Boundary layer flow of fractional Maxwell fluid over a stretching sheet with variable thickness

Accepted Manuscript Boundary layer flow of fractional Maxwell fluid over a stretching sheet with variable thickness Lin Liu, Fawang Liu

PII: DOI: Reference:

S0893-9659(17)30317-8 https://doi.org/10.1016/j.aml.2017.10.008 AML 5353

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Applied Mathematics Letters

Received date : 24 September 2017 Revised date : 18 October 2017 Accepted date : 18 October 2017 Please cite this article as: L. Liu, F. Liu, Boundary layer flow of fractional Maxwell fluid over a stretching sheet with variable thickness, Appl. Math. Lett. (2017), https://doi.org/10.1016/j.aml.2017.10.008 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.

 

Boundary layer flow of fractional Maxwell fluid over a stretching sheet with variable thickness a*

b

Lin Liu , Fawang Liu a School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China b School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Queensland 4001, Australia A novel investigation about the boundary layer flow of fractional Maxwell fluid over a stretching sheet Abstract: with variable thickness is presented. By introducing new variables, the irregular boundary changes as a regular one. Solutions of the governing equations are obtained numerically where the L1-scheme is applied. Dynamic characteristics with the effects of different parameters are shown by graphical illustrations. Three kinds of distributions versus power law parameter are presented, including monotonically increasing in nearly linear form at y=1, increasing at first and then decreasing at y=1.4 and monotonically decreasing in nearly linear form at y=2. Keywords: Fractional derivative; Maxwell fluid; Constitutive equation; Boundary layer 1. Introduction Boundary layer flow has attracted much attention due to its ever growing physical and industrial applications [1-3], such as glass blowing, metal extrusion and cooling, glass fiber production, plastic extrusion processes and paper production. In view of its great importance, many attempts [4-6] thus have been committed for its study. The previous papers mainly discuss the boundary layer flow with flat stretching sheet. Due to the acceleration or deceleration of the sheet, the thickness of the stretched sheet may decrease or increase with distance from the slot [7], which is dependent on the value of the velocity power index. Motivated by this discussion, a stretching sheet with variable thickness can be more close to the situation in practical engineering applications. And it has attracted a large number of scholars’ attention. Among the researchers, Fang et al. [8] were the first scholars to study the boundary layers over a continuously stretching sheet with a power law surface velocity revisited for a sheet with variable thickness. The key references about the stretching sheet with variable thickness are given in Refs. [9-11]. The classical constitutive equation to describe the boundary layer flow is deduced from the integer order constitutive relationship in linear form. With the progress of the scientific research, fractional calculus achieves great development due to its vast application foregrounds. As is well known, the integral order operator possesses the local nature while the time fractional operator is a nonlocal one which possesses the memory characteristic [12-13]. The effectiveness and importance of fractional derivative on the engineering applications have been verified through experiments by many researchers. Song and Jiang [14] studied the viscoelastic fluids with fractional Jeffreys model, verifying that the modified Jeffreys model was appropriate to describe the behaviors for xanthan gum and Sesbania gel. Bagley and Torvik [15] indicated that the fractional calculus models of viscoelastic material behavior were consistent with the predictions of the molecular theory of polymer solids, and provided a link between accepted microscopic theories and macroscopic observations. The fractional kinetics proved to be valuable by Zaslavsky [16] to describe some important physical phenomena, such as cooling of particles and signals, particle and wave traps, Maxwell’s Demon, etc. Meral et al. [17] verified that the fractional order Voigt model can be better simulate the surface wave response of soft tissue-like material phantoms by experiment. Jiang *Corresponding author. Email addresses: [email protected] 1 

