Scholarly article on topic 'Effects of viscous dissipation and slip velocity on two-dimensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluids'

Effects of viscous dissipation and slip velocity on two-dimensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluids Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — Umar Khan, Naveed Ahmed, Mir Asadullah, Syed Tauseef Mohyud-din

Abstract Squeezing flow of nanofluids has been taken into account under the effects of viscous dissipation and velocity slip. Two types of base fluids are used to study the behavior of Copper nanoparticles between parallel plates. Nonlinear ordinary differential equations governing the flow are obtained by imposing similarity transformations on conservation laws. Resulting equations are solved by using an efficient analytical technique the variation of parameters method (VPM). Influences of nanoparticle concentration and different emerging parameters on flow profiles are presented graphically coupled with comprehensive discussions. A numerical solution is also sought for the sake of comparison. Effect of different parameters on skin friction coefficient and Nusselt number is also discussed.

Academic research paper on topic "Effects of viscous dissipation and slip velocity on two-dimensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluids"

Propulsion and Power Research 2015;4(1):40-49

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ORIGINAL ARTICLE

Effects of viscous dissipation and slip velocity on ® two-dimensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluids

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Umar Khana, Naveed Ahmeda, Mir Asadullahb, Syed Tauseef Mohyud-dina*

aDepartment of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan COMSATS Institute of Information Technology, University Road, Abbottabad, Pakistan

Received 29 September 2014; accepted 17 November 2014 Available online 7 April 2015

KEYWORDS

Squeezing flow; Nanofluids; Variation of parameters method (VPM); Velocity slip; Numerical solution

Abstract Squeezing flow of nanofluids has been taken into account under the effects of viscous dissipation and velocity slip. Two types of base fluids are used to study the behavior of Copper nanoparticles between parallel plates. Nonlinear ordinary differential equations governing the flow are obtained by imposing similarity transformations on conservation laws. Resulting equations are solved by using an efficient analytical technique the variation of parameters method (VPM). Influences of nanoparticle concentration and different emerging parameters on flow profiles are presented graphically coupled with comprehensive discussions. A numerical solution is also sought for the sake of comparison. Effect of different parameters on skin friction coefficient and Nusselt number is also discussed.

© 2015 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V.

All rights reserved. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Squeezing flow between moving plates has gained considerable interest by the scientist due to its wide range of

"Corresponding author.: Tel.: +0092 3235577701. E-mail address: syedtauseefs@hotmail.com (Syed Tauseef Mohyud-din).

Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.

applications in many industrial and biological situations. Its applications especially in polymer processing, modeling of synthetics transportation inside living bodies, hydro-mechanical machinery and compression/injection processes are of great importance. Many studies are available for better understanding of these types of flows. The basic contribution in this regard can be named to Stefan [1]. His pioneering effort has opened new doors for researchers and a lot of studies have been carried out following him [2-8]. Homotopy perturbation

2212-540X © 2015 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. All rights reserved. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

http://dx.doi.org/10.1016/jjppr.2015.02.004

solution for Two-dimensional MHD squeezing flow between parallel plates has been determined by Siddiqui et al. [9].

Recently, much concern has been devoted to study different properties of nanofluids in different geometries. Nanoscale particle added fluids are called nanofluids. The conventional heat transfer fluids including kerosene oil, ethylene glycol and water are poor conductors of heat. Since these heating/cooling fluids play an important role in the development of energy efficient heat transfer equipment for energy supply. To enhance and control the thermal conductivity of these fluids, nano-sized conducting metal particles are added to them. This makes their proper understanding very crucial for modern industry. Applications of nanofluids include microelectronics, fuel cells, pharmaceutical processes, etc. The nanofluid model presented by Buongiorno [10] has been used as a tool to investigate different properties of various Newtonian and non-Newtonian fluids with nanoparticles by different researchers [11-22]. Choi [23] presented another model that considered different base fluids to inspect the properties of nanoparticles. Numerous studies based on Choi's model are also available, where different researchers have presented the convective flows of nanofluids [24,25]. Choi also showed that the thermal conductivity of considered fluid can be increased by adding a small amount of (less than 1% by volume) of nanoparticles to conventional heat transfer liquids.

Since most of the real life problems are inherently in the form of nonlinearity, exact solutions for these problems are very rare. Therefore many analytical approximation techniques have been developed which are being commonly used nowadays. These include variational iteration method (VIM) [26-29], adomian's decomposition method (ADM) [30-33], homotopy perturbation method (HPM) [34-37], homotopy analysis method (HAM) [38-41], etc.

