Heat transfer analysis for squeezing flow of a Casson fluid between parallel plates

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ENGINEERING PHYSICS AND MATHEMATICS

Heat transfer analysis for squeezing ﬂow

of a Casson ﬂuid between parallel plates

Umar Khan

a

, Sheikh Irfanullah Khan

a,b

, Naveed Ahmed

a

, Saima Bano

a

,

Syed Tauseef Mohyud-Din

a,*

a

Department of Mathematics, Faculty of Sciences, HITEC University, TaxilaCantt, Pakistan

b

COMSATS Institute of Information Technology, University Road, Abbottabad, Pakistan

Received 13 August 2014; revised 28 January 2015; accepted 22 February 2015 Available online 24 October 2015

KEYWORDS Squeezing ﬂows; Homotopy perturbation method (HPM); Casson ﬂuid; Heat transfer; Numerical solution

Abstract Heat transfer analysis for the squeezing ﬂow of a Casson ﬂuid between parallel circular plates has been presented. Viable mathematical model has been constructed by using conservation laws coupled with suitable similarity transforms. This model ends up on a set of two highly nonlin-ear ordinary differential equations. Resulting equations have been solved by using a well-known analytical technique homotopy perturbation method (HPM). A numerical solution using forth order Runge–Kutta method has also been sought to support our analytical solution and the com-parison shows an excellent agreement. Flow behavior under altering involved physical parameters is also discussed and explained in detail with graphical aid. For the presented problem, values of parameters are restricted. Analysis is carried out using the following ranges of parameters; squeeze

numberð4 6 S 6 4Þ, Casson ﬂuid parameter ð0:1 6 b 6 1Þ, Prandtl number ð0:1 6 Pr 6 0:7Þ,

Eckert numberð0:1 6 Ec 6 0:7Þ and 0:1 6 d 6 0:4. Increase in velocity for squeeze number and

Casson fluid parameter is observed. Temperature profile is found to be decreasing function of squeeze number and Casson fluid parameter and increasing function of Pr, Ec and d.

2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Heat transfer occurs in many physical situations. Moving bod-ies are heated not only due to some external source but their motion against other surfaces may also produce frictional heat

which is of great interest. Sustainability of mechanical systems which consist of rapidly moving pistons or parts can be increased by proper understanding of heat transfer occurring in those systems. For proper working of these machines, lubri-cants are used to reduce the friction between parts. Rheologi-cal properties of these lubricants may vary under certain thermal conditions therefore for assembling highly efﬁcient machines heat transfer analysis is very important.

Squeezing ﬂow between orthogonally moving circular plates is involved in many practical situations. Its applications especially in polymer processing, modeling of synthetics trans-portation inside living bodies, hydro-mechanical machinery

* Corresponding author. Tel.: +92 3235577701.

E-mail address:[email protected](S.T. Mohyud-Din). Peer review under responsibility of Ain Shams University.

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and compression/injection processes are of great importance. Several researchers have considered these ﬂows and have con-tributed their effort for better understanding of such types of ﬂows. The seminal contribution in this regard can be named

to Stefan[1]. His pioneering effort has opened new doors for

the researchers and lot of studies have been carried out

follow-ing him [2–8]. Homotopy perturbation solution for

two-dimensional MHD squeezing ﬂow between parallel plates has

been determined by Siddiqui et al. [9]. Domairry and Aziz

[10]investigated the same problem for the ﬂow between

paral-lel disks. Recently, Mustafa et al.[11]examined heat and mass transfer for squeezing ﬂow between parallel plates using homo-topy analysis method (HAM).

In most of realistic models the fluids involved are not simple Newtonian. Complex rheological properties of non-Newtonian fluids cannot be captured by a single model. Differ-ent mathematical models have been used to study differDiffer-ent types of non-Newtonian fluids. One of such models is known

as Casson ﬂuid model. Refs. [12,13] showed that it is most

compatible formulation to simulate blood type fluid flow. It is clear from the literature survey that squeezing flow of Casson fluid between plates moving normal to their own surface is yet to be inspected.

