Wall slip and non-integer order derivative effects on ...

Accepted Manuscript Wall slip and non-integer order derivative effects on the heat transfer flow of Maxwell fluid over an oscillating vertical plate with new definition of fractional Caputo-Fabrizio derivatives MadeehaTahir, M.A. Imran, N. Raza, M. Abdullah, Maryam Aleem PII: DOI: Reference:

S2211-3797(17)30573-9 http://dx.doi.org/10.1016/j.rinp.2017.06.001 RINP 716

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Results in Physics

Received Date: Revised Date: Accepted Date:

4 April 2017 30 May 2017 1 June 2017

Please cite this article as: MadeehaTahir, Imran, M.A., Raza, N., Abdullah, M., Aleem, M., Wall slip and non-integer order derivative effects on the heat transfer flow of Maxwell fluid over an oscillating vertical plate with new definition of fractional Caputo-Fabrizio derivatives, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp. 2017.06.001

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Wall slip and non-integer order derivative effects on the heat transfer flow of Maxwell fluid over an oscillating vertical plate with new definition of fractional Caputo-Fabrizio derivatives MadeehaTahir1, M. A. Imran*2 , N. Raza3, M. Abdullah4, Maryam Aleem2 1

Department of Mathematics, Government College University, Faisalabad, Pakistan

2

Department of Mathematics, University of Management and Technology Lahore,

Pakistan 3

Department of Mathematics, University of the Punjab Lahore, Pakistan

4

Department of Mathematics, University of Engineering and Technology Lahore,

Pakistan *

Address corresponding to M. Imran Asjad: [email protected]

Abstract: This article is focused on natural convection of unsteady flow of generalized Maxwell fluid over an oscillating vertical flat plate with constant temperature at the boundary. The Maxwell fluid with classical derivatives, describing one dimensional flow has been generalized to non-integer order derivatives known as fractional derivative with term of buoyancy. A modern definition of fractional derivative, recently introduced by Caputo and Fabrizio has been used to formulate the considered problem. Semi analytical solutions of the dimensionless problem have been obtained by using the Laplace transform. The solutions for temperature, velocity and shear stress are obtained with numerical inversion techniques of Laplace transform namely, Stehfest’s and Tzou’s algorithms. At the end, graphical illustrations for temperature, velocity, Nusselt number and shear stress are plotted. We have studied especially the influence of fractional parameter on temperature, velocity and shear stress respectively. We have observed that temperature can be enhanced for increasing the fractional parameter  while velocity and shear stress can be increased by decreasing the value of fractional parameter  . Keywords: Free convection; Slip; Maxwell fluid; Oscillation; Caputo-Fabrizio fractional derivatives; Stehfest’s and Tzou’s algorithms. 1

1. Introduction Maxwell fluid model has received much attention for being the first and one of the simplest rate type fluid models. It is still used widely specially to describe the response of some polymeric liquids. However, same like other models, Maxwell fluid model has some limitations. For example, this model does not properly describe the typical relation between shear rate and shear stress in a simple shear flow [1-2]. Most the existing studies on Maxwell fluid particularly on the analytical side are limited to momentum transfer only, see for example [3-15] and references therein. Generally, two types of boundary conditions we use in fluid problems such as no-slip boundary condition and slip boundary condition. The no-slip boundary conditions state that there is no relative motion between the wall and fluid immediately in contact with the wall. This condition is used widely perhaps due the fact that it simplifies complex situations, although it has some limitations.

There are some surfaces which are

sufficiently smooth where liquids can’t be in contact with walls and slip against the walls. For example, in capillaries the no-slip condition fails to work [16]. However, slip condition has overcome these limitations as mentioned by Navier in his pioneering work [17]. Slip condition also known as Navier condition has significant applications in lubrication, extrusion, medical sciences, especially in polishing artificial heart valves, flows through porous media, micro and nanofluids, friction studies and biological fluids [18-19]. Heat transfer due to convection has several industrial and technological applications. Their examples may be found in wire and fiber coating, manufacturing plastic films, artificial fiber, a polymer sheet, chemical processing equipment and in the design of various heat exchangers [20-22]. Therefore, our task here to extend some previous problem of classical derivatives to the fractional derivatives of any real power of the differentiation [23]. Many researchers are taking keen interest to generalize the problems of classical dynamics to fractional dynamics. However, this generalization is done using different approaches/definitions of fractional derivatives [24-38]. Based on the previous definitions and some of the difficulties therein, recently, “Caputo and 2

