J. Appl. Environ. Biol. Sci., 7(5)55-63, 2017 | ISSN: 2090-4274 |
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1, 3Department of Mathematics, NED University of Engineering Technology, Karachi, Pakistan 2Institute of Space Technology, Karachi, Pakistan 4Department of Chemical Engineering, Mehran University of Engineering Technology, Jamshoro, Pakistan
Received: December 16, 2016 Accepted: March 6, 2017
Generalized fractional Burgers fluid is analyzed for the flow of an edge in Rayleigh stokes problem. Newtonian and non-Newtonian expressions of velocity field have been established by the usage of Laplace and double Fourier Sine transforms along with their inverses. The constitutive and governing equations are solved by fractional calculus techniques. The general solutions are presented as the sum of non-Newtonian and Newtonian forms and expressed in terms of Fox-H Function. All imposed conditions (initial, natural and boundary) are fulfilled. The general solutions have been particularized for fractionalized as well as ordinary in six types of fluids such as Newtonian fluid, second grade fluid, Maxwell fluid, Oldroyd-B fluid, Burger fluid and generalized Burgers fluid. Finally the rheology is graphically described by various parameters on fluid flow. KEY WORDS: Transforms, Generalized Burger’s model, Stokes problem and graphical rheology.
At present, there is no refusing realism that characteristics of non-Newtonian fluids cannot be described by Navier-Stokes equations, this is due to realism that several technological and industrial processes frequently take place in non-Newtonian fluids. If flows of non-Newtonian fluids are assumed between two side of plates then such flows turns into several processes of industry, for instance, paper production, drawing of copper wires, glass blowing, polymer suspension, extrusion of plastic sheets, continuous stretching of plastic films and various others. In the non-Newtonian fluids models, Burgers’ fluid model [1-3], the Oldroyd-B fluid model [4-5], Maxwell fluid model [6-7] and the second grade fluid model [8-9] are known as viscoelastic models, because they recognize both characteristics viscosity as well as elasticity. The fractional Burgers’ fluid for Couette and Poiseuille flows between two parallel plates is discussed by Hyder Ali Muttaqi Shah [10]. Exact analytical solutions for temperature distributions and velocity fields for magnetohydrodynamics generalized Burgers’ fluid in the presence of radiation effects on the heat transfer is investigated in [11]. The fractional generalized Burgers’ fluid for oscillating, accelerating and rotating flows with porous medium and magnetohydrodynamics have been obtained for exact solutions along with fractional calculus approach in [12-18]. In continuation, our full attention is on the study of fluid problem between two parallel plates, in last few years, several researchers and mathematicians have diverted their research directions towards the fluid problem related between two parallel plates. However, we mention here very latest research work between two parallel plates. Exact solutions for constantly accelerating plate between two side walls perpendicular to the plate in generalized Oldroyd-B fluid is persuaded by [19]. Using fractional derivative approach, flow between two side walls perpendicular to the plate in second grade fluid is investigated in [20]. Due to suddenly moved plate the flow between two side walls in Maxwell fluid is found out by T. Hayat and et al. [21]. A constant pressure gradient between two side walls perpendicular to the plate in generalized Oldroyd-B fluid is traced out in [22]. Influence of accelerated flows between two side walls perpendicular to the plate in generalized Oldroyd-B fluid is communicated by [23]. Taza Gul and et al. analyzed impacts of magnetic field without slip boundary suppositions for third grade fluid flow using vertical belt [24]. Kashif and et al. investigated fractionalized viscoelastic fluid for magnetohydrodynamic flow by employing transformation techniques [25]. We also include here few recent work related to viscoelastic fluids [26-34]. Observing the above motivations, we traced out generalized fractional Burgers fluid for the flow of an edge in Rayleigh stokes problem. Newtonian and non-Newtonian expressions of velocity field have been established by the usage of Laplace and double Fourier Sine transforms along with their inverses. The constitutive and governing equations are solved by fractional calculus techniques. The general solutions are presented as the sum of non-Newtonian and Newtonian forms and expressed in terms of Fox-H Function. All imposed conditions (initial, natural and boundary) are fulfilled. The general solutions have been particularized for fractionalized as well as ordinary in six types of fluids such as Newtonian fluid, second grade fluid, Maxwell fluid, Oldroyd-B fluid, Burger fluid and generalized Burgers fluid. The rheology
Correspondence Author: Kashif Ali Abro, Department of Mathematics, NED University of Engineering Technology, Karachi, Pakistan. Email: kashif.abro@faculty.muet.edu.pk
is graphically described by various parameters on fluid flow and in order to discuss the effects of various parameters, we depicted figures to analyze the fluid flows.
