On some rotational flows of non-integer order rate

Ain Shams Engineering Journal xxx (2017) xxx–xxx

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Engineering Physics and Mathematics

On some rotational flows of non-integer order rate type fluids with shear stress on the boundary Azhar Ali Zafar a,b,⇑, Nehad Ali Shah a, Niat Nigar a a b

Abdus Salam School of Mathematical Sciences, GC University Lahore, Pakistan Department of Mathematics, GC University Lahore, Pakistan

a r t i c l e

i n f o

Article history: Received 17 May 2016 Revised 16 July 2016 Accepted 5 August 2016 Available online xxxx Keywords: Oldroyd-B fluid Caputo derivatives Velocity field Shear stress Circular cylinder

a b s t r a c t This article presents rotational flows of some rate type fluids with non-integer order derivatives. The fluid fills an infinite straight circular cylinder and its motion is generated by a time dependent torsion, applied to the surface of the cylinder. As novelty, the dimensionless governing equation related to the non-trivial shear tension is used and the first exact solutions analogous to a ramped shear stress on the surface are obtained using integral transforms. The results that have been obtained allow us to provide solutions for ordinary Oldroyd-B, non-integer order Maxwell and ordinary Maxwell fluids performing similar motions. In addition, the control of non-integer order framework on shear stress and velocity profiles is analyzed by numerical simulations and graphical interpretations. Ó 2017 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Rotating flows due to the shear stress is one of the important current topics in fluid dynamics because of its useful applications in meteorology, geophysics, turbo machinery and so on [1]. Rotating flows include flows caused by system rotation or swirl flows caused by swirl generators, etc. There are many examples of flow near rotating machines, for instance rotating-disc systems are used to model (experimentally and computationally) the flow and heat transfer associated with the internal-air systems of gas turbines, where discs rotate close to a rotating or a stationary surface [2]. Optimum design of the models requires an understanding of the principles of rotating flows and the development and solution of the appropriate solutions. The flow in rotating curved pipes with a constant circular cross-section have found wide applications in heat exchangers, piping systems, electric motors, chemical reactors

Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier ⇑ Corresponding author at: Abdus Salam School of Mathematical Sciences, GC University Lahore, Pakistan. E-mail addresses: [email protected] (A.A. Zafar), nehadali199@yahoo. com (N.A. Shah), [email protected] (N. Nigar).

and many other engineering systems. The flows in rotating pipes have been studied by numerous researchers. Miyazaki [3,4] analyzed the laminar boundary layer flow and heat transfer in rotating curved pipes of circular and rectangular cross-sections for the case of positive rotation. Ito and Motai [5] studied the flow with negative rotation and found the reversal of the secondary flow for the first time. During the last few years, various investigations have been made to study the different flow models of non-Newtonian fluids [6–16]. These fluids are generally used in industry and very much differ in their rheology. An Oldroyd-B (OB) model [17] is a rate type model and by the reason of the ramification of its governing equations, researchers have given significant consideration and discussed its flow in diverse geometries. Petrov and Cherepanov [18], discussed the flow of OB fluid in a circular pipe. The unsteady unidirectional transient flow of an OB fluid with non-integer order time derivatives, in an annulus region, produced by a fixed pressure gradient and a translation with constant velocity of the inner cylinder was studied by Mathur and Khandelwal [19]. Liu et al. [20] studied some helical flows of OB fluids with non-integer order time derivatives, in a space between two oscillating concentric cylinders of infinite length and within an oscillating circular cylinder of infinite length. The most existing solutions in the literature correspond to the problems with boundary conditions on the velocity. However, there are several particular problems with the specified force on the boundary [21,22]. For example in [21], Renardy

http://dx.doi.org/10.1016/j.asej.2016.08.018 2090-4479/Ó 2017 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Zafar AA et al. On some rotational flows of non-integer order rate type fluids with shear stress on the boundary. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.018

