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Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model
Alexandria Engineering Journal (2017) xxx, xxx–xxx
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Alexandria University
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ORIGINAL ARTICLE
Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model Muhammad Saqib a, Farhad Ali a,*, Ilyas Khan b, Nadeem Ahmad Sheikh a, Syed Aftab Alam Jan a, Samiulhaq c a
Department of Mathematics, City University of Science and Information Technology, Peshawar 25000, Pakistan Basic Engineering Sciences Department, College of Engineering Majmaah University, Majmaah 11952, Saudi Arabia c Department of Mathematics, Islamia College University, Peshawar 25000, Pakistan b
Received 29 December 2016; revised 22 February 2017; accepted 4 March 2017
KEYWORDS Caputo-Fabrizio fractional derivatives; Generalized Jeffrey fluid; Free convection; The Laplace transform; Exact solutions
Abstract The present article reports the applications of Caputo-Fabrizio time-fractional derivatives. This article generalizes the idea of free convection flow of Jeffrey fluid over a vertical static plate. The free convection is caused due to the temperature gradient. Therefore, heat transfer is considered for free convection. The classical model for Jeffrey fluid is written in dimensionless form with the help of non-dimensional variables. Furthermore, the dimensionless model is converted into a fractional model called as a generalized Jeffrey fluid model. The governing equations of generalized Jeffrey fluid model have been solved analytically using the Laplace transform technique. The recovery of existing solutions in the open literature is possible through this work in terms of classical Jeffrey fluid, fractional Newtonian fluid as well as classical Newtonian fluid. For various embedded parameters, the physics of velocity and temperature profiles is studied by means of numerical computation. This report provides a detailed discussion as well as a graphical representation of the obtained results. Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting 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 1695, L’Hospital raised a question in a letter [1] to Leibniz about the particular notation that he had used the nth derivatives of a function Dn gðrÞ=Drn , in his publication, what would be the result if n ¼ 1=2. Leibniz responds an apparent paradox * Corresponding author. E-mail address: [email protected] (F. Ali). Peer review under responsibility of Faculty of Engineering, Alexandria University.
from which one-day useful consequences will be drawn. In these words, fractional calculus was born [2]. In the recent past, fractional calculus has applications almost in every field of science, mathematics, and engineering. Some of the fields where fractional calculus has made a valuable impact included viscoelasticity, electrochemistry, electrical engineering, mechanics, signal and image processing, control theory, mechatronics and physics [3]. During the last 60 years, the concept of using fractional calculus in the development of constitutive equations for materials with memory has been proposed
http://dx.doi.org/10.1016/j.aej.2017.03.017 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting 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/). Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
2 many times [4]. Bagley and Torvik [5] contributed to give a review of the applications of fractional calculus to viscoelasticity. Viscoelasticity of a material is defined as when it deforms exhibits both viscous and elastic behaviors through storage of mechanical energy and simultaneous dissipation. Mainardi [6] investigated the connections among fractional calculus, wave motion, and linear viscoelasticity. Makris et al. [7], presented a general boundary element modeling for the prediction of dynamic response of viscoelastic fluids. The generalized constitutive equations are formulated with fractional order derivatives. The proposed models are valid for the fluid motions with infinitesimally small displacement gradient and for fluids with linear viscoelastic behavior. Many scholars have used the concept of fractional calculus in their study, see [8–13] and the references therein. The famous definitions of fractional derivatives in fractional calculus are Riemann-Lioville, Grunwald-Letnikov, Caputo, Oldhan and Spanier, Miller and Ross and Kolwankar and Gangal [12]. Mostly Riemann-Liouville and Caputo fractional derivative are used [13,14]. Tan and Xu [15] studied the flow of generalized second-grade fluid using the definition of Riemann-Lioville fractional derivatives and obtained the exact analytic solutions with help of the Laplace transform. Wenchang et al. [16] applied the Riemann-Liouville fractional derivative approach to the flow of generalized Maxwell fluid between two parallel plates. Shahid [17], used the approach of Caputo fractional derivatives for the study of heat and mass transfer in MHD flow over an oscillating plate. Vieru et al. [18] analyzed the time-fractional free convection flow of generalized viscous fluid using the concept of Caputo fractional derivatives. Khan et al. [19] developed a mathematical model for generalized Casson fluid with Caputo fractional operator. However, there are some discrepancies in applications of these operators. In the case of Riemann-Liouville fractional derivatives, the derivative of a constant is not zero and the Laplace transform contains terms without physical significance. On the other hand, the kernel of the Caputo fractional derivatives is a singular function. To overcome this shortcoming, Caputo and Fabrizio developed a new approach without singularities [20]. The Caputo-Fabrizio time-fractional derivatives are convenient in the application of the Laplace transform. To develop a fractional model often, in the governing equations of the classical fluid model the time derivatives can be replaced with fractional derivatives of order a. Shah and Khan [21] applied the idea of the Caputo-Fabrizio fractional derivatives to generalize the starting flow of second grade