an analytic and mathematical synchronization of micropolar...

Scientia Iranica F (2019) 26(6), 3917{3927

Sharif University of TechnologyScientia Iranica

Transactions F: Nanotechnologyhttp://scientiairanica.sharif.edu

An analytic and mathematical synchronization ofmicropolar nano uid by Caputo-Fabrizio approach

K.A. Abroa and A. Y�ld�r�mb;�

a. Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, 76062, Jamshoro,Pakistan.

b. Department of Mathematics, Faculty of Science, Ege University, 35100, Bornova-_Izmir, Turkey.

Received 10 December 2018; received in revised form 12 June 2019; accepted 10 August 2019

KEYWORDSHeat transfer ofmicropolar nano uids;Fractional derivativeof non-singular kernel;Special Fox-Hfunction;Suspension ofnanoparticles in base uid.

Abstract. Nano uids and the enhancement of heat transfer in real systems have provedto be a widely researched area of nanotechnology; other areas of particular interest toresearchers include the improvement of thermal conductivity, thermophoresis phenomenon,dispersion of nanoparticles volume fraction, and few others. Based on the touch ofnanotechnology, this study investigates the analytic and mathematical performance ofmicropolar nano uid in the enhancement of heat transfer. The base uid is taken forthe purpose of thermal conductivity subject to two types of nanoparticles: copper andsilver. The mathematical analysis of the micropolar nano uid was carried out by invokingthe non-integer order derivative and transform methods. By applying a mathematical toolto the equations of micropolar nano uid, the solutions were explored for temperature,microrotation, and velocity. In order to meet the physical aspects of the problem basedon micropolar nano uid, the comparison of velocity �eld of micropolar nano uid for thesuspension of ethylene glycol into silver and that of ethylene glycol into copper is made toenhance the rate of heat transfer.

© 2019 Sharif University of Technology. All rights reserved.

1. Introduction

Nanoparticles are small quantities of nanometer-sizedparticles that are composed of metals, e.g., copperoxide, alumina, carbides, titania, gold, copper, andseveral others. These nanoparticles have the capabil-ity to enhance the thermophysical properties of base uids including ethylene glycol, bio uids, oil, water,lubricants, polymer solutions, and many others. Next,kerosene, engine oil, water, and ethylene glycol areinsu�cient for heat transfer due to their low thermal

*. Corresponding author. Tel./Fax: +902323111747E-mail addresses: [email protected] (K.A.Abro); [email protected] (A. Y�ld�r�m)

doi: 10.24200/sci.2019.52437.2717

conductivity. In order to enhance the convective heattransfer performance of such uids, various techniqueshave been implemented such as boundary conditionsand changing ow geometries. There is no doubt that uids have lower thermal conductivity than metals.The thermal conductivity of base uids can be en-hanced by adding metals and, consequentially, such uids are characterized as nano uids [1]. Moreover,the fundamental concepts of micropolar uid originatedfrom Eringen's study [2] to characterize the dynamicsof such uids by considering the microscopic impactsrising from micro-motion and local structure of the uids. A broad assessment of the micropolar uid withits engineering applications was conducted by Arimanet al. [3,4]. Hassanien and Gorla [5] explored the e�ectsof nonisothermal stretching sheet on micropolar withheat transfer, blowing, and suction. Mohammadein

3918 K.A. Abro and A. Y�ld�r�m/Scientia Iranica, Transactions F: Nanotechnology 26 (2019) 3917{3927

and Gorla [6] analyzed internal heat generation anddissipation with heat transfer in micropolar. Hussananet al. [7] obtained a closed-form solution for Newtonianheating in micropolar uid to the problem of free con-vection ow. Choi [8] studied the application of non-Newtonian uid ow with nanoparticles to enhancethe thermal conductivity of uid. Congedo et al. [9]carried out the analysis and modeling of nano uidswith natural convection heat transfer. Ghasemi andAminossadati [10] traced the impacts of water-CuOnano uid along with natural convection heat transfer.Hussanan et al. [11] examined the e�ect of somenano uids on accelerated plate with magnetic �eld anda porous medium. The water functionalized carbonnanotube ow was analyzed over a moving/staticwedge in the magnetic �eld by Khan et al. [12].Haq et al. [13] explored the e�ects of magnetic �eldwith water functionalized metallic nanoparticles forsqueezed ow over a sensor surface. In short, thisstudy includes few latest references on nano uids [14-16], modern fractional derivatives [17-22], heat trans-fer [23-27], nanoparticles [28-31], porosity and magnetic�eld [32-37], and few di�erent circumstances [38-43].Motivated by the above research work on nano uids,the purpose of this study is to investigate the analyticand mathematical performance of micropolar nano uidto enhance heat transfer. The base uid is taken forthe purpose of thermal conductivity subject to twotypes of nanoparticles, namely copper and silver, asshown in Figure 1(a)-(c). The mathematical analysisof the micropolar nano uid has been carried out byinvoking the non-integer order derivative and transformmethods. By applying a mathematical tool to theequations of micropolar nano uid, the solutions havebeen explored in terms of temperature, microrotation,and velocity. In order to meet the physical aspectsof the problem based on micropolar nano uid, thecomparison of velocity �eld of micropolar nano uid forthe suspension of ethylene glycol into silver and thatof ethylene glycol into copper is made to enhance therate of heat transfer.