Fabrizio (CF) have introduced a modern definition of the fractional derivatives with an exponential but without singular kernel [39]. The CF derivative problems are found more suitable for the Laplace transform. Nehad and Khan [40] recently applied the CF derivatives to the heat transfer problem of viscoelastic fluid of second grade and obtained the exact solutions via Laplace transform. Farhad et al. [41], used the CF derivatives and studied MHD free convection flow of generalized Walters’-B fluid” model. The purpose of present study is to extend the idea of fractional derivatives to the heat transfer problem of generalized Maxwell fluid over an isothermal vertical plate that oscillating in its plane. We consider the slip and constant wall temperature at the boundary. Semi analytical solution of the dimensionless problem has been obtained by using the Laplace transform. The solutions for temperature, velocity and shear stress are obtained with numerical inversion techniques of Laplace transform namely, Stehfest’s and Tzou’s algorithm. At the end, graphical illustrations for temperature, velocity, Nusselt number and Skin friction are portrayed to see some physical aspects of the problem. 2. Mathematical formulation of the problem Let us “consider unsteady mixed convection flow of an incompressible Maxwell fluid over an oscillating vertical flat plate moving with oscillating velocity in its own plane as shown in figure below.

3

Physical Model Initially, at time t  0 , both the fluid and the plate are at rest with constant temperature

T . At time t  0, the plate is subjected to sinusoidal oscillation. More exactly, the plate begins to oscillate in its plane according to V  U 0 H  t  cos(t ) i; where U 0 is a constant of dimension of velocity, H (t ) is the unit step function, i is the unit vector in the vertical flow direction and  is the frequency of oscillation of the plate. It is also assumed slip at the wall. At the same time t  0, , the temperature of the plate is raised or lowered to a constant value Tw . The equations governing the Maxwell fluid flow related with shear stress and heat transfer due to mixed convection and thermal radiation are given by the following partial differential” equations [8,19]:

 

 1  1

  u  2u      1  1   g  T  T  ,  2 t  t y  t 

(1)

u  y, t    , 1  1   y, t    t  y 

(2)

 16 *T3   2T T cp  k 1  .  t 3kk *  y 2 

(3)

The appropriate initial and boundary conditions are:

4

u  y, 0   0,

u (0, t )  b

u ( y, t )  0, T  y, 0   T , t t 0

u (0, t )  U 0 H  t  cos(t ), y

u  , t   0,

(4)

T  0, t   Tw ,

T  , t   T .

(5)

(6)

Introducing the following non-dimensional quantities: yU 0 U 02t 1U 02 T  T u  *  * * *  u  , y  , t  ,   2,  ,  ,   , U0   U0 U 02 Tw  T  *

cp g  (Tw  T ) 16 *T 3  bU 0 Pr Gr  , Pr  , Pr  , Nr  ,b  eff U 03 k 1  Nr 3kk * 

(7)

into Eq. (1)-(6), we get

  u  2u     2  1    Gr , 1    t  t y  t  

(8)

 u   1     , t  y 

(9)

Pr

  2  1  Nr  2 . t y

(10)

where b, Gr , Pr, Nr , Preff , are the slip parameter, Grashof number, the Prandtl number, the radiation parameter, and effective Prandtl number respectively. The initial and boundary conditions become:

u  y, 0   0,

u (0, t )  b

u ( y, t )  0,   y, 0   0, t t 0

u (0, t )  H  t  cos(t ), y

u  y, t   0,

(11)

  0, t   1,

  y, t   0, y  .

(12) (13)

Caputo- Fabrizio time-fractional derivative model of order  (0,1), Eqs. (8), (9) and (10) are written as

u  y, t   2u  y, t  1   Dt  t  y 2  Gr 1   Dt   y, t  , 

5

(14)

u y, t 1   D   y, t    y  , 

(15)

t

Preff Dt  ( y, t ) 

 2 ( y, t ) , y 2

(16)

where Caputo-Fabrizio time-fractional derivative is defined as

Dt u ( y, t ) 

1 t   (t   )   exp   u ( y, t )d , 0    1 .” 1 0  1 

(17)

3. Solution of the problem 3.1 Temperature distribution Recently, Nehad at al. [40], find the solution for temperature with same initial and boundary conditions. We directly present the temperature field and its Laplace transform 