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has studied the motion of Maxwell fluid across a strip bounded by parallel plates and proved that, to develop a well posed problem it is necessary to impose boundary conditions on the stresses at the inflow boundary. In [22], Renardy explained how well posed boundary value problems can be established using boundary conditions on stresses. Waters and King [23] were among the first specialists who used the shear stress at the boundary to find exact solutions for motions of rate type fluids. Other remarkable problems concerning the dynamics of non-Newtonian fluids, in various geometry and boundary conditions, can be found in the Refs. [24– 27]. More recently, Fetecau et al. [28], obtained exact solutions for flows of rate type fluids in a circular domain. Nowadays, fractional calculus modeling in dynamical problems is gaining popularity. For the accurate modeling of physical and engineering processes the non-integer order derivative models and techniques are found to be best and meticulous to the experimental results [29,30],31. Basically viscoelastic fluid models for example Maxwell, Oldroyd-B, etc. are best expressed in terms of non-integer (fractional) order form. Within the context of viscoelasticity the use of non-integer order derivatives was firstly proposed by Germant [31]. After that the theory of viscoelasticity in the setting of fractional calculus was further extended by Smith and de Vries [32], Bagley and Torvik [33,34] and Koeller [35], etc. As such, these models are consistent with basic theories and are not arbitrary constructions that happen to describe experimental data. Hence a number of researchers have used fractional calculus as an empirical method of describing the properties of viscoelastic materials. A detailed bibliography is contained in the book by Mainardi [36], including an historical perspective up to 1980s. For further studies see [37,38]. Many experimental data highlighted that the state of a physical system depends not only upon its current state but, also depends of its history. Because the integer order differential operator is a local operator, the classical fluid models cannot give the best description of the fluids behavior. Since the fractional derivative operators have non-local properties, the fractional calculus has been successfully used in the description of several physical phenomena. Many researchers have proposed fractional models obtained from the classical models by replacing the integer derivative operator by the fractional derivative operator [30,39]. So the results with ordinary derivative models have marginal scientific value and definitely insufficient to warrant suitable correlation with the experimental data. Moreover, non-integer order derivatives have the elegant property that, if the limit of the fractional parameter tends to integer they coincide with the classic derivative of that order. Our goal is to investigate unsteady flow of OB fluids with noninteger order derivatives through a circular cylinder of infinite length. In the present paper the governing equation of the flow is associated to the tension and we considered the boundary conditions on the shear stress as in [28,40]. The flow of the fluid is shear driven as the consequence of the circular motion of the cylinder about its axis, under the action of a time dependent shear stress given on the boundary. The obtained solutions for the motion of OB fluids with non-integer order derivatives, which are new in the literature, allow us to recover the corresponding results for ordinary OB, fractional Maxwell and ordinary Maxwell fluids. Also the effect of the non-integer order parameters and that of Reynolds number on the profiles of shear stress and fluid’s velocity is underlined by graphical illustrations.

2. Mathematical formulation of the problem The Cauchy stress tensor T corresponding to an Oldroyd-B fluid [17] is given by the relations


 T ¼ pI þ S; S þ k S_  LS  SLT h
i ¼ l A þ kr A_  LA  ALT ;

ð1Þ

where p; I; S; k; L; l; A and kr are respectively the hydrostatic pressure, identity tensor, the extra-stress tensor, the relaxation time, the velocity gradient, the dynamic viscosity, first Rivlin-Ericksen tensor and the retardation time. The superscript T denotes the transpose operator and the superpose dot denote the material time derivative. The constitutive Eq. (1) corresponding to ordinary Oldroyd-B fluids satisfies the fundamental principles of continuum media (the symmetry and the objectivity principles, for instance). To the best of our knowledge, in the existing literature does not exist such a constitutive equation for fractional Oldroyd-B fluids. Such a constitutive equation has been provided by Palade et al. [41] for fractional Maxwell fluids only. However, for one dimensional flows, many researchers have proposed fractional models obtained from the classical models by replacing the integer order derivative operators by fractional order derivative operators into their governing equations [30,39]. Consider an infinite circular cylinder of radius R. At t 0 ¼ 0, the cylinder and fluid are at rest. After time t0 ¼ 0þ , the cylinder instigates to turn about its axis as the consequence of a time dependent torque per unit length 2pRs0 ðR; t0 Þ, where s0 is the non-trivial shear stress applied to the boundary of the cylinder. We infer that velocity and extra-stress tensor are of the form,

