March  2020, 13(3): 683-693. doi: 10.3934/dcdss.2020037

Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary

1. 

Department of Mathematics and statistics, The University of Lahore, Lahore, Pakistan

2. 

Department of Mathematics, Govt. College University, Lahore, Pakistan

3. 

Faculty of Sciences and Techniques Errachidia, Moulay Ismail University, Morocco

4. 

Department of Mathematics, University Of Management and Technology, Lahore, Pakistan

* Corresponding author: Zakia Hammouch, email: hammouch.zakia@gmail.com

Received  May 2018 Revised  November 2018 Published  March 2019

Fund Project: Prof. Zakia Hammouch was supported by the Research Project UMI2016 financed by Moulay Ismail University allowed to team E3MI.

The objective of this paper is to study the unsteady rotational flow of some non Newtonian fluids with Caputo fractional derivative through an infinite circular cylinder by means of the finite Hankel and Laplace transform. The novelty of the work is that motion is produced by applying tangential force not a specific but general function of time on the boundary. Initially the cylinder is at rest and after time $ t_{o} = 0^{+} $ it begins to rotate about its axis with an angular velocity $ \tau_{o} g(t) $. The obtained solutions of velocity field and shear stress have been presented under series form in terms of generalized $ G $-function, satisfying all imposed initial and boundary conditions. The corresponding solutions can be easily particularized to give similar solutions from existing literature for Oldroyd-B fluids, Maxwell fluids, Second grade fluids and Newtonian fluids with/without fractional derivatives performing similar motions.

Citation: Muhammad Mansha Ghalib, Azhar Ali Zafar, Zakia Hammouch, Muhammad Bilal Riaz, Khurram Shabbir. Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 683-693. doi: 10.3934/dcdss.2020037
References:
[1]

A. Atangana and J. F. Gomez Aguilar, Decolonisation of fractional calculus rules Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3.

[2]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A : Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.

[3]

H. A. AttiaM. A. Abdeen and W. A. El-Meged, Transient generalized Couette flow of viscoelastic fluid through a porous medium with variable viscosity and pressure gradient, Arab. J. Sci. Eng., 38 (2013), 3451-3458.  doi: 10.1007/s13369-013-0668-0.

[4]

I. Burdujan, The flow of a particular class of Oldroyd-B fluids, Ann. Acad. Romanian Sci. Ser. Math. Appl., 3 (2011), 23-45. 

[5]

A. Coronel EscamillaF. TorresJ. F. Gomez-AguilarF. Escobar-Jimenez and G. V. Guerrero-Ramirez, On the trajectory tracking control for an SCARA robot manipulator in a fractional model driven by induction motors with PSO tuning, Multibody System Dynamics, 43 (2018), 257-277.  doi: 10.1007/s11044-017-9586-3.

[6]

A. C. Escamilla, J. F. Gomez Aguilar, D. Baleanu, T. Cordova-Fraga, R.Jimenez, V. Peregrino and M. M. Al Qurashi, Bateman Feshbach Tikochinsky and Caldirola Kanai Oscillators with New Fractional Differentiation, Entropy, 19 (2017), 55.

[7]

C. Escamilla et al., A numerical solution for a variable-order reaction diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A: Statistical Mechanics and its Applications, 491 (2018), 406-424. doi: 10.1016/j.physa.2017.09.014.

[8]

C. FetecauMehwish RanaNiat Nigar and C. Fetecau, First exact solutions for flows of rate type fluids in a circular duct that applies a constant couple to the fluid, Z. Naturforsch, 69 (2014), 232-238.  doi: 10.5560/zna.2014-0022.

[9]

C. FetecauQ. RubbabS. Akhter and C. Fetecau, New methods to provide exact solutions for some unidirectional motions of rate type fluids, Thermal Science, 20 (2016), 7-20.  doi: 10.2298/TSCI130225130F.

[10]

M. A. Fontelos and A. Friedman, Stationary non-Newtonian fluid flows in channellike and pipe-like domains, Arch. Rational Mech. Anal., 151 (2000), 1-43.  doi: 10.1007/s002050050192.

[11]

B. J. GireeshaK. R. Madhura and C. S. Bagewadi, Flow of an unsteady dusty fluid through porous media in a uniform pipe with sector of a circle as cross-section, Int. J. Pure Appl. Math., 27 (2012), 20-38. 

