Heat transfer squeezed flow of Carreau fluid over a sensor surface with variable thermal conductivity: A numerical study

Heat transfer squeezed flow of Carreau fluid over a sensor surface

with variable thermal conductivity: A numerical study

Mair Khan

⇑

, M.Y. Malik, T. Salahuddin, Imad Khan

Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan

a r t i c l e i n f o

Article history:

Received 25 September 2016

Received in revised form 22 October 2016 Accepted 27 October 2016

Available online 4 November 2016 Keywords:

Heat transfer Squeezed flow Carreau fluid Sensor surface

Variable thermal conductivity Shooting method

a b s t r a c t

Present phenomenon is dedicated to investigate the heat transfer and squeezed flow of Carreau fluid over a sensor surface. The thermal conductivity of the fluid is assumed to be temperature dependent. After assimilating these assumptions, the appropriate transformations are used to formulate the partial differ-ential equations into non-dimensional system of ordinary differdiffer-ential equations. Solutions for the bound-ary layer momentum and heat equations are accomplished by a well-known numerical technique namely shooting method. The related essential physical parameters are visualized through graphs and tables. It is found that the velocity profile increases by increasing squeezed flow parameter b, permeable velocity parameter f0, power law index n and Weissenberg number We Similarly, temperature profile increases

by increasing small parameter

.

Ó 2016 Published 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

In recent years substantial attention has been paid on heat transfer due to its rapid progress in fluid mechanics and its signif-icance in engineering and industrial processes such as nuclear reactor cooling, space cooling, heat conduction in tissues, energy production, biomedical applications such as magnetic drug target-ing, etc. Gupta et al.[1]presented the numerical solution of flow between two parallel plates approaching each other symmetri-cally. Uddin et al.[2] discussed the numerical and experimental analysis of squeezed flow of solid sphere by using Carreau-fluid. Rohni et al.[3]investigated the two dimensional unsteady flow heat transfer flow of nanofluid towards a shrinking sheet with wall mass suction. Mahgoub[2]studied the heat transfer in a porous media with forced convection over a horizontal flat plate. Nandy

[4] studied the two dimensional heat transfer flow of a non-Newtonian fluid over a stretching sheet with thermal slip. They found that temperature field enhances with increasing values of Eckert number and magnetic parameter. Bhattacharyya[5] exam-ined the heat transfer and MHD stagnation-point flow of Casson fluid towards a stretching sheet. Akbar et al. [6] discussed the approximate solution of MHD flow of Eyering-Powell fluid over a stretching sheet. Raju et al.[7]studied the two dimensional heat transfer properties on flow of steep plats with dissipative in the

induced area. Davarnejad et al.[8]presented the concept of con-vective turbulent flow and heat transfer in the magnesium Oxide–water nanofluid. Khan et al.[9]analyzed the approximate solution of heat transfer boundary layer flow induced by nonlinear stretching sheet. Yasin et al.[10]studied the numerical solution of MHD heat transfer flow of viscous fluid with stagnation-point, par-tial slip and joule heating. They found that unique solution exists for the stretching sheet. Akbar et al.[11]presented the computa-tional study of double-diffusive natural convective boundary-layer flow of a nanofluid over a stretching sheet. Sandeep et al.

[12]studied the two dimensional MHD flow of a dusty nanofluid and heat transfer over an exponentially stretching surface. Misra et al.[13]presented the concept of chemical reaction as well as mass and heat transfer on the MHD oscillatory flow of blood. They found that the mass diffusivity decreases with the increase in mass transfer rate. Makanda et al.[14]investigated the Casson fluid over a two dimensional MHD free convection flow with partial slip in non-darcy porous medium. Raju et al.[15] studied the heat and mass transfer boundary layer flow of Casson fluid past an perme-able exponentially stretching surface. Balla et al.[16]investigated the heat transfer of a viscous fluid with unsteady Newtonian, incompressible fluid past a precipitately underway semi-infinite vertical plate. Akbar et al.[17]analyzed the MHD nanofluid flow over a stretching surface containing gyrostatic microorganisms.