Sheikholeslami et al. [42,43] presented the effects of nanoparticles concentration on squeezing flow between parallel plates using the problem developed by Wang [44]. They used water as a base fluid and studied the properties of copper (Cu) for this problem. In this study, we have used water and kerosene as base fluids and Cu is used to study the behavior of these nanofluids under the effects of viscous dissipation and velocity slip for the problem studied by Rashidi et al [8]. A relatively novel analytical technique variation of parameters method [45-51] is used to solve the governing equations. VPM is a relatively new technique that is different from VIM. VIM takes whole equation in the recurrence relation while for the VPM, highest order linear term is not taken into account. Lagrange multiplier is used in VIM while VPM uses the multiplier obtained by a Wronskian technique. This makes these multipliers different from each other. VPM is useful as it requires less computational work as compared to VIM. This is because of the exemption of highest order term in VPM. A numerical solution is also sought out for the sake of comparison with the solution obtained by VPM. An excellent agreement between the solutions is found.

2. Governing equations

Consider unsteady flow of a viscous nanofluid between two parallel plates. Distance between the plates is z — ±l(1 - at)1/2 — ± h(t), where l is the initial position (at time t — 0). They are embedded in a medium filled with water or kerosene based nanofluids containing different types of nano particles i.e. copper (Cu) or silver (Ag). The base fluids and nanoparticles are assumed to be in thermal equilibrium. It is also assumed that there is a slip between base fluids and nanoparticles. Further a>0 corresponds to squeezing motion of both plates until they touch each other at t — 1/a, for a<0 the plates leave each other and dilate (Figure 1). Also, viscous dissipation effects are retained to study the generation of heat due to friction caused by shear in the flow.

du dv 0 dx dy

1 C dU /v dU „ dU

pnf ydt \~u — -dx fv dU

1 (dv „ dv ,dvN

pnf \dt -U — -dx

,2- ¿2* , dx2 1 ay2" (2)

\dx dy '

dT „ dT „ dT _ knf id2T d2T^ ~dt + U ~ÔX + V ~dy ~ (pCp)nf U*2 + dy2

d2U\2 iô2U d2UN

+ , Vf U 'du

dx2 dy2

In above equations U, v respectively, are the velocity components in x and y-directions. T, p, pnf, pnf, (pCp)f and knf denote the temperature, pressure, effective density, effective dynamic viscosity, effective heat capacity and effective thermal conductivity of the fluid [12]; where

Pnf = (i - $)pf + 4>ps,

(pCp)nf =(1 - 4>){pCp)f + <P(pCp)s> - Vf

Pnf = ~ 77Î5 ' (5)

kf _ ks + 2kf — 2<ft(kf — ks) kf ks + 2kf + ^ ( kf — k^)

Figure 1 Geometrical description of the problem.

Boundary conditions for the flow problem are

at y — h( at y — 0

u =-Y f ; v = vw = f at y = h(t)

dû _0

dy — " ;

here y is the velocity slip parameter. Using transformation introduced by Wang [44] for a two-dimensional flow

[2(1 - at)]

— al

[2(1 - at)1'2]

F (n); F(n);

0 = -T. ;

where,

(8) (9) (10)

[1(1 - ai)1/2]

By applying above transforms, simplifying Eqs. (2)-(3) to eliminate pressure terms and using Eq. (1), we get:

/ nF(n) + 3F"(n)\

Fiv(n) - SK1 (1 - 0)2-5 +F(n)F"(n) = 0; (12)

\ - F(n)F"' (n) y

Also substituting Eqs. (8)-(12) in Eq. (6), we get a differential equation of the form

e-(n) + Prs(jK^I (F(n)e (n) - n&(n))

K 3 (1 - 0)

((F"(n))2 + 482(F'(n))2) — 0; (13)

where S — al2 '2v/ is the non-dimensional Squeeze number, Pr — h[f(P/)f is Prandtl number, Ec — ——

pfkf ™ ' ^~(pcp)f V2(1 - at)

Eckert number and 8 — I.

From Eqs. (7)-(11) boundary conditions also reduce to

F(0) — 0, F"(0) — 0, F(1) — 1, Y

F (1)—-■

(1 - 0) 0' (0) — 0; 0(1) — 1

F"(1);

Similarly for the axisymmetric case, transforms introduced by Wang [37] are

[4(1 - at)]

[4(1 - at)]

[2(1 - at)]

F (n);

F (n); F(n).