 

and Qi [18] demonstrated that fractional thermal wave models can provide a unified approach to examine the heat transfer in biological tissue. Chen et al. [19] indicated that the variable-order fractional derivative model agreed significantly better with experimental data than the traditional model. More literatures related to the introduction of the superiority for fractional model can be seen in Refs. [20-22]. For the previous studies, the boundary layer flow with flat stretching sheet using fractional constitutive relationship or the boundary layer flow over a stretching sheet with variable thickness using integer order constitutive relationship is discussed and analyzed. Because the boundary layer flow with variable thickness is a highly non-homogeneous transport process, the heat transfer for a certain position should not only be dependent on its nearby points but also considers the history dependence characteristics which can be described by the fractional constitutive relation. However, it seems that the constitutive relationship with fractional derivative has never been mentioned and explored to study the flow on a stretching sheet with variable thickness. Motivated by the discussions mentioned above, it is meaningful and important to investigate the boundary layer flow over a sheet with variable thickness with the time fractional Maxwell constitutive relationship. Solutions of the formulated governing equations are obtained numerically and the L1-scheme is used. The irregular boundary changes as a regular one by introducing new variables. Dynamic characteristics with different involved parameter effects are shown by graphical illustrations. 2. Mathematical formulation In this section, we consider a steady, two-dimensional laminar flow over a continuously stretching sheet in a quiescent fluid. The plate impulsively starts to move at t  0 with a velocity u0  x  b  . The flow n

configuration [23] is illustrated in Fig. 1. The origin is located at a slit, the x -axis runs along the stretching surface in the direction of the sheet motion while the y -axis is perpendicular to it. Here, the convection velocity along x and

y directions are defined as uw  x   u0  x  b n and v  x  , respectively. Here, u0 is a

constant, b is the physical parameter related to stretching sheet, and n is the velocity exponent parameter. The wall is impermeable with v  x   0 . The sheet is assumed not flat which is specified as [24-25]

y  A  x  b

1 n  / 2

. In order to avoid pressure gradient along the sheet, we assume the coefficient

A being

small so that the sheet is sufficiently thin. For the validity of defining the domain of boundary, we assume in this work where n  1 corresponds to the flat sheet.

n 1

The continuity equation is given by:

u v   0. x y

(1)

The time fractional Maxwell constitutive equation [26] is given by: 

 xy  1

where

 xy , 1

and



 xy t





u , y

refer to the shear stress, relaxation parameter and viscosity, the symbol

the Caputo’s time fractional derivative [27] of order

 ( 0    1 ).

Combining (2) with the following momentum equation: 2 

(2)

 refers to t 

 



u u u  xy  u  v  , t x y y

(3)

we can obtain the time factional governing equation:    1u    u    1  1  1   u   1  t t  x  t 



where

is the density and

 /

 u  u u u  2u  v y   t  u x  v y   y 2 .  

is the dynamic viscosity.



The initial conditions are chosen as u  x , y , 0   u 0  x  b   y  A  x  b  n

1  n  / 2



v  x, y,0  0 while boundary conditions are given as u  0, y, t   0 ; u x, A  x  b 



u  x, , t   0 , v x, A  x  b 



1 n  /2

(4)

, 1n  /2

u  x , y , 0  0; t



, t  u0  x  b  , n

, t  0 , where   y  is a newly introduced characteristic function, here

its value is one when y  0 while zero when y  0 . By introducing dimensionless variables for simplifying our study:

t

A 2 b 1 n



1n /2 t , x  bx  , y  Ab  y , 1  

 bn  v   v , u0  2 u0 ,  u 2 u , 1n  /2 A Ab  A

A2b1n

 , 1

and defining new variables x*  x and y*  y   x  1 symbol

1 n  /2

, the governing equations change as (here the

* is omitted for simplicity):  1  n  / 2  u 1  n  u v   0,  x  1 x 2 y y

1

(5)

    1 n  / 2  u    1u u     u 1  n            u x 1 v     1 1    1    t t   x y   t  y  2

 1 n  / 2  u 1  n  u u  u  2 u  u    v  x  1  t y  y y 2 2  x

,

(6)

subject to the dimensionless initial conditions and boundary conditions in a regular region, respectively given as:

u  x, y,0  u0  x  1   y  , n

u  x, y ,0   0 ; v  x, y,0  0 ; t

(7)

and

u  0, y, t   0 ; u  x,0, t   u0  x  1 , u  x, , t   0 , v  x,0, t   0 . n

(8)