The analysis of squeezing flow finds several industrial and prac-tical applications in different areas such as physical, chemical engi-neering, biophysical, food engiengi-neering, polymer processing, http://dx.doi.org/10.1016/j.rinp.2016.10.024

2211-3797/Ó 2016 Published 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/).

⇑Corresponding author.

E-mail address:[email protected](M. Khan).

Contents lists available atScienceDirect

Results in Physics

comparison and injection molding and many others. Hayat et al.

[18]investigated the squeezed flow of second grade fluid over a sensor surface. Akbar et al.[19]studied the incompressible flow of two dimensional viscous nanofluid over a vertical stretching sheet. Akbar et al.[20]investigated the physical analysis for vari-able thermal conductivity over a stretching sheet with convective slip boundary conditions. Ahmed et al.[21]studied the effects of MHD Casson fluid over a squeezed flow between two parallel plats. Haq et al. [22]presented the concept of MHD squeezed flow of non-Newtonian fluid namely Casson fluid over a sensor surface. Rana et al.[23]deliberated the computational study of Casson fluid over a stretching sheet in the presence parietal slip and internal heating.

Current analysis aims to deliberate the variable thermal con-ductivity effect on squeezing flow of Carreau fluid model over a sensor surface. The numerical solutions are obtained by using shooting method. The influence of embedded physical parameters temperature and velocity profiles are shown graphically.

2. Mathematical formulation

Consider incompressible, two-dimensional unsteady boundary layer flow of Carreau fluid towards a continuously permeable sen-sor surface.Fig. 1shows the closed squeezed channel with height hðtÞ assumed to be much larger than the boundary layer thickness. Further it is assumed that the squeezing free stream is supposed to be happened from the tip to the surface. Moreover, it is assumed that the lower plate is while immovable upper plate is squeezed.

The continuity, momentum and energy equations after applying the boundary layer approximation will be[22,24]

@u @xþ @

v

@y¼ 0; ð1Þ @u @tþ u @u @xþ

v

@u @y¼  1

q

@p@x
 þ

m

@ 2 u @y2þ 3 ðn  1Þ 2

C

2 @u @y
2 @2 u @y2 ! " # ; ð2Þ @U @tþ U @U @x¼  1

q

@p@x
 ; ð3Þ @T @tþ u @T @xþ

v

@T @x¼ @ @y

a

ðTÞ @T @y
 ; ð4Þ

where u and

v

are the velocity components along the x and y direc-tions respectively, U is the free stream velocity, n power law index,

m

kinematic viscosity,Cis the time constant, p is the fluid pressure,

a

ðTÞ is the variable thermal conductivity, T is temperature, t is time,

a

is the thermal diffusivity and

q

is the density. After eliminating the pressure gradient the momentum Eqs.(2) and (3)takes the form

@u @tþ u @u @xþ

v

@u @y¼ @U @tþ U @U @xþ

m

@2 u @y2þ 3 ðn  1Þ 2

C

2 @u @y
2 @2 u @y2 ! " # ; ð5Þ

with boundary conditions

uðx;0;tÞ ¼ 0;

v

ðx;0;tÞ ¼

m

0ðtÞ; uðx;1;tÞ ¼ Uðx;tÞ; k@Tðx;0;tÞ_@y ¼ qðxÞ; Tðx;1;tÞ ¼ T1:

ð6Þ

Here T1; Uðx; tÞ and qðxÞ are the free stream temperature, wall heat flux and free stream velocity, respectively. Here thermal con-ductivity

a

ðTÞ is of the form

a

¼

a

1ð1 þ

hÞ, where

is small quan-tity. Suppose that wall is a function of a heat flux qðxÞ. The velocity at the sensor surface is represented by reference velocity

v

0ðtÞ and is considered when the surface is permeable. Eqs.(4) and (5)can be transformed into differential form by using following similarity transformations:

g

¼ y ffiffia m p ; w ¼ xpffiffiffiffiffiffia

m

fð Þ; a ¼

g

1 ðsþbtÞ; u ¼ axf0ð

g

Þ;

v

¼ f ð

g

Þpffiffiffiffiffiffia

m

; U ¼ ax; hð

g

Þ ¼TT1 q0x ffiffi m a p ; ð7Þ where qðxÞ ¼ q0x and

v

0ðtÞ ¼

v

pffiffiffia.