[12]

J. F. Gomez Aguilar, T. Cordova-Fraga, J. Torres-Jimenez, R. F. Escobar-Jimenez, V. H. Olivares-Peregrino and G. V. Guerrero-Ramrez, Nonlocal transport processes and the fractional cattaneo-vernotte equation, Mathematical Problems in Engineering, 2016 (2016), Art. ID 7845874, 15 pp. doi: 10.1155/2016/7845874.

[13]

J. F. Gomez Aguilar et al., Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Advances in Difference Equations, 2017 (2017), Paper No. 68, 18 pp. doi: 10.1186/s13662-017-1120-7.

[14]

J. F. Gomez Aguilar, Chaos in a nonlinear Bloch system with Atangana Baleanu fractional derivatives, Numerical Methods for Partial Differential Equations, 34 (2018), 1716-1738.  doi: 10.1002/num.22219.

[15]

C. Guillope and J. C. Saut, Global existence and one-dimensional non-linear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Model, Math. Anal., 24 (1990), 369-401.  doi: 10.1051/m2an/1990240303691.

[16]

Z. Hammouch and T. Mekkaoui, Chaos synchronization of a fractional nonautonomous system, Nonautonomous Dynamical Systems, 1 (2014), 61-71.  doi: 10.2478/msds-2014-0001.

[17]

Z. Hammouch and T. Mekkaoui, Control of a new chaotic fractional-order system using Mittag-Leffler stability, Nonlinear Studies, 22 (2015), 565-577. 

[18]

Z. Hammouch and T. Mekkaoui, Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system., Complex and Intelligent Systems, (2015), 1-10.

[19]

T. HayatA. M. Siddiqui and S. Asghar, Some simple flows of an Oldroyd-B fluid, Int. J. Eng. Sci., 39 (2001), 135-147. 

[20]

T. HayatM. Khan and M. Ayub, Exact solutions of flow problems of an Oldroyd-B fluid, Appl. Math. Comput., 151 (2004), 105-119.  doi: 10.1016/S0096-3003(03)00326-6.

[21]

M. JamilC. Fetecau and M. Rana, Some exact solutions for Oldroyd-B fluid due to time dependent prescribed shear stress, J. Theor. Appl. Mech., 50 (2012), 549-562. 

[22]

F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana Baleanu fractional derivative, Chaos, Solitons and Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.

[23]

A. KaramiT. YousefiS. Mohebbi and C. Aghanajafi, Prediction of free convection from vertical and inclined rows of horizontal isothermal cylinders using ANFIS, Arab.J. Sci. Eng., 39 (2014), 4201-4209.  doi: 10.1007/s13369-014-1094-7.

[24]

Y. LiuF. Zong and J. Dai, Unsteady helical flow of a generalized Oldroyd-B fluid with fractional derivative, Int. J. Math. Trends Technology, 5 (2014), 66-76. 

[25]

C. F. Lorenzo and T. T. Hartley, Generalized Functions for fractional calculus, Critical Reviews in Biomedical Engineering, 36 (2008), 39-55.  doi: 10.1615/CritRevBiomedEng.v36.i1.40.

[26]

V. Mathur and K. Khandelwal, Exact solution for the flow of Oldroyd-B fluid between coaxial cylinders, Int. J. Eng. Res. Technol. (IJERT), 3 (2014), 949-954. 

[27] N. W. McLachlan, Bessel Functions for Engineers, Oxford Univeristy Press, London, 1955. 
[28]

G. Nagaraju and J. V. Ramana Murthy, MHD flow of longitudinal and torsional oscillations of a circular cylinder with suction in a couple stress fluid, Int. J. Appl. Mech. Eng., 18 (2013), 1099-1114.  doi: 10.2478/ijame-2013-0069.

[29]

J. G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc. Lond. A., 200 (1950), 523-541.  doi: 10.1098/rspa.1950.0035.

[30]

J. G. Oldroyd, The motion of an elastico-viscous liquid contained between coaxial cylinders, Q. J. Mech. Appl. Math., 4 (1951), 271-282.  doi: 10.1093/qjmam/4.3.271.

[31] I. Podlubny, Fractional Differential Equations, Academic Press, San Diege, 1999. 
[32]

K. R. Rajagopal and P. K. Bhatnagar, Exact solutions for some simple flows of an Oldroyd-B fluid, Acta Mech., 113 (1995), 233-239.  doi: 10.1007/BF01212645.