Due to inherent nonlinearity of the equations governing fluid flow most of the problems lack exact solutions. Even where exact solutions are available immense simplification assumptions have been imposed. Those overly imposed suppo-sitions may not be used for more realistic flows. However to deal with this hurdle many analytical approximation tech-niques have been developed which are commonly used nowadays.

Solution to aforementioned highly nonlinear ordinary differential equation still lacks exactness due to its abstract nature. However, different attempts have been made to approximate its solution in an acceptable and accurate way. Nowadays, several approximation techniques have been devel-oped to fulﬁll this purpose[14–31]. From them, those are used most often which are easy to apply and require less computa-tional work yet provide reliable results.

Here we present heat transfer analysis for the squeezing flow of a Casson fluid between parallel plates. Mathematical form of the problem is extracted by using conservation laws along with similarity transformations. Resulting highly nonlin-ear ordinary differential equation is then solved by using homotopy perturbation method. This method has successfully been used by numerous researchers in different scientific prob-lems[32–41]. One can also see from our work that the obtained analytical solution shows excellent compatibility with numeri-cal solution obtained by Runge–Kutta order 4 [RK-4] method. 2. Governing equations

Consider an incompressible ﬂow of a Casson ﬂuid between two

parallel plates distance z = ±l(1 at)1/2_{= ±h(t) apart,}

where l is the initial position (at time t = 0). 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 (Scheme 1). Also, viscous dissipation effects are retained to study the generation of heat due to friction caused by shear in the ﬂow. Rheological equation of Casson

ﬂuid is deﬁned as under[42–46]

sij¼ lBþ p_y 2p h i 2eij; ð1Þ

lBis dynamic viscosity of the non-Newtonian ﬂuid, pyis yield

stress of ﬂuid and p is the product of component of deforma-tion rate with itself, i.e. p = eijeij, where eijis the (i, j)th

com-ponent of the deformation rate.

The equations governing the flow are: @ û @ ^xþ @^v @ ^y¼ 0; ð2Þ @ û @tþ û @ û @ ^xþ ^v @ û @ ^y¼ 1 q @ ^p @ ^xþ t 1 þ 1 b 2@ 2 ^ u @ ^x2þ @2u^ @ ^y2þ @2^v @ ^y@ ^x ; @^v @tþ û @^v @ ^xþ ^v @^v @ ^y¼ 1 q @ ^p @ ^yþ t 1 þ 1 b 2@ 2 ^ v @ ^x2þ @2v^ @ ^y2þ @2û @ ^y@ ^x ; ð3Þ @T @tþ û @T @ ^xþ ^v @T @ ^y¼ k qCp @2T @ ^x2þ @2T @ ^y2 þ m Cp 1þ1 b 2 @ 2 ^ u @ ^x2 2 þ @ 2 ^ u @ ^y2þ @2^v @ ^x2 2 þ 2 @ 2 ^ v @ ^y2 2! :ð4Þ In the above equations ûand ^vare the velocity components in ^x and ^y-directions respectively, ^pis the pressure, T is the temperature, m is the kinematic viscosity of the fluid and b¼ l_B ffiffiffiffiffiffiffi2pc

p

=py is the Casson ﬂuid parameter. Also, q is

den-sity of the ﬂuid, Cpis the speciﬁc heat and k is thermal

conduc-tivity of the ﬂuid. Viscosity of the ﬂuid is taken as constant and it does not depend on temperature.

Boundary conditions for the ﬂow problem are ^ u¼ 0; ^v¼ vw¼dtdh; T¼ TH; at ^y¼ hðtÞ; @ ^u @ ^y¼ 0; @T @ ^y¼ 0; ^v¼ 0; at ^y¼ 0: ð5Þ We can simplify this system of equations by eliminating

pressure terms from Eqs.(2) and (3)and using Eq.(1). After

cross differentiation and introducing vorticity x, we get @x @t þ ^u @x @ ^xþ ^v @x @ ^y¼ t 1 þ 1 b @2x @ ^x2þ @2x @ ^y2 ; ð6Þ where x¼ @^v @ ^x @ ^u @ ^y : ð7Þ