^h ; V ¼ Vðr 0 ; t 0 Þ ¼ w0 ðr 0 ; t 0 Þe

S ¼ Sðr 0 ; t 0 Þ;

ð2Þ

^h is unit vector along h-direction of cylindrical coordinate where e system. For such a flow the constrain of incompressibility is fulfilled. Since the model is at rest at time t0 ¼ 0, we have

w0 ðr 0 ; 0Þ ¼ 0;

Sðr 0 ; 0Þ ¼ 0:

ð3Þ

Introducing (2) in (1) and using (3), we get Sr0 r0 ¼ Sr0 z ¼ Szh ¼ Szz ¼ 0 together with the following partial differential equation [28]

     @ @ @ 1 w0 ðr 0 ; t0 Þ; 1 þ k 0 s0 ðr 0 ; t0 Þ ¼ l 1 þ kr 0  @t @t @r 0 r 0

ð4Þ

where s0 ðr0 ; t 0 Þ ¼ Sr0 h ðr 0 ; t 0 Þ is the nonzero component of extra stress tensor. With no body force, the balance of linear momentum, reduces to [28]

q

@w0 ðr 0 ; t0 Þ ¼ @t 0



@ 2 þ @r 0 r 0



s0 ðr0 ; t0 Þ;

ð5Þ

where q is the constant density of the fluid. Driving out w0 ðr 0 ; t 0 Þ from Eqs. (4) and (5), we extract the subsequent governing equation for the shear stress [28,40]

!     @ @ s0 ðr0 ; t 0 Þ @ @2 1 @ 4 1þk 0 ¼ m 1 þ k þ  s0 ðr0 ; t0 Þ; r @t @t 0 @t 0 @r 02 r 0 @r 0 r 02

ð6Þ where m is the kinematic viscosity of the fluid. The suitable initial and boundary conditions are

s0 ðr0 ; 0Þ ¼

@ s0 ðr 0 ; t 0 Þ jt0 ¼0 ¼ 0; @t 0

s0 ðR; t0 Þ ¼ fHðt0 Þ

t0d1 d1

k

CðdÞ

;

ð7Þ

d P 1;

ð8Þ

where HðÞ is the Heaviside unit step function. By introducing the following dimensionless 0

0

0

quantities

0

w ; W o ¼ qkfR ; s ¼ sf ; c ¼ kkr , where t is the diment ¼ tk ; r ¼ rR ; w ¼ W o

sionless time, r the dimensionless radial coordinate, w the non-

Please cite this article in press as: Zafar AA et al. On some rotational flows of non-integer order rate type fluids with shear stress on the boundary. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.018

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dimensional velocity, W o is a characteristic velocity, s the nondimensional shear stress and c the non-dimensional parameter, Eqs. (5)–(8) become

  @wðr; tÞ @ 2 ¼ þ sðr; tÞ; @t @r r

ð9Þ

!     @ @ sðr; tÞ @ @2 1 @ 4 þ sðr; tÞ; ¼ 1þc  Re 1 þ @t @t @t @r 2 r @r r2

ð10Þ

t

sð1; tÞ ¼ HðtÞ

d1

CðdÞ

;

wðr; 0Þ ¼ 0;

t P 0;

ð11Þ

d P 1:

ð12Þ

The corresponding non-integer order model is characterized by

 @2 1 @   @ sðr; tÞ
4 Re 1 þ Dat þ ¼ 1 þ cDbt  @t @r2 r @r r 2 

@ 2 þ @r r

Dat wðr; tÞ ¼

!

sðr; tÞ;

ð13Þ



sðr; tÞ;