fluid over a vertical plate and obtained the exact solutions using the Laplace transform technique. Some other recent studies can be found in [22–27] and the references therein. Non-Newtonian fluids have received the attention of the researchers due to their industrial and engineering applications, for instance, in dying of paper, plastics manufacturing, textile industry, food processing, wire and blade coating and movement of biological fluids [28]. Among them, the one that is widely used is the Jeffrey fluid model. The exact analysis of free convection flow of Jeffery fluid has been performed by Khan [29]. Zin et al. [30] considered the thermal radiation effect on the convective flow of Jeffery fluid with ramped wall temperature. Zeeshan and Majeed [31], studied the magnetic dipole effect on the convective flow of Jeffery fluid, past a porous plate with suction and injection. Some interesting and recent studies can be found in [32–41] and the references
M. Saqib et al. therein. Motivated from the above discussed literature, we have chosen Jeffery fluid model for the present analysis. More exactly, the aim of the present work was to generalize the work of Khan [29]. We have converted the classical model to fractional model called as generalized Jeffrey fluid model using the fractional derivative operator of Caputo-Fabrizio. The closed form solutions have been obtained using the Laplace transform. Closed form solutions are important for many reasons. They provide a standard for checking the accuracies of many approximate methods such as numerical or asymptotic. Also, these solutions can be used as a benchmark by numerical solvers and experimentalists for checking the stability of their solutions. 2. Governing equations The momentum and energy equations for incompressible Jeffrey fluid [29] are given as under the following: @uðy; tÞ l @ @ 2 uðy; tÞ q ¼ 1 þ k2 þ qgbT ðT T1 Þ; ð1Þ @t 1 þ k1 @t @y2 qcp
@Tðy; tÞ @ 2 Tðy; tÞ ¼k ; @t @y2
ð2Þ
subjected to the following initial and boundary conditions: uðy; 0Þ ¼ 0; Tðy; 0Þ ¼ T1 ; for all y > 0;
ð3Þ
for t > 0 : for t > 0
uð0; tÞ ¼ 0; Tð0; tÞ ¼ Tw ; uð1; tÞ ¼ 0; Tðy; tÞ ¼ T1 ;
ð4Þ
Here uðy; tÞ is the velocity of Jeffrey fluid in x-direction, l is dynamic viscosity, k1 ; k2 are Jeffery fluid parameters, q is constant density, g is gravitational acceleration, bT is volumetric thermal expansion, Tðy; tÞ is the temperature, cp is the specific heat capacity, k is the thermal conductivity, Tw is the wall temperature and T1 is the ambient temperature as shown in Fig. 1. Introducing the non-dimensional variables: u U0 U2 T T1 y; t ¼ 0 t; h ¼ ; y ¼ ; m m Tw T1 U0 into Eqs. (1)(4)we obtained (‘‘*” signs are dropped to minimize
u ¼
complexity):
@uðy; tÞ 1 @ @ 2 uðy; tÞ 1þk þ Grh; ¼ @t 1 þ k1 @t @y2
ð5Þ
@hðy; tÞ 1 @ 2 hðy; tÞ ¼ ; @t Pr @y2
ð6Þ
uð0; tÞ ¼ 0; hðy; 0Þ ¼ 0;
ð7Þ
uð0; tÞ ¼ 0; hð0; tÞ ¼ 1; uð1; tÞ ¼ 0; hð1; tÞ ¼ 0; where k ¼
k2 U20 m
:
; Gr ¼ mgbT ðTUw3T1 Þ and Pr ¼ 0
ð8Þ lcp k
are the Jeffrey
fluid parameter, thermal Grashof number and Prandtl number respectively. In order to develop a time fractional model for the convective flow of an incompressible generalized Jeffrey fluid we
Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
Exact solutions for free convection flow of generalized Jeffrey fluid
3
Pr1 ¼ a0 Pr and c ¼ a0 a: Solution of Eq. (14) in light of (13) is given by the following: hðy; qÞ ¼ Wðy; q; Pr1 ; c; 0Þ; where Wðg; q; m0 ; m1 ; m2 Þ ¼
ð15Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m0 q : exp g q m2 q þ m1
ð16Þ
Inverting the Laplace transform of Eq. (16), we have the following: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m0 m2 Wðg; t; m0 ; m1 ; m2 Þ ¼ exp m2 t g m1 þ m2 Z 2m0 1 sinðgsÞ s 2 m1 t ds exp m0 þ s2 p 0 sðm0 þ s2 Þ Z 2m0 m2 1 sinðgsÞ s m p ½ m þ ðm1 þ m2 Þs2 0 2 0 s2 m1 t exp ds: ð17Þ m0 þ s2
Figure 1
replace form:
@ ð:Þ @t
Taking the Laplace transform of Eq. (9) using initial condition from (7) and boundary conditions from (8), we arrived the following: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GrðqþcÞ2 ð1þk1 Þ ð1þk1 Þa0 q uðy; qÞ ¼ Pr1 q2 fð1þa exp y ð1þa 0 kÞqcgfðqþcÞð1þk1 Þqg 0 kÞqþc ð18Þ qffiffiffiffiffiffiffi GrðqþcÞ2 ð1þk1 Þ Pr1 q Pr1 q2 fð1þa0 kÞqcgfðqþcÞð1þk1 Þqg exp y qþc ;
Physical geometry and coordinate system.
with
Dat ð:Þ
in Eqs. (5) and (6), which takes the
Dat uðy; tÞ ¼
1 @ 2 uðy; tÞ ð1 þ kDat Þ þ Grh; 1 þ k1 @y2
Dat hðy; tÞ ¼
1 @ hðy; tÞ ; Pr @y2
ð9Þ
for simplicity Eq. (18) can be written as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffi! d1 d2 d3 b0 q exp y uðy; qÞ ¼ þ q þ b1 q2 q q þ a1 sffiffiffiffiffiffiffiffiffiffiffi! d1 d2 d3 Pr1 q ; exp y 2þ qþc q q q þ a1
ð19Þ
2
ð10Þ
here Dat ð:Þ is the Caputo-Fabrizio time fractional operator [20] and can be defined as follows: Z t 1 aðt sÞ 0 f ðsÞds; for 0 < a < 1: exp Dat fðtÞ ¼ 1a 0 1a ð11Þ 3. Exact solutions Applying the Laplace transform, using corresponding initial condition from (7) Eq. (10) takes the form: 1 q 1 d2 hðy; qÞ hðy; qÞ ¼ ; ð12Þ a 1 a q þ 1a Pr dy2 with transformed boundary conditions: 1 hð0; qÞ ¼ ; hð1; qÞ ¼ 0: q
ð13Þ
with
ð1þk1 Þ cð1þk1 Þ b0 ¼ a01þa ; b1 ¼ 1þac 0 k ; d1 ¼ Gr 1þk ; 0k 1 Pr1 1 Pr1 ð1a0 kÞ ; d2 ¼ Grð1 þ k1 Þ 1þkð1þk Pr Þ2 1
d3 ¼ Grð1 þ k1 Þ ½1þk
1
ðPr1 a0 kÞ2 2 1 Pr1 ð1þa0 kÞð1þk1 Pr1 Þ
1 Pr1 ; a1 ¼ c 1þk1þk : 1 Pr1 ð1þa0 kÞ
The more suitable but equivalent form of Eq. (19) is given as under the following: uðy; qÞ ¼ d1 vðy; q; b0 ; b1 Þ þ d2 Wðy; q; b0 ; b1 ; 0Þ d3 Wðy; q; b0 ; b1 ; a1 Þ d1 vðy; q; Pr1 ; cÞ d2 Wðy; q; Pr1 ; c; 0Þ þ d3 Wðy; q; Pr1 ; c; a1 Þ; ð20Þ
where vðg; q; m0 ; m1 Þ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m0 q ; exp g q2 q þ m1
with the Laplace inverse Z t Wðg; s; m0 ; m1 ; 0Þds: vðg; t; m0 ; m1 Þ ¼
ð21Þ
ð22Þ
0
1 ; Eq. (12) implies the following: Let us denote a0 ¼ 1a
d2 hðy; qÞ Pr1 q hðy; qÞ ¼ 0; qþc dy2
where
Upon taking the Laplace inverse of Eq. (20), we have the following: ð14Þ
uðy; tÞ ¼ d1 vðy; t; b0 ; b1 Þ þ d2 Wðy; t; b0 ; b1 ; 0Þ d3 Wðy; t; b0 ; b1 ; a1 Þ d1 vðy; t; Pr1 ; cÞ d2 Wðy; t; Pr1 ; c; 0Þ þ d3 Wðy; t; Pr1 ; c; a1 Þ: ð23Þ
Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
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M. Saqib et al.