3. Numerical discretization method Firstly, we define the numerical solutions of Eqs. (5)-(6) with initial conditions (7) and boundary conditions (8) at the mesh points ( i  0,1,2,..., M ),

x , y ,t  i

j

k

as

uik, j and vik, j . The mesh point  xi , y j , tk  are expressed by xi  ihx

y j  jh y ( j  0,1, 2,..., N ), tk  k 3 

( k  0,1, ..., T ) where

hx  Xmax / M and

 

h y  Ymax /  N  1 are respectively the space steps along the x and y directions while  is the time step. At the mesh point

 x , y , t  , the final difference scheme for the continuity equation is given as follows: i

j

k

 1 n  /2   1 1  n  1  n  x  1 1n/ 2 u k  1 v k  1 v k 1  xi  1  uik, j  uik1, j   i    i , j 1 i, j i , j 1 , 2h y 2h y hx hy hy  hx 

(9)

Since the highest order of (6) is 1    1 , we discretize the governing equation at the mesh points

 x , y ,t  i

[28].

k 1/2

j

By using the L1-scheme [29] and linearizing the nonlinear term, namely the

coefficients of the nonlinear term are discretized at the mesh points

 x , y , t  , the final difference scheme for i

j

k 1

the momentum equation is given as follows (here the error term is omitted):

 r2

u ik, j 1 2hx

u

k i 1, j

 1  n  / 2  1  n  v ik, j 1 1  k k 1 u i , j  r2   r2   xi  1  u i , j 1 4hy 2 h y 2 h y2  

r  u k 1 v k 1  1  n  x  1  1 n / 2 u k 1  u k  1 u k 1   2  2  r2  i , j  i , j   r2  i  i, j  i, j i , j 1   hy 4hy 2 h y2  2hx 2h y     r  u k 1 v k 1  1  n  x  1  1 n / 2 u k 1  u k 1  r uik, j 1 u k 1 1   2  2  r2  i , j  i , j   r2  i  i, j  i, j 2 i 1, j  2hx 2h y  hy 4h y 2hx     

,

(10)

 1 1  n  x  1  1 n / 2 u k 1  r v  u k 1  1 u k 1  r1 A  r1 A   2  r2  i   i , j 1 i, j 2 i , j 1  k1 2hx k 2 4h y 2 h y  2 h y2  2 h y  1  n  / 2  1  n  / 2 r 1  n  r 1  n   1 Ak 3 c k  2 u i0, j  u i1, j  u i1, j 1  u i0, j  u i0, j 1   1  xi  1  xi  1 4hy 4h y k 1 i, j



r1 r r c k  2 v i0, j  u i1, j  u i1, j 1  u i0, j  u i0, j 1   1 c k  2 u i0, j  u i1, j  u i11, j  u i0, j  u i01, j   1 Ak 4 2hy 2hx 2hy

where c0  1 , cl   l 1

1

k 2

1

l

, r1 

1

  2    

, r2 

Ak 2    ck l  2  ck l 1  uil, j  uil,j1  uil1,1 j  uil, j  uil1, j  , l 1

1

  2    

k 1

 1 , Ak 1    ck l 1  ck l   uil, j  uil,j1  , l 1

k 2

Ak 3    ck l 2  ck l 1  uil, j  uil,j1  uil,j11  uil, j  uil, j 1  , l 1

k 2

Ak 4    ck l 2  ck l 1  vil, j  uil,j1  uil,j11  uil, j  uil, j 1  . l 1

The discretization schemes for the initial conditions and boundary conditions are respectively given as

u0  xi  1n , j  1 n 0 k k k u u  ; vi , j  0 and u1, j  0 ; uik,1  u0  x i 1 , ui , N  0 ; vi ,1  0 . j  2,3,..., N 0, 0 i, j

1 i, j

4. Results and discussion In this section, we mainly discuss the effects of the involved parameters on the spatial evolution and power law number evolution of velocity distribution at x  1 and u0  1 by graphical illustrations. The monotonically decreasing distributions versus y are presented in Figs. 2-3 and discussed in detail. 4 