ð14Þ

subject to the conditions

 @ sðr; tÞ sðr; 0Þ ¼ ¼ 0; @t t¼0

sð1; tÞ ¼ HðtÞ

td1 ; CðdÞ (

wðr; 0Þ ¼ 0;

t P 0; 1

Cð1pÞ

1 1 þ cqb rn J 02 ðr n Þ ; Re q þ qaþ1 qd

ð15Þ

d P 1:

or

sH ðrn ; qÞ ¼ 

Rt

f 0 ðsÞ 0 ðtsÞp

3.1. Calculation for shear stress Implementing Laplace transform [46] to Eq. (13) and utilizing ð15Þ1 and (16), we get

1 1 þ cqb @ 2 1 @ 4 sðr; qÞ ¼ þ  Re q þ qaþ1 @r 2 r @r r 2

r n J 02 ðr n Þ : qd

ð21Þ

1 þ cqb

1

n

n

 Reðq þ qaþ1 Þ þ r 2 ð1 þ cqb Þ r 2 q þ qaþ1 ¼  2 aþ1 ; r n ½ðq þ An Þ þ ðq þ Bn qb Þ r2

where An ¼ Ren and Bn ¼

cr2n Re

ð22Þ

, and

1 þ cq 1  Reðq þ qaþ1 Þ þ r 2n ð1 þ cqb Þ r 2n b

1 X k qX ð1Þk k!Bnkm r 2n k¼0 m¼0 m!ðk  mÞ! " # qaþmþkbmb qmþkbmb ;  þ kþ1 kþ1 ðqaþ1 þ An Þ ðqaþ1 þ An Þ

¼

ð23Þ

1 X k J 1 ðr n Þ J 1 ðr n Þ X ð1Þk k!Bkm n þ d rn q r n k¼0 m¼0 m!ðk  mÞ! " # qaþmþkbmbþ1d qmþkbmbþ1d  þ : kþ1 kþ1 ðqaþ1 þ An Þ ðqaþ1 þ An Þ

sH ðrn ; qÞ ¼ 

!

sðr; qÞ;

ð17Þ

1 ; qd

ð18Þ

where sðr; qÞ is the Laplace transform of sðr; tÞ and q is the transform variable. Applying Hankel transform [47] to Eqs. (17) and (18), we obtain

 1 1 þ cqb  r n J 02 ðr n Þsð1; qÞ  r2n sH ðrn ; qÞ ; Re q þ qaþ1

ð19Þ

R1 where sH ðr n ; qÞ ¼ 0 r sðr; qÞJ 2 ðrr n Þdr is the finite Hankel transform of the function sðr; qÞ and rn ; n ¼ 1; 2; . . .. are the positive roots of the transcendental equation J 2 ðxÞ ¼ 0; Jm ðÞ being the Bessel function of first kind of order m. Eq. (19) is equivalent to

ð24Þ

Implementing the inverse Laplace transform [48] to the last equality, we get

sH ðrn ; tÞ ¼ HðtÞ

1 X k td1 J 1 ðr n Þ J ðr n Þ X ð1Þk k!Bnkm þ HðtÞ 1 rn rn k¼0 m¼0 m!ðk  mÞ!

CðdÞ

 ½Gaþ1; aþmþkbmbþ1d; kþ1 ðAn ; tÞ þ Gaþ1; mþkbmbþ1d; kþ1 ðAn ; tÞ;

ð25Þ

where Ga;b;c ð; tÞ is the generalized Lorenzo Hartley function [48],   n b o   with L1 ðqaqdÞc ¼ Ga;b;c ðd; tÞ; Re ðac  bÞ > 0; ReðqÞ > 0; qda  < 1, and P1 d j CðcþjÞ tðcþjÞab1 Ga;b;c ðd; tÞ ¼ j¼0 CðcÞCðjþ1Þ C½ðcþjÞab : Applying the inverse Hankel transform to Eq. (25) and using the inverse formula

sðr; tÞ ¼ 2

1 X J 2 ðrr n Þ 2

n¼1

sH ðrn ; qÞ ¼

1 þ cqb

Reðq þ qaþ1 Þ þ r2 ð1 þ cqb Þ n

ð16Þ

3. Solution of the problem

sð1; qÞ ¼

ð20Þ

into Eq. (21), we get

ds; 0 6 p < 1; is the Caputo deriva0 f ðtÞ; p ¼ 1; tive operator with respect to t [42–45]. Into above relations a P b [38] and CðÞ is the Euler integral of second kind or Gamma function. For a ¼ b ¼ 1 the non-integer order OB model reduces to the usual OB model and for b ¼ 1 and kr ¼ 0 the fractional Maxwell model is obtained. where Dpt f ðtÞ ¼