2. Mathematical equations of nano uid

An unsteady ow of ethylene glycol based on mi-cropolar nano uid occupies the space lying over anoscillating plate perpendicular to the y-axis and issituated on the (x; z) plane. Initially, due to theconstant temperature of Tw, the uid is considered atrest. At t = 0+, the plate starts to oscillate with thevelocity UH(t) cos!t or U sin!t on its plane and thelevel of temperature increases up to Tw. The governingequations for the micropolar uid are as follows:

r:(�nfV) =@�nf@t

; (1)

r:p+r� (r�V)(K1 + �nf )

�r(r:V)(K1 + 2�nf ) = �nfb� �nf dVdt+K1(r�N); (2)

2K1N + nfr� (r�N)�r(r:N)( nf + �+ �)

= �nfI� �nf j dNdt +K1(r�V); (3)

where �nf , V, p, K1, �nf , b, N, I, �, �, j, and nf arethe nano uid density, velocity �eld, pressure, vortexviscosity, nano uid dynamic viscosity, body force vec-tor, microrotation vectors (gyration), body couple perunit mass vector, spin gradient viscosity coe�cients,micro-inertia density, and spin gradient viscosity, re-spectively. Under constant viscosity, Eqs. (1)-(3)can be converted in terms of Navier-Stokes equationsfor micropolar uid. Brinkman [44] expressed therelationship between base uid and dynamic viscosityof the nano uid written below:

�f = �nf (1� ')2:5: (4)

Aminossadati and Ghasemi [45] and Matin et al. [46]described the viscosity of nano uid in terms of the

Figure 1. (a) Ethylene glycol. (b) Copper metal powder. (c) Silver metal powder.

K.A. Abro and A. Y�ld�r�m/Scientia Iranica, Transactions F: Nanotechnology 26 (2019) 3917{3927 3919

following:

'�s + �f (1� ') = �nf : (5)

The expression of nf is established according to apublished study by Bourantas and Loukopoulos [47]:

j�K1

2+ �nf

�= nf : (6)

The continuity equation for incompressible ow is asfollows:

r:V = 0: (7)

By using the vector identity and neglecting body coupleforce, Eqs. (2) and (3) for free convection ow areexpressed as follows:

r:p�r(r:V)�nf �r2:V(K1 + �nf )

=�nfg � �nf dVdt +K1(r�N); (8)

2K1N� nfr2N�r(r:N)(�+ �)

=K1(r�V)� �nf j dNdt : (9)

The simpli�cation of Eqs. (8) and (9) takes place byusing mass conservation as follows:

r:p�r2:V(K1 + �nf )� �nfg = K1(r�N)

� �nf dVdt ; (10)

2K1N� nfr2N�r(r:N)(�+ �) = K1(r�V)

� �nf j dNdt : (11)

In order to apply the statement of material derivative,Eqs. (10) and (11) are expressed equivalently as follows:

r:p�r2:V(K1 + �nf )� �nf g = K1(r�N)

� �nf�

V(r:V) +dVdt

�; (12)

2K1N� nfr2N�r(r:N)(�+ �)=K1(r�V)

��nf j�

N(r:N) +dNdt

�:

(13)

For the problem considering Cartesian coordinates (x,y, z), the velocity, microrotation, and gravitational�elds are, respectively, assumed as follows:

V(w(y; t); 0; 0); N(0; 0; N(y; t)); and g(g; 0; 0):(14)

Simplifying Eqs. (12)-(14) gives:

@p@x

= �nf@w@x

+K1@N@x

+�nfg+(K1 + �nf )@2w@y2 ;

(15)

2K1N = K1@w@y� �nf j @N@x + nf j

@N@x

: (16)

Implementing Boussinesq approximation on Eq. (15)and assuming K1 = 0 in Eq. (16), we �nd that:

(T � T1)g(�T �)nf + (K1 + �nf )@2w@y2 ;+K1

@N@x

� �nf @w@x = 0; (17)

nf j@2N@y2 � �nf j @N@t = 0: (18)