Table 1 Thermo physical properties of water, kerosene and copper nanoparticles [12,52].

P' (kg'm3) Cp'(J'(kg ■ K)) k'(W'(m ■ K))

Pure water 997.1 Kerosene 783 Copper (Cu) 8933 4179 2090 385 0.613 0.145 401

Using Eqs. (16)-(18) in unsteady axisymmetric Navier-Stokes equations we get a nonlinear ordinary differential equation of the form

/ t]F(n) \ Fiv (n) - SK i (1 - <£)25 +3F"(n) = 0, (19)

V - F (n)F"(n))

Thus, for velocity profile, we have to solve non-linear ordinary differential equation of the form

(nF(n) \ +3F"(n) (n)F"(n) y - F(n)F"' (n)

Fiv(n) - SKj(1 - 0)

here, squeeze number S describes movement of the plates ( S > 0 corresponds to the plates moving apart, while S < 0 corresponds to the plates moving together

. (pCp),

K1 —(1 - 0) + 0f

K2 — (1 - 0) + 0

(PCp)/

K3 — k'^2ik For temperature distribution we have

Eq. (13) as the final one.

Physical quantities of interest are skin friction coefficient and Nusselt number defined as:

Mnf\ %) y — h(Q

Pnfvw 2

-lkf(d-l

Nu —

■f\ày

y — h(t)

In terms of Eqs. (8)-(11), we have l2'x2(1 - at)RexCf — K j (1 - 0)25 F"(1); v/(1 - at)Nu — -K30'(1).

Thermo physical properties of water, kerosene and copper nanoparticles are shown in Table 1.

3. Solution procedure

Pursuing standard procedure for VPM [45-51], we can write Eqs. (13) and (20) as

Fn+i(n) = Ci + C2n + C3 — + C4 —

rt n2s ns2 s3

,3! — 2T + ^T + 3! ' sFn(s) + 3Fn"(s) x I SK 1(1 — # 5| +ßFn(s) Fn" (s)

— Fn(s)Fn" (s)

0n+1(n)=B1 + B2n

— (n—s)

/ PrS(%) (Fn(s)drn (s)— nön' (s))

(Fn'(s))2 + 4,52( Fn' (s)):

y K3(1 —

ds, (23)

Consuming boundary conditions given in Eqs. (14) and (15), above equations can also be written as:

n in2 n2s ns2 s3

Fn+1(n) = C2n + C^ — — '±-1+ '!- + *-

3! 2! 2! 3!

/ sF(s) + 3F"(s)\ \ S +ßF'(s)F"(s) ds, F(s)F"' (s) ))

01(n) = B1 —

K1(1 — #

' — 2PrEcô2C22n2 /12 PrEcC42

— I — 3 PrEcC2C4ô-

— 3O PrEc82C4n6

•2 n

F2(n) = C2 n + C4

—K 1(1—

2. 5 30

50 SC4 12o SC2 C4 \

—K 1(1—

—K 1(1—

v + Î50 SßC2 C4 y

( 5030 SC42 — T6SS SßCA2

— 5040 S ßß C2C4

504 S^ßC2 C4 + 2® S C2 C4

V — 1680 s2c22c4 — ï2ô s2c4 /

( 20^0 S2ß2 C2 C42

— 8640 S2ßC2C42 + 22680 S2C2C42

y + 4320 S2ßC42 90720 "S^2 y

0n+1(n)=B1

— (n—s)

PrsfeM Fn(s) en (s)—nOn' (s))

(Fn'(s))2 + 4£2 (Fn' (s)):

\ K3(1 — ^

with n — 0, 1, 2, . . . .

Where C2, C4 and Bj are constants which can be computed using boundary conditions

F(1) = 1, F' (1)= —

(1 — ¿r

F"(1) and 0(1) = 1

respectively.

First few terms of the solution are given as

F0(n)= C2n + C4-6

00(n) = B1,

F1 (n) = ( C2n + C4-6

Ti/ 30 SC4 120 SC2C4\ <-

—« ( + ^Cfc4 y

— K1 (1 -#f5/'50»SC42 V.7

1680 SßC4

02(n) = B1 —

K1(1 — j)

' /30PrEcô2C4 N — 2PrEcô2C22 PrEcC42

— 1 PrEcC2C4S2 n

Similarly, other iterations of the solution can also be computed.