¼

Using the equivalent expressions

2

with Re ¼ Rkm , the Reynolds number and

 @ sðr; tÞ sðr; 0Þ ¼ ¼ 0; @t t¼0



Reðq þ qaþ1 Þ þ r 2n ð1 þ cqb Þ sH ðrn ; qÞ Reðq þ qaþ1 Þ

½J 02 ðr n Þ

sH ðrn ; tÞ;

we get

sðr; tÞ ¼ HðtÞ

td1

CðdÞ

r2 þ 2HðtÞ

1 1 X k X J 2 ðrr n Þ X ð1Þk k!Bkm n r J ðr Þ m!ðk  mÞ! n¼1 n 1 n k¼0 m¼0

 ½Gaþ1; aþmþkbmbþ1d; kþ1 ðAn ; tÞ þ Gaþ1; mþkbmbþ1d; kþ1 ðAn ; tÞ:

ð26Þ

3.2. Calculation for velocity Using Eq. (26) into Eq. (14), we obtain the following non-integer order differential equation for velocity

Please cite this article in press as: Zafar AA et al. On some rotational flows of non-integer order rate type fluids with shear stress on the boundary. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.018

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Dat wðr; tÞ ¼ 4rHðtÞ

1 1 X k X t d1 J 1 ðrr n Þ X ð1Þk k!Bkm n þ 2HðtÞ J ðr Þ CðdÞ m!ðk  mÞ! n¼1 1 n k¼0 m¼0

erated by a circular cylinder that applies a constant couple to the fluid, namely

 ½Gaþ1; aþmþkbmbþ1d; kþ1 ðAn ; tÞ þ Gaþ1; mþkbmbþ1d; kþ1 ðAn ; tÞ:

ð27Þ

sðr; tÞ ¼ HðtÞr2 þ 2HðtÞ

Implementing the Laplace transform to this equation, we have 1 1 X k X J 2 ðrr n Þ X ð1Þk k!Bkm n wðr; qÞ ¼ 4r dþa þ q J ðr Þ m!ðk  mÞ! n¼1 1 n k¼0 m¼0 " # qmþkbmbþ1d qmþkbmbadþ1 ; þ  kþ1 kþ1 ðqaþ1 þ An Þ ðqaþ1 þ An Þ

 ½Gaþ1; aþmþkbmb; kþ1 ðAn ; tÞ þ Gaþ1; mþkbmb; kþ1 ðAn ; tÞ

1

tdþa1

Cðd þ aÞ

and

wðr; tÞ ¼ 4r

1 1 X k X J 1 ðrr n Þ X ð1Þk k!Bnkm þ 2HðtÞ J ðr Þ m!ðk  mÞ! n¼1 1 n k¼0 m¼0

þ Gaþ1; mþkbmbaþ1d; kþ1 ðAn ; tÞ:

ta

Cð1 þ aÞ

1 1 X k X J 1 ðrr n Þ X ð1Þk k!Bnkm þ 2HðtÞ J ðr Þ m!ðk  mÞ! n¼1 1 n k¼0 m¼0

 ½Gaþ1; mþkbmb; kþ1 ðAn ; tÞ þ Gaþ1; mþkbmba; kþ1 ðAn ; tÞ:

 ½Gaþ1; mþkbmbþ1d; kþ1 ðAn ; tÞ ð29Þ

These last solutions, as expected, are equivalent to those of Rauf et al. [49, Eqs. (25) and (31) with x=0] 1 X J 2 ðrr n Þ J ðr Þ n¼1 1 n