The rheological equations for fractional generalized Burgers' fluid are� � �����, Λ �� Λ �Λ �1, 1 Where, ���,�,�, , , ΛΛ ,Λ ,Λ ,, and denotes indeterminate spherical stress, the extra stress
tensor, Cauchy stress tensor, first Rivlin–Ericksen tensor, dynamic viscosity, the retardation time, relaxation time, new material parameter, fractionalized parameters and fractional derivative operator respectively [10-18].We assume velocity field for such problem related between two parallel plates and extra-stress tensor as
� ,, � ,, ,���,,, 2 here is identity vector along the x-co-ordinate direction. The limitation of incompressibility is consequently fulfilled for the velocity field. The fluid being at rest up to the moment t = 0as well as considering the initial condition� ,,0�0, 3 Employing equation (2) into equation (1) and considering equation (3) for initial condition, we get the following system of equations in fractionalized form
���0, 4
1�Λ �Λ ,, �1�Λ �Λ ,,, 5
1�Λ �Λ ,, �1�Λ �Λ ,,, 6
In the above equations (5) and (6), we used the below definition so called Caputo fractional operator
′
1 � Γ 1� , � ,0 �1 1, 7
In the absence of body force the balance of linear momentum diminishes to
,, � ,, � ,, ���0, ���, 8
Assuming that there is absenteeism of pressure gradient in the flow direction and eliminating ,, and ,, between equations 5�6 and 8 . We investigate the governing equations
,, � ,,
1�Λ �Λ � 1�Λ �Λ ,9
This model contains as special cases such as Newtonian when Λ →Λ →Λ →Λ →0 , when Λ →Λ →Λ → 0 second grade, when Λ →Λ →Λ →0 Maxwell, when Λ →Λ →0 Oldroyd-B, Λ →0 Burger. It is also noted that when → →1 then above all models are termed as ordinary fluid models. In order to state the problem, we assume an incompressible fractional generalized Burger fluid possesses the space of first dial of rectangular edge as , 0;�∞ ∞. When �0 the fluid fetched to constant velocityΩ in the direction. Owing to shear, fluid slowly moved and its velocity of the form 2 . The initial and boundary conditions along with governing equations 5,6 and 9 as
For, , 0; ,, � ,, � ,,0�0, 10
For, 0; ,0,� 0,,�Ω, 11
For, � →0; ,, → ,, → ,, →0. 12
Equation (12) represents that there is no shear in the free stream when the fluid is at rest at infinity
Applying double Fourier sine transform to equation (9-11), we attain
,, 1�Λ �Λ �1�Λ �Λ � ,,�2Ω �
1�Λ �Λ ,, �0, 13
J. Appl. Environ. Biol. Sci., 7(5)55-63, 2017
Where ,, is the Fourier sine transform of ,, illustrated as
,,�2⁄ ,, , 14
has to justify imposed conditions equation (10-11),
,,0 � ,,0 �
,,0�0, , 0, 15
Applying Laplace transform to equation (13), we obtain
� 1�Λ �Λ
,,� 2Ω , 16
�Λ �Λ � �1�Λ �Λ Our aim is to express the solutions as a sum of Newtonian and non-Newtonian Part ,,� ,, , henceΛ →Λ →Λ →Λ →0 in equation (16), we have get Newtonian part as
,,, � 2٠��� , 17
Switching equation (17) into equation (16), we obtain equivalently,
,, � 2Ω 1� 1 �2Ω � 1�Λ �Λ �1�Λ �Λ � � 1�Λ �Λ � �1�Λ �Λ , 18
expressing equation (18) in terms of series form as,
,,� 2Ω 1� 1 �2Ω � 1
1Λ �Λ1�Λ�Λ1
� !Λ!! Λ!
Γ �1Γ �1Γ �1Γ �1 Γ��1Γ��1Γ Γ��1 �Λ�Λ1�Λ�Λ1
! Λ!! Λ!
à �1à �1à �1à �1��1��1à ��1 , 19
Applying inverse Laplace transform on equation (19), we get
,,� 2Ω �2Ω �� � �2Ω � �� � ∗
1Λ �Λ1�Λ�Λ1
� !Λ!! Λ!
Γ �1Γ �1Γ �1Γ � Γ��1Γ��1Γ Γ��1Γ ��2 �� ��2 �1 �Λ�Λ1�Λ�Λ1
! Λ!! Λ!
Γ �1Γ �1Γ �1Γ �
20
��1��1à ��1à � �2 �� � �2 �2 ,
Inverting equation (20) by means of Fourier sine transform, we get
,, �Ω�Ω ⁄2√ ⁄2√ �4Ω � sin sin
1Λ �Λ1
�� � �ÆÆ
� !Λ!
�Λ�Λ1Γ�1Γ�1Γ�1Γ� ! Λ !Γ��1Γ��1Γ Γ��1
��2 �� ��2 �1 �4٠�sinsin
�Λ�Λ1�� � �ÆÆ �� ! Λ!
�Λ�Λ1Γ�1Γ�1Γ�1Γ� ! Λ !Γ��1Γ��1Γ Γ��1
21 à � �2 �� � �2 �2 ,
Finally velocity field is expressed in compact form in terms of Fox-H function as,
,,� ,,�4Ω �� � �Æ
ÆÆ
1Λ �Λ1�Λ
� !Λ!!
Æ,,Æ ÆΛÆ �,0,�,0,�,0,� � ,0 ÆΛ 0,1, � ,0, � ,0,1� ,0,�,�1, � �2 � � ,� �4Ω �
�� � �Æ
�Λ�Λ1�Λ! Λ!!