[33]

A. RaufA. A. Zafar and I. A. Mirza, Unsteady rotational flows of an Oldroyd-B fluid due to tension on the boundary, Alexandria Eng J, 54 (2015), 973-979.  doi: 10.1016/j.aej.2015.09.001.

[34]

N. Raza, Unsteady rotational flow of a second grade fluid with non-integer caputo time fractional derivative, Punjab University Journal of Mathematics, 49 (2017), 15-25. 

[35]

N. RazaM. AbdullahA. Rashid ButtA. U. Awan and E. U. Haque, Flow of a second grade fluid with fractional derivatives due to a quadratic time dependent shear stress, Alexandria Eng. J., 57 (2018), 1963-1969.  doi: 10.1016/j.aej.2017.04.004.

[36]

M. Renardy, Inflow boundary condition forsteady flow of viscoelastic fluids with differential constitutive laws, Rocky Mount. J. Math., 18 (1998), 445-453.  doi: 10.1216/RMJ-1988-18-2-445.

[37]

M. Renardy, An alternative approach to inflow boundary conditions for Maxwell fluids in three space dimensions, J. Non-Newtonian Fluid Mech., 36 (1990), 419-425.  doi: 10.1016/0377-0257(90)85022-Q.

[38]

M. B. RiazM. I. Asjad and K. Shabbir, Analytic solutions of fractional Oldroyd-B fluid in a circular duct that applies a constant couple, Alexandria Eng. J., 55 (2016), 3267-3275. 

[39]

H. RudolfY. Luchko and Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal, 12 (2009), 299-318. 

[40]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048.

[41]

B. Singh and N. K. Varshney, Effect of MHD visco-elastic fluid (Oldroyd) and porous medium through a circular cylinder bounded by a permeable bed, Int. J. Math. Arch., 3 (2012), 2912-2917. 

[42]

R. Talhouk, Unsteady flows of viscoelastic fluids with inflow and outflow boundary conditions, Appl. Math. Lett., 9 (1996), 93-98.  doi: 10.1016/0893-9659(96)00080-8.

[43]

D. Vieru and I. Siddique, Axial flow of several non-Newtonian fluids through a circular cylinder, Int. J. Appl.Mech., 2 (2010), 543-556.  doi: 10.1142/S1758825110000640.

[44]

N. D. Waters and M. J. King, Unsteady flow of an elastico-viscous liquid, Rheol. Acta., 93 (1970), 345-355. 

[45]

A. A. Zafar, N. A. Shah and Niat Nigar, On some rotational flows of non-integer order rate type fluids with shear stress on the boundary, Ain Shams Eng J, 9 (2018), 1865-1876, https://www.sciencedirect.com/science/article/pii/S2090447917300138 doi: 10.1016/j.asej.2016.08.018.

[46]

A. A. ZafarM. B. Riaz and M. I. Asjad, Unsteady Rotational Flow of fractional Maxwell Fluid in a cylinder subject to shear stress on the boundary, Punjab University Journal of Mathematics, 50 (2018), 21-32. 

show all references

References:
[1]

A. Atangana and J. F. Gomez Aguilar, Decolonisation of fractional calculus rules Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3.

[2]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A : Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.

[3]

H. A. AttiaM. A. Abdeen and W. A. El-Meged, Transient generalized Couette flow of viscoelastic fluid through a porous medium with variable viscosity and pressure gradient, Arab. J. Sci. Eng., 38 (2013), 3451-3458.  doi: 10.1007/s13369-013-0668-0.

[4]

I. Burdujan, The flow of a particular class of Oldroyd-B fluids, Ann. Acad. Romanian Sci. Ser. Math. Appl., 3 (2011), 23-45. 

[5]

A. Coronel EscamillaF. TorresJ. F. Gomez-AguilarF. Escobar-Jimenez and G. V. Guerrero-Ramirez, On the trajectory tracking control for an SCARA robot manipulator in a fractional model driven by induction motors with PSO tuning, Multibody System Dynamics, 43 (2018), 257-277.  doi: 10.1007/s11044-017-9586-3.

[6]

A. C. Escamilla, J. F. Gomez Aguilar, D. Baleanu, T. Cordova-Fraga, R.Jimenez, V. Peregrino and M. M. Al Qurashi, Bateman Feshbach Tikochinsky and Caldirola Kanai Oscillators with New Fractional Differentiation, Entropy, 19 (2017), 55.