2     an brn  Re bn 1 bn sh t þ 2 ch t e2Ret ;  2 bn rn 2Re 2Re

4.1. Ordinary oldroyd-b fluid model Into Eqs. (26) and (29), letting a; b ! 1, we obtain

sðr; tÞ ¼ HðtÞ

t

CðdÞ

ð33Þ

sðr; tÞ ¼ HðtÞr2 þ 2HðtÞ

4. Limiting cases

d1

ð32Þ

ð28Þ

with the inverse Laplace transform,

wðr; tÞ ¼ 4r

1 1 X k X J 2 ðrr n Þ X ð1Þk k!Bnkm r J ðr Þ m!ðk  mÞ! n¼1 n 1 n k¼0 m¼0

r 2 þ 2HðtÞ

1 X

J 2 ðrr n Þ 1 J ðr Þ r n n¼1 1 n

1 X k X

k

ð34Þ

k!Bnkm

ð1Þ m!ðk  mÞ! k¼0 m¼0

 ½G2; 2þkd; kþ1 ðAn ; tÞ þ G2; kþ1d; kþ1 ðAn ; tÞ

ð30Þ

and 1 1 X k X td J 1 ðrrn Þ X ð1Þk k!Bkm n þ 2HðtÞ J ðr Þ Cðd þ 1Þ m!ðk  mÞ! n¼1 1 n k¼0 m¼0   G2; kþ1d; kþ1 ðAn ; tÞ þ G2; kd; kþ1 ðAn ; tÞ ;

"   2 1 X J 1 ðrrn Þ bn  an ðbr 2n  ReÞ bn sh t wðr; tÞ ¼ 4rt  HðtÞ 2 r J ðr Þ bn 2Re n¼1 n 1 n    

a bn n t e2Ret  ðan  ðbr 2n  ReÞÞ ; þ an  ðbr2n  ReÞch 2Re ð35Þ

wðr; tÞ ¼ 4r

ð31Þ

which give the expressions of the dimensionless shear stress and velocity for the same motion of an OB fluid. As an application of our results, we take d ¼ 1 in Eqs. (30) and (31) to get the solutions for the rotational flow of an OB fluid gen-

2

where an ¼ Re þ brn2 ; bn ¼ a2n  4Rer 2n and b ¼ kkr . Indeed, Fig. 1 clearly shows that the diagrams of shear stress and velocity given by Eqs. (32) and (33) tend to superpose over those corresponding to Eqs. (34) and (35) when a; b ! 1. Furthermore, as it results from this figure, the fractional fluids flow faster in comparison to the ordinary ones.

Fig. 1. Profiles of shear stress and velocity versus r corresponding to Eqs. (32) and (34), respectively (33) and (35).

Please cite this article in press as: Zafar AA et al. On some rotational flows of non-integer order rate type fluids with shear stress on the boundary. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.018

A.A. Zafar et al. / Ain Shams Engineering Journal xxx (2017) xxx–xxx

4.2. Fractional maxwell fluid model

wFM ðr; tÞ ¼ 4r

Taking kr ! 0 and b ¼ 1 into Eqs. (26) and (29), we get

5

1 1 X tdþa1 J 1 ðrr n Þ X ð1Þk þ 2HðtÞ J Cðd þ aÞ ðr Þ n 1 n¼1 k¼0

 ½Gaþ1; kþ1d; kþ1 ðAn ; tÞ

1 1 X t d1 2 J 2 ðrr n Þ 1 X sFM ðr; tÞ ¼ HðtÞ ð1Þk r þ 2HðtÞ J ðr Þ r n k¼0 CðdÞ n¼1 1 n

þ Gaþ1; kaþ1d; kþ1 ðAn ; tÞ;

ð37Þ

 ½Gaþ1; aþkþ1d; kþ1 ðAn ; tÞ þ Gaþ1; kþ1d; kþ1 ðAn ; tÞ;

ð36Þ

which are the expressions of the dimensionless shear stress and velocity for Maxwell fluid with fractional derivatives.

Fig. 2. Profiles of shear stress and velocity versus r for a variation and different values of d.