Figure 2
Velocity plot for various values of k when Gr ¼ 10 and Pr ¼ 10.
4. Special cases
4.1. Classical solutions for velocity profile
The following special cases may be recovered from the general solutions (23).
Using the property of the Caputo–Fabrizio fractional derivative, when ða ! 1 ) a0 ! 1Þ, Eq. (23) is reduced to the cor-
Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
Exact solutions for free convection flow of generalized Jeffrey fluid
Figure 3
5
Velocity plot for various values of Gr when k = 1.5 and Pr ¼ 10.
responding classical velocity profile for the Jeffrey fluid as follows: uðy; tÞ ¼ d1 vðy; t; b0 ; b1 Þ þ d2 Wðy; t; b0 ; b1 ; 0Þ d3 Wðy; t; b0 ; b1 ; a1 Þ pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi d1 /ðy Pr; tÞ d2 Uðy Pr; t; 0Þ þ d3 Uðy Pr; t; a1 Þ; ð24Þ
with 1 þ k1 1 1 þ k1 kð1 þ k1 Þ ; ; b1 ¼ a1 ¼ ; d1 ¼ ; d2 ¼ k Pr k Pr d3 ¼ kPrð1 þ k1 Þ a1 ; b0 ¼
where
Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
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M. Saqib et al.
Figure 4
/ðg; tÞ ¼
Velocity plot for various values of Pr when k = 1.5 and Gr ¼ 10.
rffiffiffi g2 g t g2 erf c pffiffi g tþ e 4t ; 2 p 2 t
Uðg; t;m4 Þ ¼
e
m4 t
2
pffiffiffiffi g m4
e
ð25Þ
pffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi g g erf c pffiffi m4 t þ eg m4 erf c pffiffi þ m4 t : 2 t 2 t ð26Þ
The obtained solutions in Eq. (24) are equivalent to the solutions obtained by Khan [29, Eq. (31)]. 4.2. Velocity profile for fractional Newtonian fluid For k1 ¼ k ¼ 0, Eq. (23) is reduced to the fractional Newtonian which is given as under the following:
Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
Exact solutions for free convection flow of generalized Jeffrey fluid
Figure 5
Temperature plot for various values of t and a.
Figure 6
Temperature profile for various Pr and t.
uðy; tÞ ¼ d1 vðy; t; b0 ; b1 Þ þ d2 Wðy; t; b0 ; b1 ; 0Þ d1 vðy; t; Pr1 ; cÞ d2 Wðy; t; Pr1 ; c; 0Þ;
ð27Þ
with b0 ¼ a0; b1 ¼ a1 ¼ c; d1 ¼
c 1 and d2 ¼ : 1 Pr1 1 Pr1
4.3. Velocity profile for classical Newtonian fluid When a ! 1 ) a0 ! 1; Eq. (27) represents the following classical Newtonian fluid: pffiffiffiffiffi uðy; tÞ ¼ Gr1 /ðy; tÞ Gr1 /ðy Pr; tÞ; ð28Þ where Gr1 ¼
7
Gr : Pr 1
5. Skin friction and Nusselt number The dimensionless form of skin friction ðfsÞ and Nusselt number ðNuÞ is given as under the following:
1 a @uðy; tÞ
fs ¼ ð1 þ kDt Þ 1 þ k1 @y y¼0 Nu ¼
@hðy; tÞ
@y y¼0
ð29Þ
ð30Þ
6. Results and discussion A new definition of fractional derivatives namely the CaputoFabrizio fractional derivatives is applied and the closed form solutions for velocity and temperature distributions are obtained using the Laplace transform method. The influence of various embedded parameters such as fractional parameter a, Jeffrey fluid parameter k, thermal Grashof number Gr and Prandtl number Pr is analyzed for different values of t on velocity distribution. Temperature profile is studied for a; Pr and t. Figs. 2–4 depict the variation of a on velocity profile. The effect of a is significantly near the static vertical plate. It is interesting to observe that for small values of t the fractional modeled fluid flows faster than the classical modeled fluid. In
Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
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M. Saqib et al.
Figure 7
Figure 8
Temperature profile for various values of Pr.
Comparison Plot for classical Jeffery fluid.
addition, when we take larger values of t the classical modeled fluid flows faster than the fractional modeled fluid. Furthermore, for short time the fluid velocity decays to zero asymptotically earlier as compared to larger time.
