower surface when S = M = 0.1, Nt = 0.2, S„ = Le = Pr = 1.0, Nb = 0.5 and K = 0.02.

Order of approximations / "(0) 6> (0) (0)

1 0.0122 1.1620 0.8120

5 0.0111 1.1350 0.7852

10 0.0111 1.1359 0.7840

15 0.0111 1.1360 0.7841

20 0.0111 1.1360 0.7841

Table 2: HAM solutions convergence at upper surface when S = M = 0.1, Nt = 0.2, Sq = Le = Pr = 1.0, Nb = 0.5 and K = 0.02.

Order of approximations / "(1) 6> (1) (1)

1 0.2580 0.9117 1.2617

5 0.2562 0.9357 1.2686

10 0.2562 0.9358 1.2687

15 0.2562 0.9358 1.2687

20 0.2562 0.9358 1.2687

5 Discussion

This portion has been arranged to explore the impacts of several pertinent variables including suction parameter S, squeezing parameter Sq, magnetic parameter M, couple stress parameter K, Prandtl number Pr, thermophoresis parameter Nt, Brownian motion parameter Nb and Lewis number Le on velocity f' ((), temperature 6 (() and concentration 0 (() profiles. Fig. 4 shows impact of suction parameter S on velocity distribution. Velocity field decreases for larger suction parameter. The velocity distribution decreases more rapidly in neighborhood of upper squeezing surface in comparison to neighborhood of the lower porous stretching surface. Because high suction causes a reduction in the velocity distribution. Behavior of squeezing parameter Sq on velocity distribution is presented in Fig. 5. It is noticed that the velocity increases for larger Sq. The larger values of Sq depreciate the contrary flow which is because of the squeezing effects of the upper plate. Figs. 6 and 7 show the influences of magnetic M and couple stress K parameters on the velocity distribution /' (() respectively. By increasing both M and K, velocity field reduces in the lower half of the channel whereas it enhances in the upper half. Velocity decreases initially with the increasing values of magnetic and couple stress parameters. However for ( > 0.45, there is an increase in the velocity distribution. Physically when we increase the magnetic parameter M the velocity decreases and so does the velocity gradient since the same mass flow rate is imposed in order to satisfy the mass conservation constraint. In MHD flow we expect that the decrease in the fluid velocity in the wall regions will be balanced by an increase in the fluid velocity near the central region giving rise to a cross-flow behavior. Impacts of suction parameter S, squeezing parameter Sq, Prandtl number Pr, thermophoresis parameter Nt and Brownian motion parameter Nb on the temperature distribution are presented in the Figs. (8) — (12) respectively. It is seen

'randtl = 0.72,

that by increasing S, Pr, Nt and Nb temperature distribution increases monotonically from ( = 0 to ( = 1 whereas it decreases by increasing Sq. It is a valuable fact to focus here that the properties of liquid metals are characterized by small values of Prandtl number (Pr < 1), which have larger thermal conductivity but smaller viscosity, whereas higher values of Prar number (Pr > 1) associate with high-viscosity oils. Particularly Prandtl number Pr = l 1.0 and 6.2 associate to air, electrolyte solution such as salt water and water respectively. Moreover it is also observed that Nt and Nb characterize the strengths of thermophoresis and Brownian motion effects respectively. The higher the values of Nt and the larger will be the strength of the corresponding effects. The change in concentration field 0 (() for different values of suction parameter S, squeezing parameter Sq, Prandtl number Pr, thermophoresis parameter Nt, Brownian motion parameter Nf, and Lewis number Le are displayed in the Figs. (13) — (18) respectively. It is observed that by increasing Sq, Pr, Nt, and Le the concentration field 0 (() decreases while it increases by increasing S. Table 3 presents the numerical values of temperature profile 6 (() for various values of magnetic parameter M when 5 = 0.1, K = 0.02, Nt = 0.2, Nb = 0.5, Sq = Le = Pr = 1.0 and C = 0.5. Here we observed that the values of temperature profile 6 (() are lower when the larger values

nperatui /"s the nr

of M are considered. Table 4 shows the numerical values of skin friction coefficients at the lower and upper surfaces for various effective parameters S, Sq, M and K. Here the values of skin friction coefficient at the lower and upper surfaces are higher for increasing values of squeezing parameter. Table 5 shows a comparison of /"(0) and /"(1) through two techniques. Numerical solutions are computed through the NDSolve. The solutions obtained by the two techniques are in good agreement up to 6 decimal places. This confirms the validity of the HAM solutions. Table 6 is computed to analyze the numerical data of local Nusselt number at the lower and upper surfaces for various values of embedding parameters. It is examined that local Nusselt number reduces at both lower and upper surfaces for larger Lewis number. Table 7 depicts numerical data of local Sherwood number at the lower and upper surfaces for several values of pertinent parameters. Here we have seen that local Sherwood number increases at both lower and upper surfaces for increasing values of Brownian motion