Λ�,0,�,0,�,0,� � ,0
Æ,,Æ ÆΛÆ0,1, � ,0, � ,0,1� ,0,�,�1, � �2 � �2,� Æ . 22
Where, Newtonian part of velocity field is , , �Ω �Ω ⁄2√ ⁄2√ and relation of Fox-H function is [35]
�Æ Æ∏ÆÆ Γ Æ�ÆÆÆ 1� ,Æ , 1� ,Æ ,…, 1� Æ,ÆÆ
�ÆÆ,Æ,Æ ÆÆÆ . 23
Æ!∏ÆÆ ΓÆÆ�ÆÆÆ 0,1,1�Æ,Æ ,1�Æ,Æ ,…,1�ÆÆ,ÆÆ
Æ
Now we can investigate shear stresses ,, and ,, by substituting equation (22) into equations (5) and
(6) as obtained by Fetecau and et al. for Oldroyd-B fluid [19].It is also noted that one can have ordinary solutions for even six models namely (Newtonian, second grade, Maxwell, Oldroyd-B, Burger and generalized Burgers fluid) if we put � �1 in equation (22).
Employing similar methodology, we retrieve the solution for simple Burger fluid for velocity field as
,,� ,,�4Ω �� � �Æ
Æ
1Λ �Λ1
� !Λ!
�,0,�,0,1� ,1 Æ,ÆΛÆ0,1, � ,0, ,0,�,�1,2 � � �1,� Æ �4Ω �
,
�� � �Æ
�Λ�Λ1! Λ!
,Λ�,0,��,0,0,1 Æ,ÆΛÆ0,1, � ,0,�,0,�,�1,2 � � �1,� Æ . 24
Employing similar methodology, we retrieve the solution for Oldroyd-B fluid for velocity field as
J. Appl. Environ. Biol. Sci., 7(5)55-63, 2017
ÆÆ ,,� ,,�4Ω �� � �Æ
1Λ ,ÆΛÆ �,0,1� ,1 Æ
� ! Æ,0,1, � ,0,1� ,0, � ,�
�4Ω � �� � �Æ ��
�Λ,ÆΛÆ �,0,1� ,1 Æ . 25 ! Æ,0,1, � ,0,1� ,0, � �2,�
Employing similar methodology, we retrieve the solution for Maxwell fluid for velocity field as
,,� ,,�4Ω �� � �Æ
Æ
1Æ,,ÆΛÆ �,0 Æ �4Ω
� 0,1,�,�1, ,
�� � �Æ
,
�� �ΛÆΛÆ �,0Æ. 26
! Æ,0,1,�,�1,1,�
Employing similar methodology, we retrieve the solution for Maxwell fluid for velocity field as
,,� ,,�4Ω �� � �Æ
ÆÆ
1Æ,,ÆΛÆ 1� ,1 Æ �4Ω
� 0,1,1� ,0, ,�
�� � �Æ
,
�� �ΛÆΛÆ 1�,1 Æ. 27
! Æ,0,1,1� ,0,� �1,�
The main purpose of this article is to investigate generalized fractional Burger fluid for the flow of an edge in Rayleigh stokes problem in Newtonian and non-Newtonian form. Velocity field have been developed by the usage of Laplace and double Fourier Sine transforms along with their inverses. The general solutions are presented in the
Æ,Æ
compact form as the sum of Newtonian and non-Newtonian forms and expressed in terms of ÆÆ,ÆÆ (Fox-H Function). All compulsory conditions (initial, natural and boundary) are fulfilled. The general solutions have been particularized for fractionalized as well as ordinary in five types of fluids such as Newtonian, second grade, Maxwell, Oldroyd-B, Burger and generalized Burgers fluid. In order to analyze the rheology of parameter for fluid flows, Comparison through graphical illustration is presented for five models namely second grade, Maxwell, Oldroyd-B, Burger and generalized Burgers fluid with and without ordinary and fractional effects at two different times. However major consequences and outcomes are:
• The general solutions (equations 22-27) for velocity field are presented in the compact form as the sum
Æ,Æ
of Newtonian and non-Newtonian fluid and expressed in terms of ÆÆ,ÆÆ (Fox-H Function).
4. In this figure, we have depicted behavior of fluid flow with and without ordinary and fractional effects. Fig. 4(a) demonstrates the fluid flow with ordinary effects which describes flow in extremely thickness situation; on the other hand Fig. 4(b) has scattering behavior of fluid flow. Meanwhile, Fig. 5 is drawn at time � 5.0, the influence of five models on fluid flow with and without ordinary and fractional effects for thickness situation and scattering behavior is good in agreement. This is due to the fact that integer-order models are not more adequate than fractional-order models, because description of the memory in fluid flow is identified by fractional derivatives due singular kernel.
The author Kashif Ali Abro is highly thankful and grateful to the NED University of Engineering and Technology, Karachi, Pakistan for facilitating this research work.
J. Appl. Environ. Biol. Sci., 7(5)55-63, 2017
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