Fig. 2 shows the influence of k on the fluid velocity. Physically, k gives rise to viscous forces or increases the nonNewtonian behavior and as a result, the thickness of momentum boundary layer increases. Due to this reason the fluid velocity decreases. The velocity of the fluid is independent of Pr when Gr ¼ 0: This case is not considered here because in the present study the velocity is just because of buoyancy forces (free convection) and we sketch our graphs with the positive values of Gr. In Fig. 3 the velocity profiles for different values of Gr are presented. Clearly, the velocity increases with an increase in Gr due to the fact that larger values of Gr enhance the buoyancy forces. Pr is a non-dimensional number which shows the relationship between thermal and momentum boundary layers. The velocity distribution for various values of Pr is plotted in Fig. 4, which shows that the velocity is a decreasing function of Pr. The physics behind these phenomena is that, larger the values of Pr, thicker will be the momentum boundary layer. Consequently, the fluid flow is reduced. In order to show the effect of a; t and Pr on the temperature profile, Figs. 5–7 are plotted. Fig. 5 shows the influence of a on the temperature profile for two different values of t: For smaller value when t ¼ 0:2 the classical temperature is less than fractional temperature. On the other hand, when we choose t ¼ 2 (larger time) then this effect reverses. The effect of Pr
Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
Exact solutions for free convection flow of generalized Jeffrey fluid Table 1
9
Variation in skin friction for different values of a.
a
t
k1
k
Pr
Gr
fs
0.2 0.5 0.7 1.00
5 5 5 5
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
1.5 1.5 1.5 1.5
10 10 10 10
3.500 1.754 1.028 0.559
Table 2
Variation in Nusselt number for different values of a.
a
t
Pr
Nu
0.2 0.5 0.7 1.00
5 5 5 5
10 10 10 10
2.082 1.208 0.977 0.400
is studied on temperature distribution as shown in Fig. 6. It is found that temperature decreases with increasing values of Pr; in addition the thickness of thermal boundary layer decreases. For smaller values of Pr the thermal conductivity increases; therefore, heat is able to defuse away from the heated plate more rapidly than for larger values of Pr. In Fig. 7 a comparison between classical and fractional temperate is provided. It is found that for larger values of t fractional temperature rapidly decays to zero asymptotically as compared to classical temperature but this behavior is opposite in case of smaller values of t. In Fig. 8 a comparison is provided between the classical solutions obtained in Eq. (24) of the present analysis to the solutions obtained by Khan [29, Eq. (31)]. The figure shows a strong agreement that both the solutions are equivalent. Table 1 exhibits variation in skin friction for different values of a when t ¼ 5: From the table it can be clearly observed that for large values of a, skin friction decreases which shows a strong agreement with that of velocity profiles. Table 2 presents variations in the rate of heat transfer in a. It is noticed that the rate of heat transfer decreases with increase in a which is quite opposite to that of temperature distribution. 7. Concluding remarks A fractional model has been developed for generalized Jeffrey fluid using the definition of Caputo-Fabrizio fractional derivatives. Exact solutions have been obtained with the help of the Laplace transform technique. The influence of various embedded parameters has been studied graphically. Based on the present analysis, the summarized key findings are as follows: Fractional fluid flows faster than classical fluid for smaller values of t whereas this effect is opposite in the case of larger values of t. The fluid velocity decreases with increasing values of k and Pr. The fluid velocity enhances with enhancement in Gr. It is noticed that temperature reduces for greater values of Pr. Skin friction and rate of heat transfer decrease with increasing a.
References [1] G.W. Leibniz, Letter from Hannover to GFA L’Hospital, 30 September 1695. Matematische Schriften, 2, 301–302. [2] S. Das, Functional fractional calculus for system identification and controls, 2008. [3] J. Sabatier, O.P Agrawal, J.T. Machado, Advances in Fractional Calculus, Vol. 4, Springer, Dordrecht, The Netherlands, 2007, No. 9. [4] R.C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51 (2) (1984) 299–307. [5] R.L. Bagley, P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol. (1978-present) 27 (3) (1983) 201–210. [6] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010. [7] N. Makris, G.F. Dargush, M.C. Constantinou, Dynamic analysis of generalized viscoelastic fluids, J. Eng. Mech. 119 (8) (1993) 1663–1679. [8] N.A. Sheikh, F. Ali, M. Saqib, I. Khan, S.A.A. Jan, A comparative study of Atangana-Baleanu and Caputo-Fabrizio fractional derivatives to the convective flow of a generalized Casson fluid, Eur. Phys. J. Plus 132 (1) (2017) 54, http://dx.doi. org/10.1140/epjp/i2017-11326-y. [9] F.C. Meral, T.J. Royston, R. Magin, Fractional calculus in viscoelasticity: an experimental study, Commun. Nonlinear Sci. Numer. Simul. 15 (4) (2010) 939–945. [10] N.A. Sheikh, F. Ali, M. Saqib, I. Khan, S.A.A. Jan, A.S. Alshomrani, M.S. Alghamdi, Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results Phys. (2017), http://dx.doi.org/ 10.1016/j.rinp.2017.01.025. [11] H. Qi, M. Xu, Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel, Mech. Res. Commun. 34 (2) (2007) 210–212. [12] N.A. Sheikh, F. Ali, I. Khan, M. Saqib. A modern approach of Caputo–Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium. Neural Comput. Appl., 1–11, http://dx.doi.org/10. 1007/s00521-016-2815-5. [13] F. Ali, N.A. Sheikh, I. Khan, M. Saqib, Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube: a fractional model, J. Magn. Magn. Mater. 423 (2017) 327–336. [14] R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Time fractional diffusion: a discrete random walk approach, Nonlinear Dyn. 29 (1–4) (2002) 129–143. [15] W.C. Tan, M.Y. Xu, The impulsive motion of flat plate in a generalized second grade fluid, Mech. Res. Commun. 29 (1) (2002) 3–9. [16] T. Wenchang, P. Wenxiao, X. Mingyu, A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates, Int. J. Non-Linear Mech. 38 (5) (2003) 645–650. [17] N. Shahid, A study of heat and mass transfer in a fractional MHD flow over an infinite oscillating plate, SpringerPlus 4 (1) (2015) 1.