1 s

 ( y, s)  exp   y 

Pr  s s  

  , 

(18)

With its inverse Laplace transform for 0    1

 ( y, t )  1 

2 Peff 



  tx 2  sin( yx) 1 exp  x( P   x 2 )   P   x 2  dx, where   1   . 0 eff  eff 



(19)

3.2 Temperature field for the ordinary case (   1 ) The temperature expression corresponding to the ordinary case is obtained based on the property of Caputo-Fabrizio fractional derivative, namely

 ( y, t )  1 

2  sin( yx)  tx 2  exp    Pr  dx. 0 x  

(20)

Using the formula below

sin(bx)   b  exp  ax 2  dx  erf  , x 2 0 2 a 





(21)

we obtain

 y Pr   y Pr   ( y, t )  1  erf    erfc   . 2 t 2 t    

(22)

The rate of heat transfer from plate to the fluid in terms of Nuseelt number is obtained 6

by introducing the Eq. (18) into the following relation

T ( y, t ) Nu   y

y 0

   T ( y, s)   L1 T ( y, s)   L1  y  0 y  y 

 e t  erf (  t ) (23)   Preff    y 0 

4 Velocity field 4.1 Maxwell fluid fractional model ( 0    1 ) Applying the Laplace transform to Eq. (14), keeping in mind the initial condition and expression from Eq. (18) we obtain the following problem for velocity in transform domain

 Preff  s     1 s  2u ( y , s ) s 1  (1   ) s    su ( y, s)  y 2  Gr 1  (1   ) s    s exp   y s    ,      

(24)



  u (0, s) s u (0, s)  b  2 , u ( y, s)  0 as y  . 2 y s  

(25)

The solution of the problem (24)-(25) is given by

  B  bBD     E   D2  A  e   1 b A 



u ( y, s ) 



1



Ay



Be yD , D2  A (26)

Preff  s     s2 Gr  s s 1 where A   s  , D and E  2 . , B     s  (1   ) s    s  2 s    (1   ) s    

Writing u ( y, s) in suitable and simple form and determine its inverse Laplace transform but Eq. (26) is complex and it is not easy to use for some practical applications. So, we have used some numerical techniques to obtained inverse Laplace transform by using 

Stehfest’s algorithm [42]. Applying the Stehfest’s formula on u ( y, s) , the solution u ( y, t ) is found to be u ( y, t ) =

ln(2) 2 m ln(2)   d j u  y, j  , where m is a positive integer t j 1 t  

d j  (1)

j m

min( j , m )



 j 1 i   2 

i m (2i)! (m  i )!i !(i  1)!( j  i )!(2i  j )!

7

and  r  denotes the integer part of the real number r. However, we obtained another 

approximation for u ( y, s) by Tzou’s algorithm for validation of our numerical inverse Laplace u (r , t ) 

e4.7 t

 1  4.7   N1 4.7  k i   k  u r ,  Re  (1) u  r ,      , t     k 1 2  t 

where Re() is the real part, i is the imaginary unit and N1

1 is a natural number [43].

5 Shear stress 5.1 Fractional Maxwell model ( 0    1 ) In order to obtain the shear stress, we use the Eq. (26) into the following relation

u  y , s    y s 1  (1   ) s       1  A   B  bBD     . E  2  e     1 b A  s  D  A  1  (1   ) s      

  y, s  

1





Ay

 BDe  yD  ,  2 D A 

(27)

For the solution of shear stress, we have found the inverse Laplace transform by using numerical techniques [42, 43].

6 Numerical results and discussion In this paper, we study the natural convection of unsteady flow of generalized Maxwell fluid over an oscillating vertical flat plate when slip and constant temperature are considered at the boundary. Semi analytical solutions of the dimensionless problem have been obtained by using the Laplace transform. The expressions for temperature, velocity and shear stress in dimensionless form are obtained with numerical inversion techniques of Laplace transform namely, Stehfest’s and Tzou’s algorithm. At the end, we studied especially the influence of the fractional parameter  on the fluid flow behavior.