4. Results and discussions

This sections present a comprehensive discussion degrading influences of different emerging parameters on velocity and temperature profiles. We split this section into two further sections to discuss two-dimensional and axisym-metric cases respectively.

4.1. Two-dimensional flow

The behavior of squeeze number S, nanoparticle volume faction slip parameter y, Eckert number Ec and S on velocity and temperature profiles are discussed. Figures 2-7 are plotted for this purpose. Effect of squeeze number S on velocity is shown in Figure 2. It is observed that for increasing values of S, there is a decrease in velocity and this change is quite prominent in center of the channel as compared to the velocity near the walls. Figure 3 depicts the

Figure 2 Behavior of velocity for varying values of S (two-dimensional case).

Figure 3 Behavior of velocity for varying values of y (two-dimensional case).

Figure 4 Behavior of (a) velocity and (b) temperature for varying values of </> (two-dimensional case).

behavior of slip parameter y on velocity profile. A clear rise in velocity near the walls is observed due to slip. In Figure 4(a), effects of nanoparticles volume fraction ^ on

Figure 5 Behavior of temperature for varying values of S (two-dimensional case).

Figure 6 Behavior of temperature for varying values of Ec (two-dimensional case).

5=1.5, ¿=0.05, k=0.01,£C=0.1

2.0 1.8 1.6 1.4 1.2 1.0

-Cu-water - " - ,

---Cu-kerpsene ..

5=0.1, 0.2, 0.3. 0.4 \

■ ' f ' f vV

Figure 7 Behavior of temperature for varying values of 5 (two-dimensional case).

velocity are portrayed. Velocity is seen to be a decreasing function of 0 for both Cu-water and Cu-kerosene nano-fluids. It is interesting to see that velocity of Cu-kerosene is slightly on the lower side as compared to Cu-water which is due to higher values of density for kerosene. Figure 4(b) is drawn to observe the behavior of 0 on temperature profile. Slight change in temperature is observed with increase in However, temperature for Cu-kerosene is seen to be on the higher side as compared to Cu-water. This is because of higher values of Prandtl number for kerosene. This shows that by adding nanoparticles to base fluid, temperature can be enhanced and this shows the applicatiosn of nanofluids.

Figure 5 shows the behavior temperature profile under varying S. Decrease in temperature is observed for

increasing values of S. Again, temperature for Cu-kerosene possess higher values as compared to Cu-water. In Figure 6, effects of Eckert number Ec on temperature profile are shown. Definite increase in temperature is observed for both Cu-water and Cu-kerosene. This is due to the presence of viscous dissipation term in the energy equation. Maximum of the nanofluid temperature is observed at the center of the channel. As in the previous cases, Cu-kerosene has higher temperature than Cu-water. In Figure 7, effects of S on temperature are shown. Increase in temperature is quite prominent at center of the channel with increasing values of S.

Same problem is solved by using well known fourth order Runge-Kutta method. A comparison among the VPM solution, numerical solution and results obtained by Wang [44] is shown in Table 2 for the two-dimensional case. An excellent agreement between the solutions is observed. It is important to check the convergence of series solutions obtained in Eq. (26) and Eq. (27). For this purpose, numerical values for unknown constants C2, C4 and B1 are computed in Table 3 for two-dimensional case. Solution is seen to be convergent after just 6 iterations for Cu-water. For Cu-kerosene, convergence of velocity profile is attained at 6 iterations while temperature profile converges after 12 iterations. A comparison between VPM and numerical solutions is also obtained to check the effectiveness of VPM.

Figure 8

case).

Behavior of velocity for varying values of S (axisymmetric

Figure 9

case).

Behavior of velocity for varying values of S (axisymmetric

Table 3 Convergence of VPM solution for S = 1.0, y = 0.01,

0=0.02, Ec = 0.1 and S = 0.1 for two-dimensional case.