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10 [18] D. Vieru, C. Fetecau, C. Fetecau, Time-fractional free convection flow near a vertical plate with Newtonian heating and mass diffusion, Therm. Sci. 19 (1) (2015) S85–S98. [19] I. Khan, N.A. Shah, D. Vieru, Unsteady flow of generalized Casson fluid with fractional derivative due to an infinite plate, Eur. Phys. J. Plus 131 (6) (2016) 1–12. [20] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl 1 (2) (2015) 1–13. [21] N.A. Shah, I. Khan, Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional CaputoFabrizio derivatives, Eur. Phys. J. C 76 (7) (2016) 1–11. [22] F. Ali, M. Saqib, I. Khan, N.A. Sheikh, Application of CaputoFabrizio derivatives to MHD free convection flow of generalized Walters’-B fluid model, Eur. Phys. J. Plus 131 (10) (2016) 377. [23] A.A. Zafar, C. Fetecau, Flow over an infinite plate of a viscous fluid with non-integer order derivative without singular kernel, Alexandria Eng. J. 55 (3) (2016) 2789–2796. [24] A. Atanganaa, I. Kocab, On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlinear Sci. Appl. 9 (2016) 2467–2480. [25] B.S.T. Alkahtani, A. Atangana, Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order, Chaos, Solitons Fractals 89 (2016) 539–546. [26] A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation, Appl. Math. Comput. 273 (2016) 948–956. [27] A. Atangana, D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. Eng. Mech. (2016) D4016005. [28] N. Dalir, M. Dehsara, S.S. Nourazar, Entropy analysis for magnetohydrodynamic flow and heat transfer of a Jeffrey nanofluid over a stretching sheet, Energy 79 (2015) 351–362. [29] I. Khan, A note on exact solutions for the unsteady free convection flow of a Jeffrey fluid, Zeitschrift fu¨r Naturforschung A 70 (6) (2015) 397–401. [30] N.A.M. Zin, I. Khan, S. Shafie, Influence of thermal radiation on unsteady MHD free convection flow of Jeffrey fluid over a vertical plate with ramped wall temperature, Math. Probl. Eng. 2016 (2016), http://dx.doi.org/10.1155/2016/6257071. [31] A. Zeeshan, A. Majeed, Heat transfer analysis of Jeffery fluid flow over a stretching sheet with suction/injection and magnetic dipole effect, Alexandria Eng. J. 55 (3) (2016) 2171–2181.
M. Saqib et al. [32] M.M. Bhatti, A. Zeeshan, Analytic study of heat transfer with variable viscosity on solid particle motion in dusty Jeffery fluid, Mod. Phys. Lett. B 30 (16) (2016) 1650196. [33] M. Turkyilmazoglu, Unsteady convection flow of some nanofluids past a moving vertical flat plate with heat transfer, J. Heat Transfer 136 (3) (2014) 031704. [34] M. Turkyilmazoglu, I. Pop, Heat and mass transfer of unsteady natural convection flow of some nanofluids past a vertical infinite flat plate with radiation effect, Int. J. Heat Mass Transf. 59 (2013) 167–171. [35] M. Turkyilmazoglu, Natural convective flow of nanofluids past a radiative and impulsive vertical plate, J. Aerospace Eng. 29 (6) (2016) 04016049. [36] Pourabdian, Majid, Mehran Qate, Mohammad Reza Morad, Alireza Javareshkian. The Jeffery-Hamel Flow and Heat Transfer of Nanofluids by Homotopy Perturbation Method and Comparison with Numerical Results. arXiv preprint arXiv: 1601.05298 (2016). [37] N.A. Sheikh, F. Ali, I. Khan, M. Saqib, A. Khan, MHD flow of micropolar fluid over an oscillating vertical plate embedded in porous media with constant temperature and concentration, Math. Probl. Eng. 2017 (2017), http://dx.doi.org/10.1155/2017/ 9402964. [38] M. Kothandapani, J. Prakash, Convective boundary conditions effect on peristaltic flow of a MHD Jeffery nanofluid, Appl. Nanosci. 6 (3) (2016) 323–335. [39] R. Ellahi, M.M. Bhatti, I. Pop, Effects of hall and ion slip on MHD peristaltic flow of Jeffrey fluid in a non-uniform rectangular duct, Int. J. Numer. Meth. Heat Fluid Flow 26 (6) (2016) 1802–1820. [40] M. Saqib, F. Ali, I. Khan, N.A. Sheikh, Heat and mass transfer phenomena in the flow of Casson fluid over an infinite oscillating plate in the presence of first-order chemical reaction and slip effect. Neural Comput. Appl. http://dx.doi.org/10.1007/ s00521-016-2810-x, pp. 1–14. [41] S.A. Gaffar, V.R. Prasad, E.K. Reddy, Computational study of Jeffrey’s non-Newtonian fluid past a semi-infinite vertical plate with thermal radiation and heat generation/absorption, Ain Shams Eng. J. (2016), http://dx.doi.org/10.1016/j. asej.2016.09.003.
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H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model Muhammad Saqib a, Farhad Ali a,*, Ilyas Khan b, Nadeem Ahmad Sheikh a, Syed Aftab Alam Jan a, Samiulhaq c a
Department of Mathematics, City University of Science and Information Technology, Peshawar 25000, Pakistan Basic Engineering Sciences Department, College of Engineering Majmaah University, Majmaah 11952, Saudi Arabia c Department of Mathematics, Islamia College University, Peshawar 25000, Pakistan b
Received 29 December 2016; revised 22 February 2017; accepted 4 March 2017
KEYWORDS Caputo-Fabrizio fractional derivatives; Generalized Jeffrey fluid; Free convection; The Laplace transform; Exact solutions
Abstract The present article reports the applications of Caputo-Fabrizio time-fractional derivatives. This article generalizes the idea of free convection flow of Jeffrey fluid over a vertical static plate. The free convection is caused due to the temperature gradient. Therefore, heat transfer is considered for free convection. The classical model for Jeffrey fluid is written in dimensionless form with the help of non-dimensional variables. Furthermore, the dimensionless model is converted into a fractional model called as a generalized Jeffrey fluid model. The governing equations of generalized Jeffrey fluid model have been solved analytically using the Laplace transform technique. The recovery of existing solutions in the open literature is possible through this work in terms of classical Jeffrey fluid, fractional Newtonian fluid as well as classical Newtonian fluid. For various embedded parameters, the physics of velocity and temperature profiles is studied by means of numerical computation. This report provides a detailed discussion as well as a graphical representation of the obtained results. Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting 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 1695, L’Hospital raised a question in a letter [1] to Leibniz about the particular notation that he had used the nth derivatives of a function Dn gðrÞ=Drn , in his publication, what would be the result if n ¼ 1=2. Leibniz responds an apparent paradox * Corresponding author. E-mail address: [email protected] (F. Ali). Peer review under responsibility of Faculty of Engineering, Alexandria University.