 

Fig. 2 shows the spatial evolution of velocity distributions with the effects of different  . The fractional parameter  refers to the memory characteristic of velocity transfer process. It can be seen from the figure, for a smaller parameter  , the distribution declines faster. The fact is that the smaller the  is, the stronger the memory characteristic will be, which will decrease the thickness of velocity boundary layer. Fig. 3 shows the velocity distribution versus

y with different  1 effects. The fractional Maxwell model

reduces to the classical Newtonian one when 1  0 . The figure shows that the influence of  1 on the

distribution is not obvious at a smaller y . With y increases, the distribution declines faster for a larger  1 . It well explains the relaxation characteristic of parameter  1 that the smaller the relaxation parameter is, the thicker the velocity boundary layer will be. Figs. 4-6 show that the velocity distributions with the effects of different power law parameters n at different positions y  1 , y  1.4 and y  2 . For y  1 , as Fig. 4 shows, the parameter n  1 corresponds to the boundary condition with a fixed value. With the increase of n , the distribution is monotonically increasing nearly linearly. It is due to the fact that the larger the parameter n is, the larger the magnitude of the initial value will be as the assumption in boundary conditions shows. Here the initial condition which is affected by n plays an important role in the distribution curve. Fig. 5 shows the distribution curves increase at first and then decrease versus n at the position y  1.4 . The fact is that the velocity distribution decreases faster for a larger n due to the concentration difference. For y  2 , the distribution is monotonically decreasing nearly linearly versus n . Similar with the interpretation as Figs. 2-3 show, for different positions, the effects of parameter 

and  1 on the distributions are derived, namely, the larger the parameter 

or the smaller the

parameter  1 is, the larger the magnitude of the distribution will be. Acknowledgement. The work is supported by the National Natural Science Foundations of China (No. 11772046, 51406008). References [1] T. Altan, S. Oh, H. Gegel, Metal forming fundamentals and applications, American society for metals, Ohio, 1979. [2] E. Fisherl, Extrusion of plastics, Wiley, New York, 1976. [3] M. Karwe, Y. Jaluria, Numerical simulation of thermal transport associated with a continuously moving flat sheet in materials processing, ASME J. Heat Transf. 113 (1991) 612-619. [4] S.H. Han, L.C. Zheng, C.R. Li, X.X. Zhang, Coupled flow and heat transfer in viscoelastic fluid with Cattaneo-Christov heat flux model, Appl. Math. Lett. 38 (2014) 87-93. [5] J. Li, L. Liu, L.C. Zheng, B. Bin-Mohsin, Unsteady MHD flow and radiation heat transfer of nanofluid in a finite thin film with heat generation and thermophoresis, J. Taiwan Inst. Chem. E. 67 (2016) 226-234. [6] M. Xenos, I. Pop, Radiation effect on the turbulent compressible boundary layer flow with adverse pressure gradient, Appl. Math. Comput. 299 (2017) 153-164. [7] T. Hayat, K. Muhammad, M. Farooq, A. Alsaedi, Melting heat transfer in stagnation point flow of carbon nanotubes towards variable thickness surface, AIP Adv. 6 (2016) 015214. [8] T.G. Fang, J. Zhang, Y.F. Zhong, Boundary layer flow over a stretching sheet with variable thickness, Appl. Math. Comput. 218 (2012) 7241-7252. 5 

 

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Figures list::

hematic of a sstretching sheeet with variab ble thicknesss. Fig. 1 Sch

ns versus y with different F Fig. 2 Velocitty distribution

ons versus y with differrent Fig. 3 Veloccity distributio

V distrib butions versuus Fig. 4 Velocity



when

1

1  0.05 and n  1.55 .

when

  0.9 and n  1..5 .

n with different  and 1 whhen d 7 

y  1.

 

utions versus Fig. 5 Vellocity distribu

n with diffferent  and 1 whenn y  1.4 . a

V distributions versuus Fig. 6 Velocity

n with different  and 1 whhen y  2 . d

8