App Cu-water (Pr=6.2) Cu-kerosene (Pr=21)

C2 C4 B1 C2 C4 B1

1 1.4109 - 2.0291 1.3733 1.4087 - 2.0032 2.2573

2 1.4017 -1.9550 1.3127 1.3989 - 1.9247 1.6391

4 1.4011 -1.9510 1.3180 1.3982 - 1.9293 1.7974

6 1.4011 -1.9510 1.3181 1.3982 - 1.9203 1.8228

8 1.4011 -1.9510 1.3181 1.3982 - 1.9203 1.8250

10 1.4011 -1.9510 1.3181 1.3982 - 1.9203 1.8251

12 1.4011 -1.9510 1.3181 1.3982 - 1.9203 1.8251

14 1.4011 -1.9510 1.3181 1.3982 - 1.9203 1.8251

Nu 1.4011 -1.9510 1.3181 1.3982 - 1.9203 1.8251

Table 2 Comparison of VPM and numerical solutions for two-

dimensional case (fi = 1) with existing results.

Sf F"(1)

VPM Numerical Wang [44]

-0.9780 - 2.1915 - 2.1915 - 2.235

- 0.4977 - 2.6193 - 2.6193 - 2.6272

- 0.09998 - 2.9277 - 2.9277 - 2.9279

0 - 3.000 - 3.000 - 3.000

0.09403 - 3.0663 - 3.0663 - 3.0665

0.4341 - 3.2943 - 3.2943 - 3.2969

1.1224 - 3.708 - 3.708 - 3.714

4.2. Axisymmetric flow

For axisymmetric flow, Figures 8-14 are plotted to discuss the behavior of different emerging parameters on velocity and temperature profile. Same behavior for all the parameters is observed in this case also. Only change is on the velocity which is observed to be slightly on the lower side as compared two-dimensional case. Also, behavior of nanoparticles volume fraction on both velocity and temperature is seen to be slower for axisymmetric flow.

Table 4 is drawn to compare the results obtained by VPM and RK-4 method with ones found by Wang [44] for axisymmetric flow. Again, an excellent agreement between the solutions is seen. Convergence analysis is also carried out for this case in Table 5. It is observed that only 6 iterations are enough to obtain a convergent solution for Cu-water nanofluid. On the other hand, Cu-kerosene requires 6 iterations for velocity profile to converge and 12 iterations for a convergent temperature profile.

Numerical values for skin friction coefficient and Nusselt number are given in Tables 6 and 7. It is clear from Table 6 that with an increase in values of S and nanoparticles fraction, there is an increase in skin friction coefficient both for two-dimensional and axisymmetric cases. However, corresponding values for two-dimensional case are on the higher side compared to axisymmetric case. On the other hand, a decrease is observed in skin friction for increasing values of slip parameter y. This decrease is observed for

Figure 10 Behavior of velocity for varying values of S (axisym-metric case).

Figure 14 Behavior of temperature for varying values of S (axisym-metric case).

Figure 11 Behavior of temperature for varying values of S (axisym-metric case).

Table 4 Comparison of VPM and numerical solutions for axi-

symmetric case (ft = 0) with existing results.

VPM Numerical Wang [44]

- 0.9952 - 2.401 - 2.401 - 2.410

- 0.4997 - 2.7151 - 2.7151 - 2.7161

- 0.1 - 2.9254 - 2.9254 - 2.9252

0 - 3.000 - 3.000 - 3.000

0.11576 - 3.0622 - 3.0622 - 3.0622

0.4138 - 3.2165 - 3.2165 - 3.2160

2.081 - 3.9610 - 3.9610 - 3.9610

Figure 12 Behavior of temperature for varying values of S (axisym-metric case).

Table 5 Convergence of VPM solution for S = 1.0, y = 0.01,

0 = 0.02, Ec = 0.1 and S = 0.1 for axisymmetric case.

App Cu-water (Pr=6.2) Cu-kerosene (Pr=21)

C2 C4 B1 C2 C4 B1

1 1.4272 - 2.2397 1.3884 1.4254 - 2.2179 2.3091

2 1.4216 - 2.1946 1.3249 1.4194 - 2.1699 1.6267

4 1.4214 - 2.1930 1.3216 1.4191 - 2.1681 1.8108

6 1.4214 - 2.1930 1.3261 1.4191 - 2.1681 1.8431

8 1.4214 - 2.1930 1.3261 1.4191 - 2.1681 1.8462

10 1.4214 - 2.1930 1.3261 1.4191 - 2.1681 1.8464

12 1.4214 - 2.1930 1.3261 1.4191 - 2.1681 1.8464

14 1.4214 - 2.1930 1.3261 1.4191 - 2.1681 1.8464

Nu 1.4214 - 2.1930 1.3261 1.4191 - 2.1681 1.8464

Figure 13 Behavior of temperature for varying values of S (axisym-metric case).

both type of flows. It is also pertinent to mention that for Cu-water the values for skin friction coefficient are in the lower side as compared to Cu-kerosene nanofluid. This increase is because of higher density of kerosene.