from which one-day useful consequences will be drawn. In these words, fractional calculus was born [2]. In the recent past, fractional calculus has applications almost in every field of science, mathematics, and engineering. Some of the fields where fractional calculus has made a valuable impact included viscoelasticity, electrochemistry, electrical engineering, mechanics, signal and image processing, control theory, mechatronics and physics [3]. During the last 60 years, the concept of using fractional calculus in the development of constitutive equations for materials with memory has been proposed
http://dx.doi.org/10.1016/j.aej.2017.03.017 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting 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/). Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
2 many times [4]. Bagley and Torvik [5] contributed to give a review of the applications of fractional calculus to viscoelasticity. Viscoelasticity of a material is defined as when it deforms exhibits both viscous and elastic behaviors through storage of mechanical energy and simultaneous dissipation. Mainardi [6] investigated the connections among fractional calculus, wave motion, and linear viscoelasticity. Makris et al. [7], presented a general boundary element modeling for the prediction of dynamic response of viscoelastic fluids. The generalized constitutive equations are formulated with fractional order derivatives. The proposed models are valid for the fluid motions with infinitesimally small displacement gradient and for fluids with linear viscoelastic behavior. Many scholars have used the concept of fractional calculus in their study, see [8–13] and the references therein. The famous definitions of fractional derivatives in fractional calculus are Riemann-Lioville, Grunwald-Letnikov, Caputo, Oldhan and Spanier, Miller and Ross and Kolwankar and Gangal [12]. Mostly Riemann-Liouville and Caputo fractional derivative are used [13,14]. Tan and Xu [15] studied the flow of generalized second-grade fluid using the definition of Riemann-Lioville fractional derivatives and obtained the exact analytic solutions with help of the Laplace transform. Wenchang et al. [16] applied the Riemann-Liouville fractional derivative approach to the flow of generalized Maxwell fluid between two parallel plates. Shahid [17], used the approach of Caputo fractional derivatives for the study of heat and mass transfer in MHD flow over an oscillating plate. Vieru et al. [18] analyzed the time-fractional free convection flow of generalized viscous fluid using the concept of Caputo fractional derivatives. Khan et al. [19] developed a mathematical model for generalized Casson fluid with Caputo fractional operator. However, there are some discrepancies in applications of these operators. In the case of Riemann-Liouville fractional derivatives, the derivative of a constant is not zero and the Laplace transform contains terms without physical significance. On the other hand, the kernel of the Caputo fractional derivatives is a singular function. To overcome this shortcoming, Caputo and Fabrizio developed a new approach without singularities [20]. The Caputo-Fabrizio time-fractional derivatives are convenient in the application of the Laplace transform. To develop a fractional model often, in the governing equations of the classical fluid model the time derivatives can be replaced with fractional derivatives of order a. Shah and Khan [21] applied the idea of the Caputo-Fabrizio fractional derivatives to generalize the starting flow of second grade fluid over a vertical plate and obtained the exact solutions using the Laplace transform technique. Some other recent studies can be found in [22–27] and the references therein. Non-Newtonian fluids have received the attention of the researchers due to their industrial and engineering applications, for instance, in dying of paper, plastics manufacturing, textile industry, food processing, wire and blade coating and movement of biological fluids [28]. Among them, the one that is widely used is the Jeffrey fluid model. The exact analysis of free convection flow of Jeffery fluid has been performed by Khan [29]. Zin et al. [30] considered the thermal radiation effect on the convective flow of Jeffery fluid with ramped wall temperature. Zeeshan and Majeed [31], studied the magnetic dipole effect on the convective flow of Jeffery fluid, past a porous plate with suction and injection. Some interesting and recent studies can be found in [32–41] and the references
M. Saqib et al. therein. Motivated from the above discussed literature, we have chosen Jeffery fluid model for the present analysis. More exactly, the aim of the present work was to generalize the work of Khan [29]. We have converted the classical model to fractional model called as generalized Jeffrey fluid model using the fractional derivative operator of Caputo-Fabrizio. The closed form solutions have been obtained using the Laplace transform. Closed form solutions are important for many reasons. They provide a standard for checking the accuracies of many approximate methods such as numerical or asymptotic. Also, these solutions can be used as a benchmark by numerical solvers and experimentalists for checking the stability of their solutions. 2. Governing equations The momentum and energy equations for incompressible Jeffrey fluid [29] are given as under the following: @uðy; tÞ l @ @ 2 uðy; tÞ q ¼ 1 þ k2 þ qgbT ðT T1 Þ; ð1Þ @t 1 þ k1 @t @y2 qcp
@Tðy; tÞ @ 2 Tðy; tÞ ¼k ; @t @y2
ð2Þ
subjected to the following initial and boundary conditions: uðy; 0Þ ¼ 0; Tðy; 0Þ ¼ T1 ; for all y > 0;
ð3Þ
for t > 0 : for t > 0
uð0; tÞ ¼ 0; Tð0; tÞ ¼ Tw ; uð1; tÞ ¼ 0; Tðy; tÞ ¼ T1 ;
ð4Þ
Here uðy; tÞ is the velocity of Jeffrey fluid in x-direction, l is dynamic viscosity, k1 ; k2 are Jeffery fluid parameters, q is constant density, g is gravitational acceleration, bT is volumetric thermal expansion, Tðy; tÞ is the temperature, cp is the specific heat capacity, k is the thermal conductivity, Tw is the wall temperature and T1 is the ambient temperature as shown in Fig. 1. Introducing the non-dimensional variables: u U0 U2 T T1 y; t ¼ 0 t; h ¼ ; y ¼ ; m m Tw T1 U0 into Eqs. (1)(4)we obtained (‘‘*” signs are dropped to minimize
u ¼
complexity):
@uðy; tÞ 1 @ @ 2 uðy; tÞ 1þk þ Grh; ¼ @t 1 þ k1 @t @y2
ð5Þ
@hðy; tÞ 1 @ 2 hðy; tÞ ¼ ; @t Pr @y2
ð6Þ
uð0; tÞ ¼ 0; hðy; 0Þ ¼ 0;
ð7Þ
uð0; tÞ ¼ 0; hð0; tÞ ¼ 1; uð1; tÞ ¼ 0; hð1; tÞ ¼ 0; where k ¼
k2 U20 m
:
; Gr ¼ mgbT ðTUw3T1 Þ and Pr ¼ 0
ð8Þ lcp k
are the Jeffrey
fluid parameter, thermal Grashof number and Prandtl number respectively. In order to develop a time fractional model for the convective flow of an incompressible generalized Jeffrey fluid we
Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
Exact solutions for free convection flow of generalized Jeffrey fluid
3
Pr1 ¼ a0 Pr and c ¼ a0 a: Solution of Eq. (14) in light of (13) is given by the following: hðy; qÞ ¼ Wðy; q; Pr1 ; c; 0Þ; where Wðg; q; m0 ; m1 ; m2 Þ ¼
ð15Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m0 q : exp g q m2 q þ m1
ð16Þ
Inverting the Laplace transform of Eq. (16), we have the following: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m0 m2 Wðg; t; m0 ; m1 ; m2 Þ ¼ exp m2 t g m1 þ m2 Z 2m0 1 sinðgsÞ s 2 m1 t ds exp m0 þ s2 p 0 sðm0 þ s2 Þ Z 2m0 m2 1 sinðgsÞ s m p ½ m þ ðm1 þ m2 Þs2 0 2 0 s2 m1 t exp ds: ð17Þ m0 þ s2
Figure 1
replace form:
@ ð:Þ @t
Taking the Laplace transform of Eq. (9) using initial condition from (7) and boundary conditions from (8), we arrived the following: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GrðqþcÞ2 ð1þk1 Þ ð1þk1 Þa0 q uðy; qÞ ¼ Pr1 q2 fð1þa exp y ð1þa 0 kÞqcgfðqþcÞð1þk1 Þqg 0 kÞqþc ð18Þ qffiffiffiffiffiffiffi GrðqþcÞ2 ð1þk1 Þ Pr1 q Pr1 q2 fð1þa0 kÞqcgfðqþcÞð1þk1 Þqg exp y qþc ;
Physical geometry and coordinate system.