Energy equation with thermal radiation as the previ-ously published papers [48,49] is de�ned as follows:

Knf@2T@y2 � (�C)nf

@T@t� @qr@y

= 0; (19)

where Knf is the thermal conductivity of nano uids,and (�C)nf is the heat capacity under constant pres-sure described by Khan et al. [50]:

'(�Cp)s + (�Cp)f (1� ') = (�Cp)nf ;

Knf

Kf=

2Kf + 2'(Ks �Kf ) +Ks

2Kf + '(Ks �Kf ) +Ks: (20)

The imposed conditions are set as follows:

T (y; 0)=T1; T (0; t)=Tw; T (1; t)=T1; (21)

N(y; 0)=C1; N(0; t)= t; N(1; t)=0; (22)

w(y; 0) = 0; w(0; t) = UH(t) cos(!t) or

U sin(!t); w(1; t) = 0: (23)

The energy equation (19) can be implemented usingRosseland approximation [51,52]:

Knf

�16T 3��3Knfk�

�@2T@y2 � @T

@t(�C)nf = 0: (24)

Inserting the following dimensionless quantities intoEqs. (17)-(19) gives:

T =T � T1Tw � T1 ; N� =

vfNU2 ; w� =

wU;

t� =U2tvf

; y� =Uyvf;

and:


K =k1

�f; R =

16T 31��3Kfk�

;

Gr =(Tw � T )1g(�T )f

U3 ; Pr =(Cp)f�fKf

:

(*: symbol is dropped for simplicity).The governing partial di�erential equations for

temperature distribution, microroration �eld, and ve-locity �eld are obtained in the Appendix (Eqs. (A.1)to (A.6)), respectively.

@2T@y2 = @0pr@�1

1@T@t; (25)

@2N@y2 = @2@�1

3@N@t

; (26)

@2w@y2 = @2@�1

5@w@t� @�1

5 Gr@4T � @�15 k

@N@y

: (27)

Here, the assumptions for Eqs. (25)-(27) are as follows:

T (y; 0) = 0; T (0; t) = t; T (1; t) = 0; (28)

N(y; 0) = 0; N(0; t) = t; N(1; t) = 0; (29)

w(y; 0) = 0; w(0; t) = UH(t) cos(!t) or

U sin(!t); w(1; t) = 0: (30)

Finally, expressing the governing equations (25)-(27)in terms of Caputo-Fabirizio fractional derivative, wehave:

@2T@y2 = @�1

1 pr@0@�

@t�T; (31)

@2N@y2 = @2@�1

3@�

@t�N; (32)

@2w@y2 = @2@�1

5@�

@t�w � @�1

5 Gr@4T � @�15 k

@N@y

: (33)

A fractional di�erential operator is de�ned for Eqs.(31)-(33). We have:

CF�@�

@t�

�= CF

�D�t

�=

tZ0

G0(�)1� � exp

��(t� �)

1� ��d�; 0 � � � 1;

(34)

where D�t or @�

@t� represents the fractional operatorof Caputo-Fabirizio having order 0 � � � 1 [53-55] de�ned at the normalization functions, which areM(1) = M(0) = 1.

3. Investigation of temperature distributionand microroration �eld

Using Laplace transform for Caputo-Fabirizio fraction-alized di�erential equations (31)-(32) and utilizing thefact that � = 1

(1��) , we obtain the following:

@2 �T@y2 = @�1

1 pr@0�s

(s+ ��)�T ; (35)

@2 �N@y2 = @�1

3 @2�s

(s+ ��)�N: (36)

Expressing Eqs. (35)-(36) in a more suitable formatequivalently, we have:

�T = s�2 exp

"�yr

pr@0�s@1s+ @1��

#; (37)

�N = s�2 exp

"�yr @2�s@3s+ @3��

#: (38)

Reworking on Eqs. (37)-(38), we obtain the summationform as follows:

�T =1s2 +

1Xl=1

1l!

�yr�pr@0

@1

!l1Xm=0

(��)m��m+ l

2

�m!�

� l2

�sm+2

; (39)

�N =1s2 +

1Xl=1

1l!