Numerical values for Nusselt number are tabulated in Table 7. For increasing values of S, Nusselt number is observed to be decreasing function. However, with increase in nanoparticles volume fraction Eckert number Ec and S, there appears to be an increase in Nusselt number. Also, values of Cu-kerosene are again on the higher side and this is because of the higher values of Prandtl number for kerosene.

Table 6 Numerical values for skin friction coefficient.

S <t> Y Cu-water Cu-kerosene Cu-water Cu-kerosene

(two-dimensional) Cf (two-dimensional) Cf (axisymmetric) Cf (axisymmetric) Cf

-1.0 0.05 0.01 2.3256 2.3774 2.6822 2.8178

-0.5 3.0123 3.2173 3.1565 3.3910

0.5 4.0373 4.4324 3.9312 4.3083

1.0 4.4760 4.9103 4.2588 4.6910

1.0 0.00 0.01 5.1343 5.9942 4.9425 5.7703

0.05 5.2855 6.2638 5.0611 5.9841

0.10 5.3645 6.4184 5.1217 6.1062

0.15 5.3837 6.4811 5.1333 6.1532

1.0 0.05 0.01 4.5147 4.8371 4.2588 4.6320

0.02 4.2760 4.7183 4.1001 4.5142

0.03 4.1169 4.5405 3.9526 4.3501

0.04 3.9689 4.3753 3.8152 4.1973

Table 7 Numerical values for Nusselt number.

S * Ec S Cu-water Cu-kerosene Cu-water Cu-kerosene

(two-dimensional) Nu (two-dimensional) Nu (axisymmetric) Nu (axisymmetric) Nu

-1.0 0.05 0.05 0.1 1.4716 14.787 1.3473 11.259

-0.5 1.1414 5.3794 1.1224 5.1384

0.5 0.9473 2.8250 0.9427 2.7870

1.0 0.9250 2.6287 0.9096 2.5338

1.0 0.00 0.05 0.1 1.2209 3.3708 1.2024 3.2658

0.05 1.2204 3.4711 1.2001 3.3454

0.10 1.2228 3.5497 1.2022 3.4156

0.15 1.2293 3.6198 1.2096 3.4844

1.0 0.05 0.05 0.1 0.9250 2.6287 0.9096 2.5338

0.10 1.8501 5.2574 1.1819 5.0676

0.15 2.7752 7.8861 2.7290 7.6014

0.20 3.7003 10.574 3.6387 10.135

0.1 0.9250 2.6287 0.9096 2.5338

0.2 0.9575 2.6723 0.9418 2.5755

0.3 1.0115 2.7457 0.9954 2.6450

0.4 1.0872 2.8469 1.0704 2.7424

5. Conclusions

Two types of squeezing flows namely two-dimensional and axisymmetric flows are considered between parallel plates under the effects of velocity slip. Appropriate steps have been taken to obtain governing nonlinear ordinary differential equation. Solution to the problem is sought using VPM and one numerical technique (RK-4 coupled with shooting). Comparison between VPM, numerical solution and already existing solutions obtained to prove the efficiency of VPM. It can be observed that our solutions agree exceptionally well with the solutions obtained by

Wang [44]. Influence of different emerging parameters on velocity and temperature profiles for both flows is discussed with the help of graphs coupled with comprehensive explanations. Also, numerical values for skin friction coefficient and Nusselt number are also tabulated and discussed.

Acknowledgement

Authors are thankful to the anonymous reviews and editor for their worthy comments that really helped to improve the quality of presented work.

References

[1] M.J. Stefan, Versuch Uber die scheinbare adhesion, Sitzungsberichte der Akademie der Wissenschaften in Wien, Mathematik-Naturwissen 69 (1874) 713-721.

[2] O. Reynolds, On the theory of lubrication and its application to Mr. Beauchamp Tower's experiments, including an experimental determination of the viscosity of olive oil, Philosophical Transactions of the Royal Society of London 177 (1886) 157-234.

[3] F.R. Archibald, Load capacity and time relations for squeeze films, Journal of Lubrication Technology 78 (1956) A231-A245.

[4] R.J. Grimm, Squeezing flows of Newtonian liquid films: an analysis include the fluid inertia, Applied Scientific Research 32 (2) (1976) 149-166.