with
Dat ð:Þ
in Eqs. (5) and (6), which takes the
Dat uðy; tÞ ¼
1 @ 2 uðy; tÞ ð1 þ kDat Þ þ Grh; 1 þ k1 @y2
Dat hðy; tÞ ¼
1 @ hðy; tÞ ; Pr @y2
ð9Þ
for simplicity Eq. (18) can be written as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffi! d1 d2 d3 b0 q exp y uðy; qÞ ¼ þ q þ b1 q2 q q þ a1 sffiffiffiffiffiffiffiffiffiffiffi! d1 d2 d3 Pr1 q ; exp y 2þ qþc q q q þ a1
ð19Þ
2
ð10Þ
here Dat ð:Þ is the Caputo-Fabrizio time fractional operator [20] and can be defined as follows: Z t 1 aðt sÞ 0 f ðsÞds; for 0 < a < 1: exp Dat fðtÞ ¼ 1a 0 1a ð11Þ 3. Exact solutions Applying the Laplace transform, using corresponding initial condition from (7) Eq. (10) takes the form: 1 q 1 d2 hðy; qÞ hðy; qÞ ¼ ; ð12Þ a 1 a q þ 1a Pr dy2 with transformed boundary conditions: 1 hð0; qÞ ¼ ; hð1; qÞ ¼ 0: q
ð13Þ
with
ð1þk1 Þ cð1þk1 Þ b0 ¼ a01þa ; b1 ¼ 1þac 0 k ; d1 ¼ Gr 1þk ; 0k 1 Pr1 1 Pr1 ð1a0 kÞ ; d2 ¼ Grð1 þ k1 Þ 1þkð1þk Pr Þ2 1
d3 ¼ Grð1 þ k1 Þ ½1þk
1
ðPr1 a0 kÞ2 2 1 Pr1 ð1þa0 kÞð1þk1 Pr1 Þ
1 Pr1 ; a1 ¼ c 1þk1þk : 1 Pr1 ð1þa0 kÞ
The more suitable but equivalent form of Eq. (19) is given as under the following: uðy; qÞ ¼ d1 vðy; q; b0 ; b1 Þ þ d2 Wðy; q; b0 ; b1 ; 0Þ d3 Wðy; q; b0 ; b1 ; a1 Þ d1 vðy; q; Pr1 ; cÞ d2 Wðy; q; Pr1 ; c; 0Þ þ d3 Wðy; q; Pr1 ; c; a1 Þ; ð20Þ
where vðg; q; m0 ; m1 Þ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m0 q ; exp g q2 q þ m1
with the Laplace inverse Z t Wðg; s; m0 ; m1 ; 0Þds: vðg; t; m0 ; m1 Þ ¼
ð21Þ
ð22Þ
0
1 ; Eq. (12) implies the following: Let us denote a0 ¼ 1a
d2 hðy; qÞ Pr1 q hðy; qÞ ¼ 0; qþc dy2
where
Upon taking the Laplace inverse of Eq. (20), we have the following: ð14Þ
uðy; tÞ ¼ d1 vðy; t; b0 ; b1 Þ þ d2 Wðy; t; b0 ; b1 ; 0Þ d3 Wðy; t; b0 ; b1 ; a1 Þ d1 vðy; t; Pr1 ; cÞ d2 Wðy; t; Pr1 ; c; 0Þ þ d3 Wðy; t; Pr1 ; c; a1 Þ: ð23Þ
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Figure 2
Velocity plot for various values of k when Gr ¼ 10 and Pr ¼ 10.
4. Special cases
4.1. Classical solutions for velocity profile
The following special cases may be recovered from the general solutions (23).
Using the property of the Caputo–Fabrizio fractional derivative, when ða ! 1 ) a0 ! 1Þ, Eq. (23) is reduced to the cor-
Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
Exact solutions for free convection flow of generalized Jeffrey fluid
Figure 3
5
Velocity plot for various values of Gr when k = 1.5 and Pr ¼ 10.
responding classical velocity profile for the Jeffrey fluid as follows: uðy; tÞ ¼ d1 vðy; t; b0 ; b1 Þ þ d2 Wðy; t; b0 ; b1 ; 0Þ d3 Wðy; t; b0 ; b1 ; a1 Þ pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi d1 /ðy Pr; tÞ d2 Uðy Pr; t; 0Þ þ d3 Uðy Pr; t; a1 Þ; ð24Þ
with 1 þ k1 1 1 þ k1 kð1 þ k1 Þ ; ; b1 ¼ a1 ¼ ; d1 ¼ ; d2 ¼ k Pr k Pr d3 ¼ kPrð1 þ k1 Þ a1 ; b0 ¼
where
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Figure 4
/ðg; tÞ ¼
Velocity plot for various values of Pr when k = 1.5 and Gr ¼ 10.
rffiffiffi g2 g t g2 erf c pffiffi g tþ e 4t ; 2 p 2 t
Uðg; t;m4 Þ ¼
e
m4 t
2
pffiffiffiffi g m4
e
ð25Þ
pffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi g g erf c pffiffi m4 t þ eg m4 erf c pffiffi þ m4 t : 2 t 2 t ð26Þ
The obtained solutions in Eq. (24) are equivalent to the solutions obtained by Khan [29, Eq. (31)]. 4.2. Velocity profile for fractional Newtonian fluid For k1 ¼ k ¼ 0, Eq. (23) is reduced to the fractional Newtonian which is given as under the following:
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Exact solutions for free convection flow of generalized Jeffrey fluid
Figure 5
Temperature plot for various values of t and a.