�yr�@2

@3

!l1Xm=0

(��)m��m+ l

2

�m!�

� l2

�sm+2

: (40)

Inverting Eqs. (39)-(40), we expressed the generalsolutions of temperature and microrotation �eld interms of Fox-H function as follows:

T =t+ (l!)�1

tZ0

(t� �)1Xl=1

�yr�pr@0

@1

!l

H1;11;3

24(��t)

��1� l

2 ; 1�

(0; 1);�1� l

2 ; 0�; (0; 1)

35 d�; (41)

N =t+ (l!)�1

tZ0

(t� �)1Xl=1

�yr�@2

@3

!l

H1;11;3

24(��t)

��1� l

2 ; 1�

(0; 1);�1� l

2 ; 0�; (0; 1)

35 d�; (42)


1Xf

(��)fQpj=1 �(aj +Ajf)

f !Qqj=1 �(bj +Bjf)

= H1;pp;q+1

�� (1� a1; A1); (1� a2; A2); (1� a3; A3); � � � ; (1� ap; Ap)(0; 1); (1� b1; B1); (1� b2; B2); (1� b3; B3); � � � ; (1� bq; Bq)

�: (43)

Box I

where the special function is de�ned by Eq. (43) asshown in Box I [56-59].

4. Investigation of velocity �eld

Using Laplace transform for Caputo-Fabirizio fraction-alized di�erential equation (33) and utilizing the factthat � = 1

(1��) , we obtain the following:

@2 �w@y2 =

@2�s(@5s+ @5��)

�w � @�15 Gr@4 �T� @�1

5 k@ �N@y

:(44)

Solving the partial di�erential equation (44) and usinginitial and boundary conditions (28-30), we get:

�w =Us

s2 + !2 exp

(�ys @2�s

(@5s+ @5��)

)

+kq@2�@3h@5@2�@3� @2

i 1s2 exp

(�ys @1�s

(@3s+ @3��)

)

� Gr@4hpr@0�@5@1

�@2

i 1s(s+��)

exp

(�yr@0�prs

@1

):

(45)

Reworking on Eq. (45), we obtain the summation formas follows:

�w =Us

s2 + !2 + U1Xl=1

1l!

��yp@2�@5

�l 1Xm=0

(��)��m+ l

2

�m!�

� l2

� ssm(s2 + !2)

+kq@2�@3h@2@5�@3� @2

i� 1s2 exp

(�ys @2�s@3(s+ ��)

)� Gr@4h

pr@5@0�@1�@2

i 1s(s+��)

exp

(�yrpr@0�s@1

):

(46)

Inverting Eq. (46), we expressed the general solution of

velocity as follows:

wc =UH(t) cos!t

+ UH(t)1Xl=1

1l!

��yp@2�@5

�l tZ0

cos!(t� �)

H1;11;3

24��t

� ��1� l

2 ; 1�

(0; 1);�1� l

2 ; 0�; (0; 1)

35 d�+

kq@2�@3h@5@2�@3� @2

i tZ0

��y; �;

@2�@3

; ��d�

� Gr@4h@5@0�pr@1� @2

i tZ0

��y; �;

@0�pr@1

; ��

exp (��(t� �)) d�; (47)

where:

L�1

(1s2 exp

�yr

Ass+B

!)=

tZ0

�(y; �; A;B)d�;

and:

L�1

(1s

exp

�yr

Ass+B

!)= �(y; �; A;B):

The case of sine oscillations has been establishedby applying a similar algorithm:

ws =U sin!t+ U1Xl=1

1l!

��yp@2�@5

�l tZ0

sin!(t� �)

�H1;11;3

24��t

� ��1� l

2 ; 1�

(0; 1);�1� l

2 ; 0�; (0; 1)

35 d�+

kq@2�@3h@2@5�@3� @2

i tZ0

��y; �;

@2�@3

; ��d�


� Gr@4h@5@0pr�@1� @2

i tZ0

��y; �;

@0pr�@1

; ��

exp (��(t� �)) d�: (48)

5. Limiting cases

5.1. Investigation of regular or conventionalnano uid, K1 = 0

The solutions for velocity �eld to the regular or conven-tional nano uid are established by assuming K1 = 0 (inthe absence of microrotation parameter) in Eqs. (47)and (48), as shown in the following:

wc =UH(t) cos!t

+ UH(t)1Xl=1

1l!

��yp@2�@5

�l tZ0

cos!(t� �)

H1;11;3

��t

� �� 1� l

2 ; 1�

(0; 1);�1� l

2 ; 0�; (0; 1)

�d�

� Gr@4h@5@0pr�@1� @2

i tZ0

��y; �;

@0pr�@1

; ��

exp (��(t� �)) d�: (49)

ws =U sin!t

+ U1Xl=1

1l!

��yp@2�@5

�l tZ0

sin!(t� �)

H1;11;3

24��t

� ��1� l

2 ; 1�

(0; 1);�1� l

2 ; 0�; (0; 1)

35 d�� Gr@4h@5@0pr�@1

� @2

i tZ0

��y; �;

@0pr�@1

; ��

exp(��(t� �))d�: (50)

5.2. Investigation of regular or conventionalNewtonian uid, K1 = � = 0

It is also pointed out that the analytic solutions to theregular or conventional Newtonian uid can be recov-ered from Eqs. (47) and (48) by assuming K1 = � = 0(in the absence of microrotation parameter). In whatfollows, one can also transform the analytic solutionsinto an ordinary di�erential operator by substituting� = 1.