S, = 0.0, , ,1.2,1.6

5 = 0.00, 15,1 ,0.45,0.60

Fig. 4. Plots of /' (C) for

Fig. 5. Plots of /' (C) for St

M = 0.0, , ,2.7,3.0

K = 0.0, , ,6.0,8.0

Fig. 6. Plots of /' (C) for M.

Fig. 7. Plots of /' (C) for K.

5 = 0.0,0.5, ,1.5,2.0

Sq = 0.0, , ,1.5,2.0

Fig. 8. Plots of e (C) for

Fig. 9. Plots of 6 (C) for St

parameter.

f' (£) f'(0

Fig. 10. Plots of 6 (C) for Pr.

Nh 0.5, , ,2.0,2.5

Fig. 11. Plots of d (C) for Nt.

Fig. 12. Plots of 6 (C) for Nb.

0.2 0.4 0.6 0.8 1.0

Fig. 13. Plots of 0 (C) for

0.2 0.4 0.6 0.8 1.0

Fig. 14. Plots of 0 (C) for Sq.

M = N, = 0.2, Sq = Le = Pr = 1.0, Nb = 0.5, K = 0.02

J = 0.0, , ,1.2,1.6

Pr = 0.5, , ,2.0,2.5

Fig. 15. Plots of ^ (C) for Pr .

Fig. 16. Plots of 0 (C) for Nt.

Fig. 17. Plots of 0 (C) for Nb. Fig. 18. Plots of 0 (C) for Le.

Table 3. Numerical values of temperature profile 9 (() for various values of M when S = 0.1, K = 0.02, Nt = 0.2, Nh = 0.5, Sq = Le = Pr = 1.0 and C = 0.5.

M 0.0 0.5 1.0 1.5 2.0 2.5 3.0

9 (C) 0.53660 0.53653 0.53633 0.53600 0.53551 0.53483 0.53393

Table 4. Numerical data for skin friction coefficients at the lower and upper surfaces for S, Sq, M and K.

S Sn M K

0.0 1.0 0.2 0.02 0.9151 -0.9549

0.1 0.2

1.4739 -0.3307 1.9103 0.3427

0.2 0.0 0.2 0.02 5.2000 3.2000

0.3 0.6

4.9181 2.5369 3.0431 1.6914

0.2 1.0 0.0 0.02 1.8744 0.3450

0.5 1.0

0.2 1.0 0.2 0.00 2.( )9 0.2575 0.01 l.i L9 0.3003 3371 0.3423

Table 5. Comparative values of f'' and M when S„ = K = 0.

2.0741 0. 2.2439

at both surfaces for different values of S

0.5 0.0 -7.411153

1.0 -7.591618

2.0 -8.110334

3.0 -8.910096

0.0 2.0 -4.587891

0.3 -6.665662

0.6 -8.851444

1.0 —11.948584

f "(1) f "(0) f "(1)

4.713303 -7.411153 4.713303

4.739017 -7.591618 4.739017

4.820251 -8.110334 4.820251

4.964870 -8.910096 4.964870

1.842447 -4.587891 1.842447

3.653695 -6.665662 3.653695

5.391248 -8.851444 5.391248

7.593426 -11.948584 7.593426

Table 6. Numerical data for local Nusselt numbers at the lower and upper surfaces for S,

Sq, M, Pr, Nt, Nb and Le.