Using transform introduced by [11,14] for a

two-dimensional ﬂow ^

u¼ a^x

½2ð1 atÞF

0_ðgÞ; _ð8Þ

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^ v¼ al ½2ð1 atÞ1=2FðgÞ; ð9Þ h¼ T TH ; ð10Þ where g¼ y^ ½lð1 atÞ1=2: ð11Þ

Substituting Eqs.(8–11)in(5)accompanied by Eq.(7), we

obtain a nonlinear ordinary differential equation for Casson ﬂuid ﬂow as 1þ1 b Fiv_{ðgÞ SðgFðgÞ þ 3F}00_ðgÞ þ F0_ðgÞF00_{ðgÞ FðgÞF}000_{ðgÞÞ ¼ 0;} _ð12Þ

Also, substituting Eqs.(7–11) in(4), we get a differential equation of the form

h00ðgÞ þ Pr S FðgÞhð 0ðgÞ gh0ðgÞÞ Pr Ec 1 þ1 b ðF00_ðgÞÞ2 þ 4d2_ðF0_ðgÞÞ2 ¼ 0; ð13Þ

where S = al2/2m is the non-dimensional Squeeze number,

Pr¼lCp k is Prandtl number, Ec¼ 1 Cp a^x 2ð1atÞ 2 is Eckert number and d¼l ^

x. Here, squeeze number S describes movement of the

plates (S > 0 corresponds to the plates moving apart, while

S< 0 corresponds to the plates coming together). Also,

Ec= 0 corresponds to the case when viscous dissipation

effects are neglected.

Using Eqs.(8–11)boundary conditions Eq.(6)also reduce

to

Fð0Þ ¼ 0; F00_{ð0Þ ¼ 0; Fð1Þ ¼ 1; F}0_{ð1Þ ¼ 0;} _ð14Þ

h0ð0Þ ¼ 0; hð1Þ ¼ 1: ð15Þ

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

Cf¼ m 1 þ 1 b @ ^u @ ^y ^ y¼hðtÞ v2 w ; Nu¼ lk @T @ ^y ^ y¼hðtÞ kTH :

In terms of Eqs.(8–11), we have

l2=x2_{ð1 atÞRe} xCf¼ 1 þ1_b F00ð1Þ; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 atÞ p Nu¼ h0ð1Þ: ð16Þ 3. Solution procedure

Using standard procedure for HPM [27–33], we construct a

Homotopy as for Eqs.(13) and (14)as:

HðFðgÞ; pÞ ¼ ð1 pÞðFiv_{ðgÞÞ þ p F}iv_{ðgÞ S} b bþ 1 ðgFðgÞ þ 3F00_{ðgÞ þ F}0_ðgÞF00_{ðgÞ FðgÞF}000_ðgÞÞÞ; _ð17Þ and HðhðgÞ; pÞ ¼ ð1 pÞðh00ðgÞÞ þ p h00ðgÞ þ Pr SðFðgÞh0ðgÞ gh0ðgÞÞ Pr Ec 1 þ1 b ðF00_ðgÞÞ2 þ 4d2 F0ðgÞ ð Þ2 : ð18Þ

Here p is the embedding parameter and belongs to [0, 1]. The changing process of p from zero to unity is just that of solution from F0(g) to F(g).

Here, F(g) and h(g) are considered in a series form as FðgÞ ¼ F0ðgÞ þ pF1ðgÞ þ p

2

F2ðgÞ þ . . . ; ð19Þ

and

hðgÞ ¼ h0ðgÞ þ ph1ðgÞ þ p2h2ðgÞ þ . . . ð20Þ

Substituting Eqs.(19) and (20) into(17) and (18)

respec-tively and applying the same procedure on boundary

condi-tions(14) and (15)after comparing the like powers of p, we

get system of nonlinear equations as follows: F₀ivðgÞ ¼ 0;

h₀00ðgÞ ¼ 0; ð21Þ

With boundary conditions

F0ð0Þ ¼ 0; F000ð0Þ ¼ 0; F0ð1Þ ¼ 1; F00ð1Þ ¼ 0;

h₀0ð0Þ ¼ 0; h0ð1Þ ¼ 1:

ð22Þ First order deformation equations:

F1ivðgÞ S b bþ1 gF0ðgÞ þ 3F000ðgÞ þ F 0 0ðgÞF 00 0ðgÞ F0ðgÞF0000ðgÞ ¼ 0; h₁00ðgÞ þ PrSðF0ðgÞh00ðgÞ gh 0 0ðgÞÞ Pr Ec 1 þ1 b F000ðgÞ 2 þ 4d2 ðF0 0ðgÞÞ 2 ¼ 0: ð23Þ With F1ð0Þ ¼ 0; F100ð0Þ ¼ 0; F1ð1Þ ¼ 0; F00ð1Þ ¼ 0; h₁0ð0Þ ¼ 0; h1ð1Þ ¼ 0: ð24Þ Second order deformation equations:

F₂ivðgÞ S b bþ1 gF1ðgÞ þ 3F100ðgÞ þ ðF 0 1ðgÞF 00 0ðgÞ þ F 0 0ðgÞF 00 1ðgÞÞ F1ðgÞF0000ðgÞ þ F0ðgÞF1000ðgÞÞ ¼ 0; h₂00ðgÞ þ PrSððF1ðgÞh00ðgÞ þ F0ðgÞh10ðgÞÞ gh 0 1ðgÞÞ Pr Ec 1 þ1 b ð2F00 0ðgÞF 00 1ðgÞ þ 8d 2_ðF0 1ðgÞF 0 0ðgÞÞÞ ¼ 0: ð25Þ With F2ð0Þ ¼ 0; F200ð0Þ ¼ 0; F2ð1Þ ¼ 0; F20ð1Þ ¼ 0; h₂0ð0Þ ¼ 0; h2ð1Þ ¼ 0: ð26Þ And so on.

Solving Eqs.(21, 23, and 25)with associated boundary

con-ditions we get solutions of the form: Zeroth order solutions:

F0ðgÞ ¼ 1₂g3þ3₂g;

h0ðgÞ ¼ 1:

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First order solutions: F1ðgÞ ¼ S _bþ1b 1 280g 71 10g 5_þ53 280g 313 40g ; h1ðgÞ ¼ 9Pr Ec _bþ1b 1 2d 2_g2_þ1 12g 41 6d 2_g4_þ1 30d 2_g6 þ b bþ1 3Pr Ecþ24 5Pr Ecd 2 g b bþ1 9 4Pr Ec 3 2Pr Ecd 2 : ð28Þ Second order solutions:

F2ðgÞ ¼ 3 70S 2 b bþ 1 1 3960g 11_þ 5 432g 9193 280g 7_þ53 60g 5 b bþ 1 ₁₈₇₁ 38808S 2 g3_þ 25477 1293600S 2 g ; h2ðgÞ ¼ 9 10Pr Ec b bþ 1 5 56Pr Sg 8_þ1 30S Prd 2_g10 13 56S Prd 2_g8_þ5 6S Prd 2_g6 1 6S Prg 6_{þ d}2 g61 4S Prg 5 2 5S Prd 2 g55 4S Prd 2 g5 þ5 2g 4_5d2_g4_þ5 4S Prg 3 þ4 3S Prd 2_g3_{þ 15d}2_g2 0 B B B B B B B B B B @ 1 C C C C C C C C C C A þ b bþ 1 81 280S Pr 2 Ecþ1 4S Pr 2_Ecd2 þ 3 Pr2 Ecþ24 5 Pr Ecd 2 g b bþ 1 11 80Pr 2_Ecd 19 400S Pr Ecd 2 9 4Pr Ec 3 2Pr Ecd 2 :

ð

29Þ Other iterations can be obtained in a similar manner. Finally solution can be written as:

FðgÞ ¼ F0ðgÞ þ F1ðgÞ þ F2ðgÞ þ . . . ;

hðgÞ ¼ h0ðgÞ þ ph1ðgÞ þ p2h2ðgÞ þ . . .