6. Results and conclusion

In this research, micropolar nano uid veri�ably showedbetter thermal performance than conventional uidsbased on the mathematical tools of non-integer orderfractional derivative and transform. The analysisresults showed vivid e�ects on the enhancement of highthermal conductivity subject to suspended nanopar-ticles in the base uid. The graphical illustrationsof the investigated solutions, which recti�ed physicalconditions, were discussed. Various graphs are depictedin Table 1 for highlighting the e�ects of nanoparticlesand embedded parameters of micropolar nano uid.However, the key results are enumerated below:

(i) The analytic solutions were explored in termsof temperature, microrotation, and velocity; inaddition, similar solutions for velocity �eld andtemperature distribution to the regular or conven-tional nano uid, K1 = 0, and Newtonian uid,K1 = � = 0:, were also recovered as the limitingcases;

(ii) Figure 2 depicts the e�ect of nanoparticles basedon the two types of solutions, i.e., fractionalizednano uids, � = 0:4, and ordinary nano uids, � =1:0, in which the velocity �eld of copper-ethyleneglycol is higher than that of pure ethylene glycoland silver-ethylene glycol. It is noted that thevelocity �eld of fractionalized nano uids, � = 0:4,has reciprocal behavior with ordinary nano uids,� = 1:0. This may result from the exponentialkernel in the Caputo-Fabrizio fractional deriva-tive;

(iii) Figure 3 depicts temperature distribution withand without Caputo-Fabrizio fractional derivativefor the in uence of pure ethylene glycol, copper-

Table 1. Fundamental thermo-physical properties.

Base uid/nanoparticles � (kg/m3) Cp (J/kgK) k (W/m)

Cu 8933 385 401

Ag 10500 235 429

Ethylene glycol 1.115 0.58 0.1490


Figure 2. Pro�le of velocity �eld with and without Caputo-Fabrizio derivative.

Figure 3. Pro�le of temperature distribution with and without Caputo-Fabrizio derivative.

Figure 4. Pro�le of microrotation with and without Caputo-Fabrizio derivative.

ethylene glycol, and silver-ethylene glycol. Inthis �gure, copper-ethylene glycol has scatteringe�ects on the fractionalized temperature distri-bution and has opposite impacts on the ordinarytemperature distribution;

(iv) The in uence of nanoparticles on microrotationis underlined in Figure 4, in which the e�ect ofmicrorotation is observed to be opposite near theplate. It is also pointed out that copper-ethylene

glycol has accelerating behavior in comparison toall others; on the contrary, copper-ethylene glycolis of decelerating nature near the plate. This mayresult from the e�ective fractional parameter, �;

(v) Figures 5(a), 5(b), and 5(c) are plotted to inves-tigate the e�ects of di�erent values of volume � =0:0; 0:01; 0:02 on velocity �eld. Here, nanoparti-cles are suspended as copper-ethylene glycol andsilver-ethylene glycol for the velocity �eld with


Figure 5(a). Pro�le of velocity �eld with and without Caputo-Fabrizio derivative when nanoparticle volume fraction is� = 0:0.

Figure 5(b). Pro�le of velocity �eld with and without Caputo-Fabrizio derivative when nanoparticle volume fraction is� = 0:01.

Figure 5(c). Pro�le of velocity �eld with and without Caputo-Fabrizio derivative when nanoparticle volume fraction is� = 0:02.

and without Caputo-Fabrizio fractional deriva-tive. An increase in volume fraction results inthe scattering behavior of copper-ethylene glycolwith Caputo-Fabrizio fractional operator and thesequestrating behavior of copper-ethylene glycolwithout Caputo-Fabrizio fractional operator. It issigni�cantly noted that the converse phenomenonis observed as the values of volume fractionincrease; in simple words silver-ethylene glycol forthe velocity �eld with Caputo-Fabrizio fractionalderivative has scattering behavior and copper-ethylene glycol has sequestrating behavior. Thisis due to the fact that when temperature is lowerthan 180�C, then an increase in volume fractiongenerates an increase in thermal conductivity.

The same phenomenon can also be observed fortemperature distribution and microrotation, too.

Acknowledgement

The author Kashif Ali Abro is highly thankful andgrateful to Mehran University of Engineering and Tech-nology, Jamshoro, Pakistan for the generous supportand providing facilities for this research work.