S Sq M Pr Nt Nb Le 6' (0) 6' (1)

0.7676

0.0 1.0 0.2 1.0 0.2 0.5 1.0 1.2122 0.9534

0.2 1.2949 0.9358

0.4 1.3814 0.9188

0.2 0.0 0.2 1.0 0.2 0.5 1.0 1.5593 0.6446

0.2 1.5004 0.70

0.4 1.4447

0.2 1.0 0.0 1.0 0.2 0.5 1.0 1.2950 0.9358

0.5 1.2947 0.9359

1.0 1.2938 0.9363

0.2 1.0 0.2 0.4 0.2 0.5 1.0 1.1185 0.9687

0.7 1.2070 0.9504

1.0 1.2949 0.9358

0.2 1.0 0.2 1.0 0.0 0.5 1.0 1.1857 1.0133

0.2 1.2949 0.9358

0.4 1.4106 0.8641

2 1.0 0.2 0.4 1.0 1.2423 0.9763

0.7 1.4040 0.8595

1.0 1.5768 0.7562

0.2 1.0 0.2 1.0 0.2 0.5 0.5 1.3010 0.9409

1.0 1.2949 0.9358

1.5 1.2893 0.9310

0.2 1.0 0 /

Table 7. Numerical data for local Sherwood numbers at the lower and upper surfaces for various values of S, Sq, M, Pr, Nt, Nb and Le.

S Sq M Pr Nt Nb Le

0.2 0.4

0.2 0.4

0.5 1.0

0.2 1.0 0.2

Le (0) ^ (1)

1.0 0.8066 1.2886

0.8349 1.2687

0.8633 1.2493

1.0 0.8921 1.0789

0.8813 1.1172

0.8702 1.1554

1.0 0.8349 1.2687

0.8348 1.2688

0.8344 1.2693

1.0 0.9322 1.1137

X 0.8829 1.1933

0.8349 1.2687

1.0 0.9496 1.2484

0.8349 1.2687

0.6304 1.3407

1.0 0.8317 1.2553

0.8374 1.2828

0.8376 1.2911

0.5 0.8553 1.1480

1.0 0.8349 1.2687

1.5 0.8157 1.3842

6 Conclusions

Magnetohydrodynamic (MHD) unsteady squeezed flow of couple stress nanomaterial between two parallel walls is investigated. Main observations of present analysis are listed below:

• Larger values of squeezing parameter lead to higher velocity field while the opposite trend is seen for temperature and concentration fields.

Both the temperature and concentration fields show opposite behavior for increasing values of Prandtl number.

Effects of thermophoresis parameter on the temperature and concentration fields are

quite opposite.

Both the temperature and concentration fields show similar behavior for Brownian motion parameter.

References

[1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, USA, ASME, FED 231/MD 66 (1995) 99-105.

.995) 99-10E

[2] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer, 128 (2006) 240-250.

[3] M. Mustafa, T. Hayat, I. Pop, S. Asghar and S. Obaidat, Stagnation-point flow of a nanofluid towards a stretching sheet, Int. J. Heat Mass Transfer, 54 (2011) 5588-5594.

[4] O.D. Makinde and A. Aziz, Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition, Int. J. Thermal Sei., 50 (2011) 1326-1332.

[5] M. Turkyilmazoglu, Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids, Chem. Eng. Sei., 84 (2012) 182-187.

[6] M.M. Hashmi, T. Hayat and A. Alsaedi, On the analytic solutions for squeezing flow of nanofluid between parallel disks, Nonlinear Analysis: Model. Control, 17 (2012) 418-430.

[7] M.M. Rashidi, S. Abelman and N.F. Mehr, Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid, Int. J. Heat Mass Transfer, 62 (2013) 515-525.

[8] A. Zeeshan, M. Baig, R. Ellahi and T. Hayat, Flow of viscous nanofluid between the concentric cylinders, J. Comp. Theoretical Nanoscience, 11 (2014) 646-654.

[9] M. Sheikholeslami, M. Hatami and D.D. Ganji, Nanofluid flow and heat transfer in a rotating system in the presence of a magnetic field, J. Mol. Liq., 190 (2014) 112-120.

[10] T. Hayat, T. Muhammad, S.A. Shehzad and A. Alsaedi, A mathematical study for three-dimensional boundary layer flow of Jeffrey nanofluid, Z. Naturforsch. A, 70a (2015) 225-233.

[11] A. Malvandi, M.R. Safaei, M.H. Kaffash and D.D. Ganji, MHD mixed convection in a vertical annulus filled with Al203-water nanofluid considering nanoparticle migration, J. Magn. Magn. Mater., 382 (2015) 296-306.