Where b is a squeezed flow index, s is a arbitrary constant, a is strength squeezing flow parameter, q0is heat flux and k is thermal conductivity.

Note that Eq.(1)is identically satisfied while Eqs.(4) and (5)

takes the form

f000þ f þb

g

2
 f000 f 0 2 þ bðf0_{ 1Þ þ 3}ðn  1Þ 2 We 2_ðf00_Þ2 ðf000_{Þ þ 1 ¼ 0;} ð8Þ ð1 þ

hÞh00_{þ Pr f þ}b

g

2
 h0_{þ Pr}

_ðh0_Þ2  Pr f0_þb 2
 h ¼ 0; ð9Þ

and the corresponding transformed boundary conditions are

f ð0Þ ¼ f0; f 0_{ð0Þ ¼ 0; h}0_{ð0Þ ¼ 1;} f0ð1Þ ! 1; h 1ð Þ ! 0; ð10Þ where Pr¼t ris Prandtl number, f0¼ ffiffiffi

t

p

is the permeable velocity and We2¼a3_x2_C2

m is Weissenberg number.

Using the boundary approximations, the wall shear stress at the sensor surface is given by

s

w¼@u_@yþ

C

2ðn  1Þ₂ @u_@y

3

; qw¼ k @T

@y; at y ¼ 0: ð11Þ

the coefficient of skin friction and Nusselt number are defined as

Cf ¼

s

w ax ffiffia m p ; Nu ¼qwk qxK: ð12Þ

In the dimensionless form, the skin friction and Nusselt number are defined as ffiffiffiffiffiffiffi Rex p cf¼ f00ð

g

Þ þðn1ÞWe 2 2 ðf 00_ð

_g

_ÞÞ3 h i g¼0; Nu ffiffiffiffiffiffiffiRex p ¼ ½h0_ð

_g

_Þ g¼0; ð13Þ where Rex¼ xpffiffiam.

Flow

Flow

L

h(t)

x,u

y,v

V

V

Squeezing

3. Numerical solutions

The non-linear ordinary differential equations for momentum and heat i.e. Eqs.(8) and (9)subject to the boundary conditions (10) are solved numerically using fifth order Runge–Kutta method along with shooting technique for unlike values of parameters. Let u1¼ f ; u2¼ f0; u3¼ f00; u4¼ h and u5¼ h00.

Hence the leading equations become

u01¼ u2; ð11Þ u02¼ u3; ð12Þ u03¼ 2 2þ 3ðn  1Þu2 3We 2 h i  u1þb

g

2
 u3þ u22 bðu2 1Þ  1   ; ð13Þ u04¼ u5; ð14Þ u05¼ 1 ð1 þ

u4Þ Pr u1þb

g

2
 u5 Pr

u25þ Pr u2þb 2
 u4   ¼ 0; ð15Þ

with the analogous initial conditions:

u1ð0Þ ¼ f0;u2ð0Þ ¼ 0; u4ð0Þ ¼ 1; u2ð

g

Þ ! 1; u4ð

g

Þ ! 0 when

g

! 1;

ð16Þ

In order to solve this modified system of five first order ordinary differential Eqs.(11)–(15)by Runge–Kutta method five initial con-ditions are compulsory whereas only three of them are given. Two more initial conditions are required to replace u2ð

g

Þ ! 1; u4ð

g

Þ ! 0 when

g

! 1. Thus it is assumed that u2ð0Þ ¼ s1 and u4ð0Þ ¼ s2. These unknown initial conditions u2ð0Þ and u4ð0Þ are first pre-dicted and consequently resolute using Newton–Raphson’s method for individual parameters with respect to the given free stream conditions. These slopes are modified such that the bound-ary conditions are satisfied for

g

! 1. The resulting initial value problem is attempted numerically using a fifth order Runge– Kutta-Fehlberg integration scheme. The convergence criteria and step size are taken to be 106and 0:1 for all numerical solutions.