Please cite this article in press as: Zafar AA et al. On some rotational flows of non-integer order rate type fluids with shear stress on the boundary. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.018

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4.3. Ordinary maxwell fluid model

wM ðr; tÞ ¼ 4r

Into Eqs. (30) and (31), letting kr ¼ 0, we have

 ½G2; kþ1d; kþ1 ðAn ; tÞ þ G2; kd; kþ1 ðAn ; tÞ;

1 1 X t d1 2 J 2 ðrr n Þ 1 X sM ðr; tÞ ¼ HðtÞ ð1Þk r þ 2HðtÞ J ðr Þ r n k¼0 CðdÞ n¼1 1 n

 ½G2; 2þkd; kþ1 ðAn ; tÞ þ G2; kþ1d; kþ1 ðAn ; tÞ;

1 1 X tdþa1 J 1 ðrrn Þ X ð1Þk þ 2HðtÞ J Cðd þ 1Þ ðr Þ n 1 n¼1 k¼0

ð38Þ

ð39Þ

the expressions of shear stress and velocity for ordinary Maxwell fluid.

Fig. 3. Profiles of shear stress and velocity versus r for b variation and different values of d.

Please cite this article in press as: Zafar AA et al. On some rotational flows of non-integer order rate type fluids with shear stress on the boundary. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.018

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5. Numerical results and discussion In this article, unsteady rotational flows of an OB fluid with noninteger order derivatives which fills a straight circular cylinder of radius R and of infinite length are studied. Flows are produced by a time dependent torque applied to the boundary of the cylinder. As novelty, the governing equation related to the dynamic torsion is used. Closed form solutions of dimensionless shear stress and velocity fields are obtained by utilizing integral transforms. These solutions that satisfy all prescribed initial and boundary condition, allow us to contribute the exact solutions for the motion of a rate

7

type fluid produced by the circular cylinder that exerts a constant or time dependent shear stress to the fluid. These solutions can easily be step down to the analogous solutions for ordinary OB, non-integer order Maxwell fluids and ordinary Maxwell fluids as limiting cases. Further, the control of non-integer order parameters on the flow is investigated numerically and by graphical interpretation as follows. In Fig. 2, we have prepared graphs in order to study the control of the non-integer order parameter a on the shear stress and fluid velocity for d ¼ 1 (corresponding to uniform shear stress), d ¼ 2 and 3 (for time dependent shear stress). The profiles corresponding

Fig. 4. Profiles of shear stress and velocity versus r for Re variation and different values of d.

Please cite this article in press as: Zafar AA et al. On some rotational flows of non-integer order rate type fluids with shear stress on the boundary. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.018

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to shear stress and velocity are plotted versus r for small time and miscellaneous values of the non-integer order parameter a, namely, a 2 f0:3; 0:5; 0:7g and b ¼ 0:2; Re ¼ 3. The curves show that the effect of the non-integer order parameter a is significant only near the boundary of the cylinder. It is observed that near the boundary, the shear stress and velocity decreases by increasing values of a. It is also observed that the magnitude of the shear stress and velocity decrease by increase the value of d. In Fig. 3, we presented the influence of the non-integer order parameter b on the shear stress and fluid velocity. The profiles corresponding to shear stress and velocity are plotted versus r for different values of non-integer order parameter b, namely,

b 2 f0:2; 0:4; 0:6g; a ¼ 0:8, and Re ¼ 3. The curves show that the influence of the fractional parameter b is also significant near the boundary of the cylinder. It is found that both shear stress and velocity increases by increasing the values of b, but very close to the surface of cylinder there is a critical value of r where the trend of velocity is reversed. It is observed that the magnitude of shear stress and velocity decreases by increase the value of d. In Fig. 4, we presented the effect of the Reynolds number on the shear stress and velocity of the fluid. The shear stress and velocity profiles are plotted versus r for separate values of Reynolds number Re, namely, Re 2 f0:5; 1:5; 3:5g; a ¼ 0:7 and b ¼ 0:2. The curves show that the shear stress along with velocity decreases with

Fig. 5. Profiles of shear stress and velocity versus d for a ¼ 0:7; b ¼ 0:5 and Re ¼ 15.