[12] T. Hayat, T. Muhammad, A. Alsaedi and M.S. Alhuthali, Magnetohydrodynamic three-dimensional flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation, J. Magn. Magn. Mater., 385 (2015) 222-229.

[13] R. Ellahi, M. Hassan and A. Zeeshan, Shape effects of nanosize particles in Cu-H20 nanofluid on entropy generation, Int. J. Heat Mass Transfer, 81 (2015) 449-456.

[14] B.J. Gireesha, R.S.R. Gorla and B. Mahanthesh, Effect of suspended nanoparticles on three-dimensional MHD flow, heat and mass transfer of radiating Eyring-Powell fluid over a stretching sheet, J. Nanofluids, 4 (2015) 474-484.

[15] Y. Lin, L. Zheng, X. Zhang, L. Ma and G. Chen, MHD pseudo-plastic nanofluid unsteady flow and heat transfer in a finite thin film over stretching surface with internal heat generat

[16] T. I

ition, Int. J. Heat Mass Transfer, 84 (2015) 903-911.

[16] T. Hayat, M. Imtiaz and A. Alsaedi, Impact of magnetohydrodynamics in bidirectional flow of nanofluid subject to second order slip velocity and homogeneous-heterogeneous reactions, J. Magn. Magn. Mater., 395 (2015) 294-302.

[17] A. Chamkha, S. Abbasbandy and A.M. Rashad, Non-Darcy natural convection flow for non-Newtonian nanofluid over cone saturated in porous medium with uniform heat and volume fraction fluxes, Int. J. Numer. Methods Heat Fluid Flow, 25 (2015) 422-437.

[18] M. Azimi and R. Riazi, Flow and heat transfer of MHD graphene oxide-water nanofluid between two non-parallel walls, Thermal Sci., (2015), D01:10.2298/TSCI150513100A

[19] M. Azimi and R. Riazi, Heat transfer analysis of magnetohydrodynamics graphene oxide-water nanofluid flow through convergent-divergent channels, J. Comp. Theoretical Nanoscience, 13 (2016) 659-665.

[20] M. Azimi and R. Riazi, Analytical solution of unsteady GO-water nanofluid flow and heat transfer between two parallel moving plates, Indian J. Chem. Tech., 23 (2016) 47-52.

[21] M. Azimi and R. Riazi, MHD copper-water nanofluid flow and heat transfer through convergent-divergent channel, J. Mech. Sci. Tech., 30 (2016) 4679-4686.

[22] T. Hayat, T. Muhammad, A. Alsaedi and B. Ahmad, Three-dimensional flow of nanofluid with Cattaneo-Christov double diffusion, Results Phys., 6 (2016) 897-903.

md D.D.

[23] M. Sheikholeslami, M.M. Rashidi, T. Hayat ; and D.D. Ganji, Free convection of magnetic nanofluid considering MFD viscosity effect, J. Mol. Liq., 218 (2016) 393-399.

[24] T. Hayat, Z. Hussain, T. Muhammad and A. Alsaedi, Effects of homogeneous and heterogeneous reactions in flow of nanofluids over a nonlinear stretching surface with variable surface thickness, J. Mol. Liq., 221 (2016) 1121-1127.

[25] T. Hayat, T. Muhammad, S.A. Shehzad and A. Alsaedi, On magnetohydrodynamic flow of nanofluid due to a rotating disk with slip effect: A numerical study, Comp. Methods Appl. Mech. Eng., 315 (2017) 467-477.

[26] M.J. Stefan, Versuch Uber die scheinbare adhesion, Akademie der Wissenschaften in fien. Mathematik-Naturwissen 69 (1874) 713.

[27] P.J. Leider, R. B. Bird, Squeezing flow between parallel disks, I. Theoretical analysis, Ind. Eng. Chem. Fundam. 13 (1974) 336-341.

[28] E.A. Hamza and D.A. MacDonald, A fluid film squeezed between two plane surfaces, J. Fluid Mech., 109 (1981) 147-160.

[29] M. Mahmood, S. Asghar and M.A. Hossain, Squeezed flow and heat transfer over a porous surface for viscous fluid, Heat Mass Transfer, 44 (2007) 165-173.

[30] A.M. Siddiqui, S. Irum and A.R. Ansari, Unsteady squeezing flow of a viscous MHD fluid between parallel plates, Math. Model. Anal., 13 (2008) 565-576.