Figure 6
Temperature profile for various Pr and t.
uðy; tÞ ¼ d1 vðy; t; b0 ; b1 Þ þ d2 Wðy; t; b0 ; b1 ; 0Þ d1 vðy; t; Pr1 ; cÞ d2 Wðy; t; Pr1 ; c; 0Þ;
ð27Þ
with b0 ¼ a0; b1 ¼ a1 ¼ c; d1 ¼
c 1 and d2 ¼ : 1 Pr1 1 Pr1
4.3. Velocity profile for classical Newtonian fluid When a ! 1 ) a0 ! 1; Eq. (27) represents the following classical Newtonian fluid: pffiffiffiffiffi uðy; tÞ ¼ Gr1 /ðy; tÞ Gr1 /ðy Pr; tÞ; ð28Þ where Gr1 ¼
7
Gr : Pr 1
5. Skin friction and Nusselt number The dimensionless form of skin friction ðfsÞ and Nusselt number ðNuÞ is given as under the following:
1 a @uðy; tÞ
fs ¼ ð1 þ kDt Þ 1 þ k1 @y y¼0 Nu ¼
@hðy; tÞ
@y y¼0
ð29Þ
ð30Þ
6. Results and discussion A new definition of fractional derivatives namely the CaputoFabrizio fractional derivatives is applied and the closed form solutions for velocity and temperature distributions are obtained using the Laplace transform method. The influence of various embedded parameters such as fractional parameter a, Jeffrey fluid parameter k, thermal Grashof number Gr and Prandtl number Pr is analyzed for different values of t on velocity distribution. Temperature profile is studied for a; Pr and t. Figs. 2–4 depict the variation of a on velocity profile. The effect of a is significantly near the static vertical plate. It is interesting to observe that for small values of t the fractional modeled fluid flows faster than the classical modeled fluid. In
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Figure 7
Figure 8
Temperature profile for various values of Pr.
Comparison Plot for classical Jeffery fluid.
addition, when we take larger values of t the classical modeled fluid flows faster than the fractional modeled fluid. Furthermore, for short time the fluid velocity decays to zero asymptotically earlier as compared to larger time.
Fig. 2 shows the influence of k on the fluid velocity. Physically, k gives rise to viscous forces or increases the nonNewtonian behavior and as a result, the thickness of momentum boundary layer increases. Due to this reason the fluid velocity decreases. The velocity of the fluid is independent of Pr when Gr ¼ 0: This case is not considered here because in the present study the velocity is just because of buoyancy forces (free convection) and we sketch our graphs with the positive values of Gr. In Fig. 3 the velocity profiles for different values of Gr are presented. Clearly, the velocity increases with an increase in Gr due to the fact that larger values of Gr enhance the buoyancy forces. Pr is a non-dimensional number which shows the relationship between thermal and momentum boundary layers. The velocity distribution for various values of Pr is plotted in Fig. 4, which shows that the velocity is a decreasing function of Pr. The physics behind these phenomena is that, larger the values of Pr, thicker will be the momentum boundary layer. Consequently, the fluid flow is reduced. In order to show the effect of a; t and Pr on the temperature profile, Figs. 5–7 are plotted. Fig. 5 shows the influence of a on the temperature profile for two different values of t: For smaller value when t ¼ 0:2 the classical temperature is less than fractional temperature. On the other hand, when we choose t ¼ 2 (larger time) then this effect reverses. The effect of Pr
Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017
Exact solutions for free convection flow of generalized Jeffrey fluid Table 1
9
Variation in skin friction for different values of a.
a
t
k1
k
Pr
Gr
fs
0.2 0.5 0.7 1.00
5 5 5 5
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
1.5 1.5 1.5 1.5
10 10 10 10
3.500 1.754 1.028 0.559
Table 2
Variation in Nusselt number for different values of a.
a
t
Pr
Nu
0.2 0.5 0.7 1.00
5 5 5 5
10 10 10 10
2.082 1.208 0.977 0.400
is studied on temperature distribution as shown in Fig. 6. It is found that temperature decreases with increasing values of Pr; in addition the thickness of thermal boundary layer decreases. For smaller values of Pr the thermal conductivity increases; therefore, heat is able to defuse away from the heated plate more rapidly than for larger values of Pr. In Fig. 7 a comparison between classical and fractional temperate is provided. It is found that for larger values of t fractional temperature rapidly decays to zero asymptotically as compared to classical temperature but this behavior is opposite in case of smaller values of t. In Fig. 8 a comparison is provided between the classical solutions obtained in Eq. (24) of the present analysis to the solutions obtained by Khan [29, Eq. (31)]. The figure shows a strong agreement that both the solutions are equivalent. Table 1 exhibits variation in skin friction for different values of a when t ¼ 5: From the table it can be clearly observed that for large values of a, skin friction decreases which shows a strong agreement with that of velocity profiles. Table 2 presents variations in the rate of heat transfer in a. It is noticed that the rate of heat transfer decreases with increase in a which is quite opposite to that of temperature distribution. 7. Concluding remarks A fractional model has been developed for generalized Jeffrey fluid using the definition of Caputo-Fabrizio fractional derivatives. Exact solutions have been obtained with the help of the Laplace transform technique. The influence of various embedded parameters has been studied graphically. Based on the present analysis, the summarized key findings are as follows: Fractional fluid flows faster than classical fluid for smaller values of t whereas this effect is opposite in the case of larger values of t. The fluid velocity decreases with increasing values of k and Pr. The fluid velocity enhances with enhancement in Gr. It is noticed that temperature reduces for greater values of Pr. Skin friction and rate of heat transfer decrease with increasing a.
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Please cite this article in press as: M. Saqib et al., Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.017