8

Fig. 1 presents the influence of non-integer order fractional parameter  . It is observed that by increasing the value of  temperature increases and thermal boundary layer thickness increases for increased in time. Fig. 2 explains the effect of Prandtl number on temperature field and observed that for larger value of the Preff reduced the temperature field. Physically, it is due to the fact that fluid viscosity becomes large and reduces the thickness of thermal boundary layer. Fig. 3 plotted between velocity and spatial variable y. It is clearly seen that velocity is a decreasing function of fractional parameter  but for large value of time magnitude of fluid velocity increases. Fig. 4 depicts the influence of Preff on fluid velocity and seen fluid velocity increases and vicinity changes for y  0.5 . Physically, the fluid viscosity is large and thickness of velocity boundary layer becomes large. Fig. 5 shows that velocity in an increasing function of Gr parameter. Fig. 6 velocity is a deceasing function of relaxation time and as we increased the value of time the magnitude of velocity increases. Fig. 7 shows the velocity field with and without slip influence. In the absence of slip fluid velocity increases while by increasing the value of slip parameter velocity reduces. Fig. 8 presented the rate of heat transfer in terms of Nusselt number for three values of Preff parameter. It can be seen that Nusselt number increases if the fractional parameter  decreases. Therefore, the heat transfer is enhanced in fluids modeled with fractional derivatives of order  . Also, the Nusselt number varies with different values of Preff and observed that Nusselt number increases if the fractional parameter  decreases. So, the memory effect described by the fractional derivative leads to an improved heat transfer analysis. Fig. 9 Plotted between shear stress and spatial variable y and it can be seen that shear stress increases if the value of fractional parameter decreases. In Fig. 10 for larger the value of Preff parameter shear stress also increases. In Fig. 11 and Fig. 12 shear stress is an increasing function of Grashof number and decreasing for relaxation time respectively. Fig. 13 presented shear stress with and without slip and can be seen for no slip at the wall shear stress increases while by increasing the slip parameter shear stress decreases. To validate our solutions obtained by means of numerical inversion Laplace transforms namely, Stehfest’s and Tzou’s algorithm. We presented equivalence relation between those techniques in Fig. 14 and Tables. 1-2. In 9

tables 3-5 we especially studied the influence of fractional parameter  on temperature, velocity and shear stress and found that temperature can be enhanced by increasing the value of fractional parameter while velocity and shear stress can be increased as we decreased the value of  .

10

Fig.1 Profiles of temperature for variation of

Fig.2 Profiles of temperature for variation

 and different time.

of Prandtl number and different time. 11

Fig.3 Profiles of velocity for variation of  and different time.

Fig.4Profiles of velocity for variation of Prandtl number and different time. 12

Fig.5 Profiles of velocity for variation of Grashof number and different time.

Fig.6 Profiles of velocity for variation of  and different time. 13

Fig.7 Profiles of velocity for variation

Fig.8 Profiles of Nusselt number for variation

of slip parameter and different time.

of Prandtl number and different time. 14

Fig.9 Profiles of shear stress for variation

Fig.10 Profiles of shear stress for variation

of  and different time.

of Prandtl number and different time.

15

Fig.11 Profiles of velocity for variation

Fig.12 Profiles of velocity for variation of  and different time.

of Grashof number and different time. 16

Fig.13 Profiles of shear stress for variation of slip parameter and different time. 17

Fig. 14 Comparison of inverse Laplace numerical algorithms for the solution of velocity and shear stress

u (y, s)

u (y, s)

y

[Stehfest’s]

[Tzou’s]

0

0.999263

0.999263

0.2

0.753465

0.753465

0.4

0.585234

0.585234

0.6

0.466973

0.466973

0.8

0.379886

0.379886

1.0

0.312532

0.312532

1.2

0.258506

0.258506

1.4

0.214275

0.214275

1.6

0.177734

0.177734

1.8

0.147448

0.147448

2.0

0.122323

0.122323

Table. 1 18

y

Shear stress

Shear stress

[Stehfest’s]

[Tzou’s]

0

0.582348

0.582301

0.2

0.392801

0.392848

0.4

0.270441

0.270518

0.6

0.194456

0.194529

0.8

0.147344

0.147399

1.0

0.116624

0.116662

1.2

0.094828

0.094854

1.4

0.078111

0.078129

1.6

0.064663

0.064677

1.8

0.053614

0.053626

2.0

0.044471

0.044481

Table. 2

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

 ( y, t )  0

 ( y, t )   0.2

 ( y, t )   0.4

 ( y, t )   0.6

 ( y, t )   0.8

 ( y, t )  1

1 0.778801 0.606531 0.472367 0.367879 0.286505 0.223130 0.173774 0.135335 0.105399 0.082085