Please cite this article in press as: Zafar AA et al. On some rotational flows of non-integer order rate type fluids with shear stress on the boundary. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.018

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increasing values of Re, but very near to the cylinder there is a critical value of r after which the velocity increases with the increasing values of Re. It is observed that the magnitude of the shear stress and velocity decreases by increasing the value of d. In all the graphs, we have chosen the dimensionless material parameters kr ¼ 1:3 and k ¼ 1:8 and dimensionless time t ¼ 0:2. In Fig. 5, we presented the influence of d-the parameter of the external torsion, it is observed that both, the shear stress and velocity admit a maximum value for certain values of d at each

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moment of time t. These maximum values increase with the time t. Also, there are values of the d for which the shear stress and velocity are zero. In the studied case, if d > 20, both, the shear stress and velocity are zero for t P 10. In Fig. 6, we have prepared graphs to study the influence of Reynolds number on the shear stress and velocity. It is observed that the variation of the shear stress and of velocity with the Reynolds number is small and approaches some constant value for each moment of the time t. It is noted that influence of the Reynolds

Fig. 6. Profiles of shear stress and velocity versus Reynolds number for d ¼ 2; a ¼ 0:7; b ¼ 0:5 and c ¼ 0:2.

Please cite this article in press as: Zafar AA et al. On some rotational flows of non-integer order rate type fluids with shear stress on the boundary. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.018

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Fig. 7. Profiles of shear stress and velocity versus b for Re ¼ 15; d ¼ 2; a ¼ 1 and c ¼ 0:2.

number on velocity of the fluid and shear stress is significant only for small values of time t. In Fig. 7, we have presented the influence of the fractional parameter b on the shear stress and fluid velocity. It is observed that, for a given time t, in each position of the flow domains, the shear stress and fluid velocity have constant values when b varies and t > 1. Furthermore to approximate the positive roots of the Bessel function J 2 ðxÞ ¼ 0 we used the computer software Mathcad.

6. Conclusions Unsteady rotational flows of an OB fluid with non-integer order derivatives, which fills a straight circular cylinder of finite radius are studied. Flows are produced by a time dependent torque applied to the surface of the cylinder. As novelty, the governing equation related to the dynamic torsion is used to produce exact solutions analogous to motions due to a time dependent shear on the boundary. These solutions, which are new in the literature,

Please cite this article in press as: Zafar AA et al. On some rotational flows of non-integer order rate type fluids with shear stress on the boundary. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.018

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allow us to recover exact solutions for the motion of rate type fluid produced by a circular cylinder that applies constant or ramp type shear stress to the fluid. All solutions can conveniently be reduced to the corresponding solutions for ordinary OB, fractional Maxwell and ordinary Maxwell fluids performing the similar motion. To study the control of non-integer order framework and of Reynolds number on the fluid motion, we plotted some graphs for profiles of dimensionless shear stress and velocity versus r and summarize our investigation as;  The effects of the non-integer order parameters a and b on the fluid motion is significant especially in the vicinity of the cylinder.  The shear stress is a decreasing function with respect to the fractional parameter a and Reynolds number Re but it increases with regard to b.  The fluid velocity decreases for increasing values of a on the whole domain. It is an increasing function with respect to b and a decreasing one with respect to the Reynolds number Re in the vicinity of the cylinder. Opposite effects appear for r greater than a critical value r c .  The magnitude of the shear stress as well as that of velocity decreases with increasing value of d.  The shear stress and velocity admit a maximum value for certain values of d at each moment of time t. These maximum values increase with the time t. Also, there are values of the d for which the shear stress and velocity are zero.  The variation of the shear stress and of velocity with the Reynolds number is small and approaches some constant value for each moment of the time t. It is noted that influence of the Reynolds number on velocity of the fluid and shear stress is significant only for small values of time t.  For a given time t, in each position of the flow domains, the shear stress and fluid velocity have constant values when b varies and t > 1.  Ordinary fluids flow slower in contrast with non-integer order/ fractional fluids.

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Dr. Azhar Ali Zafar is a Post Doctoral Fellow at Abdus Salam School of Mathematical Sciences, GC University Lahore, Pakistan. He did his PhD in the field of Fluid Dynamics in 2014.

Please cite this article in press as: Zafar AA et al. On some rotational flows of non-integer order rate type fluids with shear stress on the boundary. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.018