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13. Haq, R.U., Nadeem, S., Khan, Z.H., and Noor, N.F.M.\MHD squeezed ow of water functionalized metallicnanoparticles over a sensor surface", Physica E: LowDimensional Systems and Nanostructures, 73, pp. 45-53 (2015).

14. Kashif, A.A., Mohammad, M.R., Ilyas, K., Irfan, A.A.,and Asifa, T. \Analysis of stokes' second problemfor nano uids using modern fractional derivatives",Journal of Nano uids, 7, pp. 738-747 (2018).

15. Abro, K.A., Mukarrum, H., and Mirza, M.B. \Ananalytic study of molybdenum disul�de nano uids us-ing modern approach of Atangana-Baleanu fractionalderivatives", Eur. Phys. J. Plus, 132, pp. 439-450(2017). DOI 10.1140/epjp/i2017-11689-y

16. Qasem, A.M.l., Kashif, A.A., and Ilyas, K. \Analyticalsolutions of fractional Walter's-B uid with applica-tions", Complexity, Article ID 8918541 (2018).

17. Kashif, A.A., Shaikh. H.S., Norzieha, M., Ilyas, K.,and Asifa, T. \A mathematical study of magnetohy-drodynamic Casson uid via special functions withheat and mass transfer embedded in porous plate",Malaysian Journal of Fundamental and Applied Sci-ences, 14(1), pp. 20-38 (2018).

18. Kashif, A.A. and Ilyas, K. \Analysis of heat and masstransfer in MHD ow of generalized Casson uid in aporous space via non-integer order derivative withoutsingular kernel", Chinese Journal of Physics, 55(4),pp. 1583-1595 (2017).

19. Zafar, A.A. and Fetecau, C. \Flow over an in�niteplate of a viscous uid with non-integer order deriva-tive without singular kernel", Alexandria EngineeringJournal, 55(3), pp. 2789-2796 (2016).

20. Muza�ar, H.L., Kashif, A.A., and Asif, A.S. \Heli-cal ows of fractional viscoelastic uid in a circularpipe", International Journal of Advanced and AppliedSciences, 4(10), pp. 97-105 (2017).

21. Kashif, A.A., Mukarrum, H., and Mirza, M.B. \An-alytical solution of MHD generalized Burger's uidembedded with porosity", International Journal ofAdvanced and Applied Sciences, 4(7), pp. 80-89 (2017).

22. Arshad, K., Kashif, A.A., Asifa, T., and Ilyas, K.\Atangana-Baleanu and Caputo Fabrizio analysis offractional derivatives for heat and mass transfer of sec-ond grade uids over a vertical plate: A comparativestudy", Entropy, 19(8), pp. 1-12 (2017).

23. _Ibics, B. and Bayram, M. \Numerical compari-son of methods for solving fractional di�erential-algebraic equations (FDAEs)", Computers & Mathe-matics with Applications, 62(8), pp. 3270-3278 (2011).https://doi.org/10.1016/j.camwa.2011.08.043

24. Kashif, A.A., Ilyas, K., and Asifa, T. \Applicationof Atangana-Baleanu fractional derivative to con-vection ow of MHD Maxwell uid in a porousmedium over a vertical plate", Mathematical Mod-elling of Natural Phenomena, 13, pp. 1-18 (2018).https://doi.org/10.1051/mmnp/2018007


25. Ilyas, K. and Kashif, A.A. \Thermal analysisin Stokes' second problem of nano uid: Ap-plications in thermal engineering", Case Studiesin Thermal Engineering, 10, pp. 271-275 (2018).https://doi.org/10.1016/j.csite.2018.04.005

26. Wakif, A., Boulahia, A.A., Animasaun, I.L., Afridi,M.I., Qasim, M., and Sehaqui, R. \Magneto-convection of alumina-water nano uid within thinhorizontal layers using the revised generalized Buon-giorno's model", Front. Heat Mass Transf., 12, pp. 1-15 (2019).

27. Doungmo, E.F.G. \Evolution equations with a param-eter and application to transport-convection di�eren-tial equations", Turkish Journal of Mathematics, 41,pp. 636-654 (2017). DOI:10.3906/mat-1603-107

28. Abderrahim, W., Zoubair, B., and Rachid, S. \A semi-analytical analysis of electro-thermo-hydrodynamicstability in dielectric nano uids using Buongiorno'smathematical model together with more realisticboundary conditions", Results in Physics, 9, pp. 1438-1454 (2018).