ð30Þ

4. Results and discussions

This section highlights the behavior of different emerging

parameters on radial velocity (F0_{(g)) and temperature proﬁle}

(h(g)). For this purposeFigs. 1–12are displayed.Fig. 1depicts the effects of increasing values for squeeze number S on radial velocity F0_{(g). Decrease in F}0_{(g) is observed for an increase in}

squeeze number S for g 6 0:4: A sudden change in the behav-ior is seen in F0_{(g) for 0:4 < g 6 1: That is, there is an increase}

in velocity for these values of g. InFig. 2behavior of S on tem-perature proﬁle h(g) is shown. A clear increase in temtem-perature proﬁle is observed for increasing values of squeeze number S. It is pertinent to mention that reduction in velocity is directly

Figure 1 Effects of S on F0_(g).

Figure 2 Effects of S on h(g).

Figure 3 Effects of b on F0_{(g) for S > 0.}

Figure 4 Effects of b on h(g) for S > 0.

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linked with a thinner boundary layer. Also, thermal boundary layer is a decreasing function for increasing S.

Figs. 3–6are presented to study the behaviors of velocity and temperature profiles for two different cases with increasing values of Casson fluid parameter b, one is when plates are moving apart coming together (S > 0) and second when plates are moving apart (S < 0). It can be observed fromFig. 3that for S > 0, increase in b increases the velocity profile for g 6 0:4: On the other hand for 0:4 < g 6 1; a sudden decrease in F0_{(g) can be also be observed. In}_{Fig. 4}_{, behavior of}

temper-ature proﬁle is depicted for increasing b when S > 0. A clear decrease in h(g) is seen for increasing values of Casson ﬂuid parameter. Totally opposite behavior is witnessed for increas-ing values of b when plates are movincreas-ing apart (S < 0) for Figure 6 Effects of b on h(g) for S < 0.

Figure 7 Effects of Pr on h(g) for S > 0.

Figure 8 Effects of Pr on h(g) for S < 0.

Figure 9 Effects of Ec on h(g) for S > 0.

Figure 10 Effects of Ec on h(g) for S < 0.

Figure 11 Effects of d on h(g) for S > 0.

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velocity proﬁle. That is velocity decreases for g 6 0:4 and sud-denly starts increasing for 0:4 < g 6 1: However, this change is quite rapid as compared to the case when S > 0. For temper-ature h(g), there is not much difference for the cases when S> 0 or S < 0 with increasing Casson ﬂuid parameter. Here,

it is important to mention that b =1 gives us the case for a

simple Newtonian ﬂuid.

Effects of Prandtl number Pr, Eckert number Ec and d on

temperature proﬁle are depicted inFigs. 7–12. Almost similar

behavior is seen for all these parameters both for S > 0 and

S< 0. That is h(g) is an increasing function for all these

parameters. Only change here is that, for the case S < 0 change in temperature profile is slightly rapid. Increase in tem-perature is significantly due the presence of viscous dissipation term. Also, thermal boundary layer thickness decreases with increase in all these parameters. The range of Prandtl number is taken in a way that it incorporates some fluids in real life (Pr = 0.44 (Mercury), Pr = 0.7 (Air)).

To check the convergence of the solution obtained by HPM,Table 1is drawn. It can be observed that only 6 itera-tions of the analytical solution are enough for a convergent solution. For the temperature proﬁle, 7 iterations are used to obtain a convergent solution. It is pertinent to mention here that the solutions obtained by Homotopy perturbation method are convergent and the convergence of the said tech-nique is already proved in the literature.[34–36].

Same problem is solved by a well-known numerical tech-nique Runge–Kutta coupled with shooting. A comparison of

all solutions is shown inTable 2for radial velocity temperature proﬁle. It can be observed that the solutions agree well with

each other. Table 3 gives a comparison of current study to

already existing solutions obtained by Wang [14]. It can be

observed that our results are in excellent agreement with the solutions obtained by Wang.

Numerical values for skin friction coefﬁcient and Nusselt

number are presented in Tables 4 and 5. Observations show

that magnitude of skin friction coefﬁcient is increasing func-tion of S while effects of b are quite opposite to that of S. Also, magnitude of Nusselt number is decreasing function for increasing values of S and b. On the other hand, for increasing values of Pr, Ec and d, magnitude of Nusselt number is observed as increasing function.