[5] W.A. Wolfe, Squeeze film pressures, Applied Scientific Research 14 (1965) 77-90.

[6] D.C. Kuzma, Fluid inertia effects in squeeze films, Applied Scientific Research 18 (1968) 15-20.

[7] J.A. Tichy, W.O. Winer, Inertial considerations in parallel circular squeeze film bearings, Journal of Lubrication Technology 92 (1970) 588-592.

[8] M.M. Rashidi, H. Shahmohamadi, S. Dinarvand, Analytic approximate solutions for unsteady two-dimensional and axisymmetric squeezing flows between parallel plates, Mathematical Problems in Engineering (2008) 1-13.

[9] A.M. Siddiqui, S. Irum, A.R. Ansari, Unsteady squeezing flow of a viscous MHD fluid between parallel plates, a solution using the homotopy perturbation method, Mathematical Modelling and Analysis 13 (2008) 565-576.

[10] J. Buongiorno, Convective transport in nanofluids, Journal of Heat Transfer (American Society of Mechanical Engineers) 128 (3) (2010) 240.

[11] W.A. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, International Journal of Heat and Mass Transfer 53 (2010) 2477-2483.

[12] S. Nadeem, R.U. Haq, Z.H. Khan, Heat transfer analysis of water-based nanofluid over an exponentially stretching sheet, Alexandria Engineering Journal 53 (2014) 219-224.

[13] D. Domairry, M. Sheikholeslami, H.R. Ashorynejad, R.S.R. Gorla, M. Khani, Natural convection flow of a non-Newtonian nanofluid between two vertical flat plates, Journal of Nanoengineering and Nanosystems 225 (3) (2012) 115-122.

[14] D.A. Nield, A.V. Kuznetsov, The Cheng Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid, International Journal of Heat and Mass Transfer 52 (2009) 5792-5795.

[15] U. Khan, N. Ahmed, S.I.U. Khan, S.T. Mohyud-Din, Thermo-diffusion effects on MHD stagnation point flow towards a stretching sheet in a nanofluid, Propulsion and Power Research 3 (3) (2014) 151-158.

[16] N.S. Akbar, A.W. Butt, CNT suspended nanofluid analysis in a flexible tube with ciliated walls, The European Physical Journal Plus 129 (8) (2014) 1-10.

[17] N.S. Akbar, A.W. Butt, N.F.M. Noor, Heat transfer analysis on transport of copper nanofluids due to metachronal waves of cilia, Current Nanoscience 10 (6) (2014) 807-815.

[18] F. Mabood, W.A. Khan, A.I.M. Ismail, MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet, a numerical study, Journal of Magnetism and Magnetic Materials 374 (2015) 569-576.

[19] O.D. Makinde, Fluid dynamics of parallel plates viscometer: a case study of series summation technique, Quaestiones Mathematicae 26 (4) (2013) 405-417.

[20] O.D. Makinde, Collapsible tube flow - a mathematical model, Romanian Journal of Physics 50 (2005) 493-506.

[21] S.S. Motsa, O.D. Makinde, S. Shateyi, Application of successive linearisation method to squeezing flow with bifurcation, Advances in Mathematical Physics, Article ID 410620, 6 pages, 2014.

[22] O.D. Makinde, S. Khamis, M.S. Tshehla, O. Franks, Analysis of heat transfer in Berman flow of nanofluids with Navier slip, viscous dissipation, and convective cooling, Advances in Mathematical Physics, Article ID 809367, 13 pages, 2014.

[23] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, International Mechanical Engineering Congress and Exposition, ASME, FED 231/MD, San Francisco, USA99-105.

[24] N.S. Akbar, S. Nadeem, R.U. Haq, Z.H. Khan, Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface with convective boundary condition, Chinese Journal of Aeronautics 26 (6) (2013) 1389-1397.

[25] S. Nadeem, R.U. Haq, Effect of thermal radiation for megnetohydrodynamic boundary layer flow of a nanofluid past a stretching sheet with convective boundary conditions, Journal of Computational and Theoretical Nanoscience 11 (2013) 32-40.

[26] S. Abbasbandy, A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials, J. Comput. Appl. Math. 207 (2007) 59-63.

[27] S. Abbasbandy, Numerical solutions of nonlinear KleinGordon equation by variational iteration method, Internat. J, Numer. Meth. Engrg 70 (2007) 876-881.