29. Abro, A.K., Ali, D.C., Irfan, A.A., and Ilyas, K.\Dual thermal analysis of magnetohydrodynamic owof nano uids via modern approaches of Caputo-Fabrizio and Atangana-Baleanu fractional derivativesembedded in porous medium", Journal of Ther-mal Analysis and Calorimetry, 18, pp. 1-11 (2018).https://doi.org/10.1007/s10973-018-7302-z

30. Kashif, A.A., Anwer, A.M., Shahid, H.A., and Ilyas,I. Tlili \Enhancement of heat transfer rate of solarenergy via rotating Je�rey nano uids using Caputo-Fabrizio fractional operator: An application to so-lar energy", Energy Reports, 5, pp. 41-49 (2019).https://doi.org/10.1016/j.egyr.2018.09.00

31. Abderrahim, W., Zoubair, B., and Rachid, S. \Numer-ical study of the onset of convection in a Newtoniannano uid layer with spatially uniform and non-uniforminternal heating", Journal of Nano uids, 6(1), pp. 136-148 (2017).

32. Doungmo, E.F.G. \Strange attractor existence for non-local operators applied to four-dimensional chaoticsystems with two equilibrium points", Chaos, 29,023117 (2019). https://doi.org/10.1063/1.5085440.

33. Ullah, H., Islam, S., Khan I., Sharidan, S., and Fiza,M. \MHD boundary layer ow of an incompressibleupper-convected Maxwell uid by optimal homotopyasymptotic method", Scientia Iranica B, 24(1), pp.202-210 (2017).

34. Ambreen, S., Kashif, A.A., and Muhammad,A.S. \Thermodynamics of magnetohydrodynamicBrinkman uid in porous medium: Applicationsto thermal science", Journal of Thermal Analysisand Calorimetry 136(6), pp. 2295-2304 (2018). DOI:10.1007/s10973-018-7897-0

35. Kashif, A.A., Ilyas, K., and G�omez-Aguilar, J.F. \Amathematical analysis of a circular pipe in rate type uid via Hankel transform", Eur. Phys. J. Plus, pp.133-397 (2018). DOI 10.1140/epjp/i2018-12186-7

36. Abderrahim, W., Zoubair, B., Mishra, S.R., Mo-hammad, M.R., and Rachid, S. \In uence of auniform transverse magnetic �eld on the thermo-hydrodynamic stability in water-based nano uids withmetallic nanoparticles using the generalized Buon-giorno's mathematical model", Eur. Phys. J. Plus,133(5), pp. 1-17 (2018).

37. Doungmo, E.F.G. \Solvability of chaotic fractional sys-tems with 3D four-scroll attractors", Chaos, Solitons& Fractals, 104, pp. 443-451 (2017).

38. Kashif, A.A. and Ahmet, Y. \Fractional treat-ment of vibration equation through modern anal-ogy of fractional di�erentiations using integral trans-forms", Iranian Journal of Science and Technol-ogy, Transaction A: Science, 43, pp. 1-8 (2019).https://doi.org/10.1007/s40995-019-00687-4

39. Abro, K.A. and Gomez-Aguilar, J.F. \Dual fractionalanalysis of blood alcohol model via non-integer orderderivatives", Fractional Derivatives with Mittag-Le�erKernel, Studies in Systems, Decision and Control,194, pp. 69-79 (2019). https://doi.org/10.1007/978-3-030-11662-0 5

40. Abro, K.A., Ali, A.M., and Anwer, A.M. \Function-ality of circuit via modern fractional di�erentiations",Analog Integrated Circuits and Signal Processing, 18,pp. 1-11 (2018). https://doi.org/10.1007/s10470-018-1371-6

41. Kashif, A.A., Mukarrum, H., and Mirza, M.B. \Amathematical analysis of magnetohydrodynamic gen-eralized Burger uid for permeable oscillating plate",Punjab University Journal of Mathematics, 50(2), pp.97-111 (2018).

42. Kashif, A.A. and Muhammad, A.S. \Heat transfer inmagnetohydrodynamics second grade uid with porousimpacts using Caputo-Fabrizoi fractional derivatives",Punjab University Journal of Mathematics, 49(2), pp.113-125 (2017).

43. Doungmo E.F.G., Khan, Y., and Mugisha, S. \Con-trol parameter & solutions to generalized evolu-tion equations of stationarity, relaxation and di�u-sion", Results in Physics, 9, pp. 1502-1507 (2018).https://doi.org/10.1016/j.rinp.2018.04.051.

44. Brinkman, H.C. \The viscosity of concentrated sus-pensions and solution", Journal of Chemical Physics,20, pp. 571-581 (1952).

45. Aminossadati, S.M. and Ghasemi, B. \Natural convec-tion cooling of a localised heat source at the bottomof a nano uid-�lled enclosure", European Journal ofMechanics B/Fluids, 28, pp. 630-640 (2009).