Table 1 Convergence of HPM solution for b = 0.5, Pr = 0.5,

Ec= 0.5, d = 0.2. Number of iterations S= 1 S=1 F0₍₀₎ _h(0) _F0₍₀₎ _h(0) 1 1.500000 1.00000 1.500000 1.00000 2 1.469097 1.597320 1.530952 1.521081 3 1.471235 1.608531 1.533140 1.717126 4 1.471077 1.608492 1.533299 1.729715 5 1.471088 1.608403 1.533311 1.729220 6 1.471088 1.608405 1.533312 1.729217 7 1.471088 1.608405 1.533312 1.729214 8 1.471088 1.608405 1.533312 1.729214 10 1.471088 1.608405 1.533312 1.729214 15 1.471088 1.608405 1.533312 1.729214

Table 2 Comparison between numerical and HPM solutions for b = 0.5, Pr = 0.5, Ec = 0.5, d=0.2.

g F0_{(g) HPM} _S_{= 1 numerical} _{h(g) HPM} _Numerical _F0_{(g) HPM} _S₌_{1 numerical} _{h(g) HPM} _Numerical

0 1.471088 1.471088 1.608405 1.608405 1.533312 1.533312 1.729214 1.729214 0.1 1.457816 1.457816 1.607067 1.607067 1.516264 1.516264 1.727735 1.727735 0.2 1.417826 1.417826 1.602574 1.602574 1.465347 1.465347 1.722486 1.722486 0.3 1.358507 1.358507 1.593464 1.593464 1.381234 1.381234 1.711061 1.711061 0.4 1.255219 1.255219 1.577207 1.577207 1.265029 1.265029 1.689567 1.689567 0.5 1.130494 1.130494 1.550065 1.550065 1.118232 1.118232 1.652786 1.652786 0.6 0.974841 0.974841 1.506874 1.506874 0.942704 0.942704 1.594420 1.594420 0.7 0.786351 0.786351 1.440742 1.440742 0.740614 0.740614 1.507393 1.507393 0.8 0.562793 0.562793 1.342629 1.342629 0.514381 0.514381 1.384217 1.384217 0.9 0.301621 0.301621 1.200787 1.200787 0.266604 0.266604 1.217417 1.217417 1.0 0 0 0 0 0 0 0 0

Table 3 Comparison of HPM and numerical solutions with

existing results. Sﬂ Present results (HPM) Present results (RK-4) Wang [14] 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

Table 4 Numerical values for skin friction coefﬁcient.

S b ð1 þ1 bÞF 00_ð1Þ 1.5 0.4 9.364200 0.5 10.136144 0.5 10.850814 1.5 11.517511 1.0 0.1 32.277264 0.3 12.263610 0.5 8.253052 1.0 0.1 33.706146 0.3 13.694188 0.5 9.685823

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5. Conclusions

Heat transfer analysis for the Squeezing flow of a non-Newtonian fluid namely Casson fluid is considered between parallel circular plates. Non-linear ordinary differential equa-tions governing the flow is obtained by using conservation laws along with viable similarities transformations. Homotopy per-turbation method has been used then to obtain analytical solu-tion to the problem. A numerical solusolu-tion has also been sought by using RK-4 coupled with shooting method. Comparison between both the solutions exposes their great compatibility. Effects of different parameters on the flow behavior are shown with the help of graphical figures accompanied by comprehen-sive discussion. Also, numerical values of skin friction coeffi-cient and Nusselt number are tabulated and influence of involved parameters is discussed.

Acknowledgement

The authors are very grateful to the anonymous reviews whose kind suggestions made this work more presentable.

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Table 5 Numerical values for Nusselt number.

S b Pr Ec d h0₍₁₎ 1.5 0.1 0.4 0.2 0.1 2.735159 0.5 2.698735 0.5 2.666799 1.5 2.638843 0.5 0.1 2.666799 0.3 1.050966 0.5 0.727899 0.1 0.1 0.669604 0.3 2.002994 0.5 3.328689 0.4 0.1 1.333399 0.2 2.666799 0.3 4.000199 0.2 0.1 2.666799 0.2 2.790951 0.3 2.997870

(8)

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