[28] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burger's and coupled Burger's equations, J. Comput. Appl. Math. 181 (2005) 245-251.

[29] M.A. Noor, S.T. Mohyud-Din, Variational iteration technique for solving higher order boundary value problems, Appl. Math. Comp 189 (2007) 1929-1942.

[30] M.A. Abdou, A.A. Soliman, New applications of variational iteration method, Phys, D 211 (1-2) (2005) 1-8.

[31] G. Adomian, Solving frontier problems of physics: the decomposition method, Vol. 60, Kluwer Academic, Boston, Mass, USA, 1994.

[32] G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Computer and Mathematics with Applications 13 (1990) 17-43.

[33] M. Asadullah, U. Khan, N. Ahmed, R. Manzoor, S.T. Mohyud-Din, MHD flow of a Jeffery fluid in converging and diverging channels, International Journal of Modern Mathematical Sciences 6 (2013) 92-106.

[34] Y. Cherruault, G. Adomian, Decompositon method, a new proof of convergence, Mathematical and Computer Modeling 18 (1993) 103 -106.

[35] S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method, Applied Mathematics and Computation 172 (1) (2006) 485-490.

[36] S. Abbasbandy, Modified homotopy perturbation method for nonlinear equations and comparison with adomian decomposition method, Applied Mathematics and Computation 172 (1) (2006) 431-438.

[37] S.T. Mohyud-Din, A. Yildirim, S.A. Sezer, Analytical approach to a slowly deforming channel flow with weak permeability, Zeitschrift fur Naturforschung A, A Journal of Physical Sciences (2010) 1033-1038.

[38] S. Abbasbandy, The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation, Physics Letters A 361 (2007) 478-483.

[39] A. Hussain, S.T. Mohyud-Din, T.A. Cheema, Analytical and numerical approaches to squeezing flow and heat transfer between two parallel disks with velocity slip and temperature jump, Chinese Physics Letters 29 (2012) 114705.

[40] R. Ellahi, The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: analytical solutions, Applied Mathematical Modeling 37 (2013) 1451-1467.

[41] R. Ellahi, M. Raza, K. Vafai, Series solutions of non-Newtonian nanofluids with Reynolds' model and Vogel's model by means of the homotopy analysis method, Mathematical and Computer Modelling 55 (2012) 1876-1891.

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

[43] M. Sheikholeslami, D.D. Ganji, H.R. Ashorynejad, Investigation of squeezing unsteady nanofluid flow using ADM, Powder Technology 239 (2013) 259-265.

[44] C.Y. Wang, The squeezing of fluid between two plates, Journal of Applied Mechanics 43 (4) (1976) 579-583.

[45] M.A. Noor, S.T. Mohyud-Din, A. Waheed, Variation of parameters method for solving fifth-order boundary value problems, Applied Mathematics and Information Sciences 2 (2008) 135-141.

[46] U. Khan, N. Ahmed, Z.A. Zaidi, S.U. Jan, S.T. Mohyud-Din, On Jeffery-Hamel flows, International Journal of Modern Mathematical Sciences 7 (2013) 236-247.

[47] U. Khan, N. Ahmed, S.I.U. Khan, Z.A. Zaidi, X.J. Yang, S.T. Mohyud-Din, On unsteady two-dimensional and axisymmetric squeezing flow between parallel plates, Alexandria Engineering Journal 53 (2014) 463-468.

[48] U. Khan, N. Ahmed, Z.A. Zaidi, M. Asadullah, S.T. Mohyud-Din, MHD squeezing flow between two infinite plates, Ain Shams Engineering Journal 5 (2014) 187-192.

[49] N. Ahmed, U. Khan, S. Ali, M. Asadullah, Y.X. Jun, S.T. Mohyud-Din, Non-Newtonian fluid flow with natural heat convection through vertical flat plates, International Journal of Modern Mathematical Sciences 8 (2013) 166-176.

[50] J.I. Ramos, On the variational iteration method and other iterative techniques for nonlinear differential equations, Applied Mathematics and Computation 199 (2008) 39-69.

[51] S.T. Mohyud-Din, A. Yildirim, Ma's variation of parameters method for Fishers equation, Advances in Applied Mathematics and Mechanics 2 (2010) 379-388.

[52] W.A. Khan, Z.H. Khan, M. Rahi, Fluid flow and heat transfer of carbon nanotubes along a flat plate with Navier slip boundary, Applied Nanoscience, 10.1007/s13204-013-0242-9.