46. Matin, M.H. and Pop, I. \Forced convection heatand mass transfer ow of a nano uid through aporous channel with a �rst order chemical reaction onthe wall", International Communications in Heat andMass Transfer, 46, pp. 134-141 (2013).


47. Bourantas, G.C. and Loukopoulos, V.C. \MHDnatural-convection ow in an inclined square enclo-sure �lled with a micropolar-nano uid", InternationalJournal of Heat and Mass Transfer, 79, pp. 930-944(2014).

48. Turkyilmazoglu, M. and Pop, I. \Heat and masstransfer of unsteady natural convection ow of somenano uids past a vertical in�nite at plate with radi-ation e�ect", International Journal of Heat and MassTransfer, 59, pp. 167-171 (2013).

49. Hussanan, A., Khan, I., Hashim, H., Mohamed,M.K.A., Ishak, N., Sarif, N.M., and Salleh, M.Z.\Unsteady MHD ow of some nano uids past an accel-erated vertical plate embedded in a porous medium",Journal Teknologi, 78, pp. 121-126 (2016).

50. Khan, W.A., Khan, Z.H., and Haq, R.U. \Flow andheat transfer of ferro uids over a at plate withuniform heat ux", The European Physical JournalPlus, 130, pp. 1-10 (2015).

51. Hussanan, A., Khan, I., and Sha�e, S. \An exactanalysis of heat and mass transfer past a verticalplate with Newtonian heating", Journal of AppliedMathematics, 12, pp. 1-9 (2013).

52. Hussanan, A., Ismail, Z., Khan, I., Hussein, A.G.,and Sha�e, S. \Unsteady boundary layer MHD freeconvection ow in a porous medium with constantmass di�usion and Newtonian heating", The EuropeanPhysical Journal Plus, 129, pp. 1-16 (2014).

53. Caputo, M. and Fabrizio M. \A new de�nition offractional derivative without singular kernel", Progr.Fract. Di�er. Appl., 1(2), pp. 73-85 (2015).

54. Shah, N.A. and Khan, I. \Heat transfer analysis in asecond grade uid over and oscillating vertical plateusing fractional Caputo-Fabrizio derivatives", Eur.Phys. J. C, 4, pp. 362-376 (2016).

55. Kashif, A.A., Mukarrum, H., and Mirza, M.B. \Slip-page of fractionalized oldroyd-B uid with magnetic�eld in porous medium", Progr. Fract. Di�er. Appl.,3(1), pp. 69-80 (2017).

56. Mathai, A.M., Saxena, R.K., and Haubold, H.J., TheH-Functions: Theory and Applications, Springer, NewYork (2010).

57. Kashif, A.A. and Gomez-Aguilar, J.F. \A compar-ison of heat and mass transfer on a Walter's-B uid via Caputo-Fabrizio versus Atangana-Baleanufractional derivatives using the Fox-H function",Eur. Phys. J. Plus, 134, pp. 101-114 (2019). DOI10.1140/epjp/i2019-12507-4

58. Debnath, L. and Bhatta, D., Integral Transforms andTheir Applications, 2nd Ed., Chapman & Hall/CRC(2007).

59. Abro, K.A., Ilyas, K., Jos�e, F.G.A. \Thermal e�ectsof magnetohydrodynamic micropolar uid embeddedin porous medium with Fourier sine transform tech-nique", Journal of the Brazilian Society of MechanicalSciences and Engineering, 41, pp. 174-181 (2019).

Appendix

@0 =(Cp�)s'(Cp�)f

+ (1� '); (A.1)

@1 = R+knfkf

; (A.2)

@2 ='�s�f

+ (1� '); (A.3)

@3 =1

(1� ')2:5 +K2; (A.4)

@4 =(�T �)s'(�T �)f

+ (1� '); (A.5)

@5 =1

(1� ')2:5 + k: (A.6)

Biographies

Kashif Ali Abro was born in 1988 in Dadu, Sindh,Pakistan. He is currently an Assistant Professor inMehran University of Engineering and Technology,Jamshoro, Pakistan. He received his Bachelor's degreein Mathematics from University of Sindh Jamshoro,Pakistan and MPhil and PhD in Applied Mathematicsfrom the NED University of Engineering and Tech-nology, Karachi, Pakistan. His research interests in-clude fractional calculus (modern fractional di�erentialoperators), heat and mass transfer, nano uids, uidmechanics, and modern control engineering.

Ahmet Y�ld�r�m was born in 1980. He received hisMSc degree in Mathematics from Ege University, Izmir,Turkey in 2002. His �eld of research includes appliedmathematics and mathematical physics.

an analytic and mathematical synchronization of micropolar...

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