1 0.861549 0.739747 0.633208 0.540489 0.460157 0.390834 0.331226 0.280136 0.236477 0.199268

1 0.900279 0.806586 0.719478 0.639210 0.565816 0.499157 0.438975 0.384929 0.336624 0.293637

1 0.920701 0.843806 0.770045 0.699952 0.633890 0.572081 0.514625 0.461529 0.412724 0.368079

1 0.932818 0.866733 0.802325 0.740067 0.680337 0.623418 0.569515 0.518757 0.471209 0.426883

1 0.940784 0.882154 0.824532 0.768290 0.713743 0.661155 0.610736 0.562646 0.517000 0.473871

Table 3: Effect of fractional parameter on dimensionless temperature 19

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

u ( y, t )

u ( y, t )

u ( y, t )

u ( y, t )

u ( y, t )

u ( y, t )

 0

  0.2

  0.4

  0.6

  0.8

 1

0.907899 0.804131 0.716142 0.641340 0.577485 0.522654 0.475224 0.433843 0.397408 0.365030 0.336008

0.921766 0.805540 0.708253 0.626665 0.557957 0.499720 0.449934 0.406944 0.369426 0.336340 0.306885

0.939153 0.804756 0.694290 0.603514 0.528648 0.466437 0.414179 0.369714 0.331369 0.297889 0.268344

0.962003 0.798167 0.666901 0.562723 0.480044 0.413764 0.359675 0.314590 0.276258 0.243154 0.214265

0.896499 0.801913 0.720932 0.651403 0.591466 0.539519 0.494196 0.454351 0.419029 0.387455 0.359002

0.995074 0.788074 0.615076 0.481639 0.396441 0.336717 0.283323 0.235626 0.195408 0.161784 0.133329

. Table 4: Effect of fractional parameter on dimensionless velocity.

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

 ( y, t )  0

 ( y, t )   0.2

 ( y, t )   0.4

 ( y, t )   0.6

 ( y, t )   0.8

 ( y, t )  1

0.567706 0.485549 0.416233 0.358029 0.309420 0.269033 0.235609 0.208001 0.185176 0.166226 0.150374

0.574621 0.486308 0.412445 0.351059 0.300382 0.258791 0.224798 0.197050 0.174349 0.155662 0.140121

0.577008 0.481231 0.402039 0.337157 0.284471 0.242001 0.207914 0.180563 0.158514 0.140565 0.125743

0.570277 0.465156 0.379602 0.311024 0.256810 0.214391 0.181354 0.155561 0.135212 0.118875 0.105467

0.544774 0.427125 0.333439 0.261338 0.207468 0.167964 0.139060 0.117556 0.101027 0.087809 0.076843

0.487095 0.348678 0.233975 0.151091 0.112578 0.095567 0.079492 0.064006 0.051670 0.042102 0.034242

Table 5: Effects of fractional parameter on dimensionless shear stress.

20

7 Conclusions In this paper, a modern approach of fractional derivative of Caputo and Fabrizio was used to find analytic solutions of fractional Maxwell fluid. Heat transfer analysis was also added. Numerical inversion Laplace transform technique namely, Stehfest’s and Tzou’s techniques are used to find inverse Laplace transform for temperature, velocity and shear stress. Some important key findings of this study are outlined in the following: 1. Temperature was increased for larger value of fractional parameter 2. Temperature was decreased for larger value of Prandtl number 3. Nusselt number was increased with decreasing fractional parameter. 4. Rate of heat transfer was increased with increasing the Prandtl number. 5. Fluid velocity is an increasing function for decreasing the values fractional parameter  , relaxation time  and slip parameter b respectively. 6. Magnitude of fluid velocity increases for larger values of time t. 7. Shear stress was increased for decreasing values of fractional parameter and larger values of Prandtl number. 8. Shear stress increases by increasing the Grashof number. 9. Shear stress decreasing for relaxation time and slip parameter respectively. 10. Our obtained solutions through inversion algorithms namely, Stehfest’s and Tzou’s are equivalent. Acknowledgments: The authors would like to thank reviewers for their careful assessment and pertinent observations. The authors would also acknowledge the University of Management and Technology Lahore, Pakistan for the financial support for this research.

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