[31] G. Domairry and A. Aziz, Approximate analysis of MHD squeeze flow between two parallel disks with suction or injection by homotopy perturbation method, Math. Prob. Eng., 2009 (2009) 603916.

[32] T. Hayat, A. Yousaf, M. Mustafa and S. Obaidat, MHD squeezing flow ■-K1 fluid between two parallel disks, Int. J. Numer. Meth. Fluids, 69

[33] M. Sheikholeslami and D.D. Ganji, Heat transfer of Cu-water n nofluid flow between parallel plates, Powder Technology, 235 (2013) 873-879.

[34] G. Domairry and H. Hatami, Squeezing Cu-water nanofluid flow analysis between parallel plates by DTM-Pade Method, J. Mol. Liquids, 193 (2014) 37-44.

[35] T. Hayat, T. Muhammad, A. Qayyum, A. Alsaedi and M. Mustafa, On squeezing flow of nanofluid in the presence of magnetic field effects, J. Mol. Liquids, 213 (2016) 179-185.

[36] V. K. Stokes, Couple stresses in fluids, Phys. Fluids, 9 (1966) 1709-1715.

[37] D. Srinivasacharya and K. Kaladhar, Mixed convection flow of couple stress fluid in a non-darcy porous medii lm with Soret and Dufour effects, J. Appl. Sci. Eng., 15 (2012) 415-422.

ledium wit!

[38] A. Alsaedi, N. Ali, D. Tripathi and T. Hayat, Peristaltic flow of couple stress fluid

through uniform porous medium, Appl. Math. Mech., 35 (2014) 469-480.

cyilmazoglu, Exact solutions for two-dimensional laminar flow over a continuously tretching or shrinking sheet in an electrically conducting quiescent couple stress fluid, t. J. Heat Mass Transfer, 72 (2014) 1-8.

[40] T. Hayat, A. Aziz, T. Muhammad and B. Ahmad, Influence of magnetic field in three-dimensional flow of couple stress nanofluid over a nonlinearly stretching surface with convective condition, Plos One, 10 (2015) e0145332.

[41] K. Ramesh, Influence of heat and mass transfer on peristaltic flow of a couple stress fluid through porous medium in the presence of inclined magnetic field in an inclined asymmetric channel, J. Mol. Liq., 219 (2016) 256-271.

[42] S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004) 499-513.

[43] M. Dehghan, J. Manafian and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Meth. Partial Eq., 26 (2010) 448-479.

[44] T. Hayat, R. Sajjad, Z. Abbas, M. Sajid and A.A. Hendi, Radiation effects on MHD flow of Maxwell fluid in a channel with porous medium, Comm. Nonlinear Sei. Numer. Simulat., 54 (2011) 854-862.

[45] A. Malvandi, F. Hedayati and G. Domairry, Stagnation point flow of a nanofluid toward an exponentially stretching sheet with nonuniform heat generation/absorption, J. Thermodynamics, 2013 (2013) 764827.

[46] M. Turkyilmazoglu, An effective approach for evaluation of the optimal convergence control parameter in the homotopy analysis method, Filomat, 30 (2016) 1633-1650.

[47] M. Ramzan and M. Bilal, Three-dimensional flow of an elastico-viscous nanofluid with chemical reaction and magnetic field effects, J. Mol. Liq., 215 (2016) 212-220.

[48] T. Hayat, T. Abbas, M. Ayub, T. Muhammad and A. Alsaedi, On squeezed flow of Jeffrey nanofluid between two parallel disks, Appl. Sei., 6 (2016) 346.

[49] T. Hayat, T. Muhammad, S. Al-Mezal and S.J. Liao, Darcy-Forchheimer flow with variable thermal conductivity and Cattaneo-Christov heat flux, Int. J. Numer. Methods Heat Fluid Flow, 26 (2016) 2355-2369.

[50] T. Hayat, T. Muhammad, S.A. Shehzad and A. Alsaedi, An analytical solution for magnetohydrodynamic Oldroyd-B nanofluid flow induced by a stretching sheet with heat generation/absorption, Int. J. Thermal Sei., Ill (2017) 274-288.

Highlights

• Squeezing flow of couple stress nanofluid is constructed.

• Time-dependent magnetic field is accounted.

• Brownian motion and thermophoresis effects are present.

• Development of convergent series solutions is made.