4. Results and discussion

Numerical computations are carried out for various sets of per-tinent parameters on skin friction coefficient, local Nusselt num-ber, velocity and temperature profiles. The behavior of power law index n on velocity profile is illustrated inFig. 2. It is examined that the increase in power law index n causes reduction in both velocity of the fluid and boundary layer thickness. This proves that the increment in power law index n gives the field reduction and pre-vents the motion of the fluid. The dimensionless velocity profile for different values of Weissenberg number Weis sketched inFig. 3. It is noticed that large values of Weissenberg number We dimension-lies velocity profile for n > 1.Fig. 4shows the velocity distribution for different values of permeable velocity f0. It is noticed that, velocity profile reduces for suction case (i.e f0> 0). Because suc-tion causes the fluid to be more attached to the sensor surface, so velocity decreases. The velocity and temperature profiles against

g

for different values of squeezed flow index b are shown inFigs. 5 and 6. It is clear that increment in squeezed flow index

Fig. 2. Variation of f0ðgÞ with n Fig. 3. Variation of f0ðgÞ with We.

b causes reduction in both velocity and temperature distributions.

Fig.7shows the behavior of Prandtl number Pr on temperature pro-file. It is clear from figure that an increase in the values of Prandtl number Pr, the temperature profile reduces. This is due to the fact that by increasing Prandtl number Pr the thermal diffusivity of the fluid reduces, so temperature decreases.Fig. 8demonstrates the influence of small parameter

on temperature profile. An increase in small parameter

corresponds to increase in kinetic energy of the fluid particles which increases the variation of thermal charac-teristics, so the fluid temperature and boundary layer thickness increases.Fig.9shows the effect of skin friction coefficient against squeezed flow parameter b for different values of power law index n. It is observed that skin friction coefficient increases for increas-ing values of power law index n.Fig.10displays the skin friction coefficient against squeezed flow parameter b for various values of Weissenberg number We. It is noticed that the skin friction increases for large values of Weissenberg number We.Fig. 11 illus-trates the effect of local Nusselt number against squeezed flow parameter for different values of small parameter

. It is clear that,

Fig. 5. Variation of f0ðgÞ with b.

Fig. 6. Variation of h(g) with b.

Fig. 7. Variation of hðgÞ with Pr.

Fig. 8. Variation of hðgÞ with
.

Fig. 9. Variation of skin friction coefficient against squeezed flow parameter b for different values of power index n.

local Nusselt number increases on increasing values of small parameter

.Tables 1 and 2shows the behaviors of local Nusselt number and skin friction coefficient for different values of perti-nent parameters.

5. Conclusions

Two dimensional squeezed flow of Carreau fluid in presence of variable thermal conductivity is considered. The existing investiga-tion are mostly useful in applicainvestiga-tions involving solar collectors heat transfer, oil recover, etc. The main points are summarized below:

 The temperature profile reduces for large values of Prandtl number Pr.

 The temperature profile enhances by increasing small parame-ter

.

 The increasing values of squeezed flow parameter b enhances the velocity and temperature profiles.

 The skin friction coefficient enhances for large values of Weis-senberg number We and power law index n.

 The increasing values the power law index n and Weissenberg number We enlarges the velocity profile.

 The velocity profile increases for increasing values of permeable velocity parameter f0.

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Fig. 10. Variation of skin friction coefficient against squeezed flow parameter b for different values of Weissenberg number We.

Fig. 11. Variation of local Nusselt number against squeezed flow parameter b for different values of variable thermal conductivity.

Table 1

Variation of the skin friction coefficient Re12

xCfagainst different values of Weissenberg

We and power law index n.

We n CfRex1=2 0.1 1.2 0.7234 0.5 1.2 0.7320 0.9 1.2 0.7394 0.1 1.5 0.7238 0.1 1.8 0.7241 0.1 2.1 0.7244 Table 2

Variation of the Nusselt numberhð0Þ against various values of Prandtl number Pr and variable thermal conductivity.

Pr
h0_ðOÞ 0.1 0.5 0.2500 0.5 1.2 0.2514 0.9 1.2 0.2563 0.1 1.5 0.3463 0.1 1.8 0.3482 0.1 2.1 0.3514

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