[7]

C. Escamilla et al., A numerical solution for a variable-order reaction diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A: Statistical Mechanics and its Applications, 491 (2018), 406-424. doi: 10.1016/j.physa.2017.09.014.

[8]

C. FetecauMehwish RanaNiat Nigar and C. Fetecau, First exact solutions for flows of rate type fluids in a circular duct that applies a constant couple to the fluid, Z. Naturforsch, 69 (2014), 232-238.  doi: 10.5560/zna.2014-0022.

[9]

C. FetecauQ. RubbabS. Akhter and C. Fetecau, New methods to provide exact solutions for some unidirectional motions of rate type fluids, Thermal Science, 20 (2016), 7-20.  doi: 10.2298/TSCI130225130F.

[10]

M. A. Fontelos and A. Friedman, Stationary non-Newtonian fluid flows in channellike and pipe-like domains, Arch. Rational Mech. Anal., 151 (2000), 1-43.  doi: 10.1007/s002050050192.

[11]

B. J. GireeshaK. R. Madhura and C. S. Bagewadi, Flow of an unsteady dusty fluid through porous media in a uniform pipe with sector of a circle as cross-section, Int. J. Pure Appl. Math., 27 (2012), 20-38. 

[12]

J. F. Gomez Aguilar, T. Cordova-Fraga, J. Torres-Jimenez, R. F. Escobar-Jimenez, V. H. Olivares-Peregrino and G. V. Guerrero-Ramrez, Nonlocal transport processes and the fractional cattaneo-vernotte equation, Mathematical Problems in Engineering, 2016 (2016), Art. ID 7845874, 15 pp. doi: 10.1155/2016/7845874.

[13]

J. F. Gomez Aguilar et al., Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Advances in Difference Equations, 2017 (2017), Paper No. 68, 18 pp. doi: 10.1186/s13662-017-1120-7.

[14]

J. F. Gomez Aguilar, Chaos in a nonlinear Bloch system with Atangana Baleanu fractional derivatives, Numerical Methods for Partial Differential Equations, 34 (2018), 1716-1738.  doi: 10.1002/num.22219.

[15]

C. Guillope and J. C. Saut, Global existence and one-dimensional non-linear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Model, Math. Anal., 24 (1990), 369-401.  doi: 10.1051/m2an/1990240303691.

[16]

Z. Hammouch and T. Mekkaoui, Chaos synchronization of a fractional nonautonomous system, Nonautonomous Dynamical Systems, 1 (2014), 61-71.  doi: 10.2478/msds-2014-0001.

[17]

Z. Hammouch and T. Mekkaoui, Control of a new chaotic fractional-order system using Mittag-Leffler stability, Nonlinear Studies, 22 (2015), 565-577. 

[18]

Z. Hammouch and T. Mekkaoui, Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system., Complex and Intelligent Systems, (2015), 1-10.

[19]

T. HayatA. M. Siddiqui and S. Asghar, Some simple flows of an Oldroyd-B fluid, Int. J. Eng. Sci., 39 (2001), 135-147. 

[20]

T. HayatM. Khan and M. Ayub, Exact solutions of flow problems of an Oldroyd-B fluid, Appl. Math. Comput., 151 (2004), 105-119.  doi: 10.1016/S0096-3003(03)00326-6.

[21]

M. JamilC. Fetecau and M. Rana, Some exact solutions for Oldroyd-B fluid due to time dependent prescribed shear stress, J. Theor. Appl. Mech., 50 (2012), 549-562. 

[22]

F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana Baleanu fractional derivative, Chaos, Solitons and Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.

[23]

A. KaramiT. YousefiS. Mohebbi and C. Aghanajafi, Prediction of free convection from vertical and inclined rows of horizontal isothermal cylinders using ANFIS, Arab.J. Sci. Eng., 39 (2014), 4201-4209.  doi: 10.1007/s13369-014-1094-7.

[24]

Y. LiuF. Zong and J. Dai, Unsteady helical flow of a generalized Oldroyd-B fluid with fractional derivative, Int. J. Math. Trends Technology, 5 (2014), 66-76. 

[25]

C. F. Lorenzo and T. T. Hartley, Generalized Functions for fractional calculus, Critical Reviews in Biomedical Engineering, 36 (2008), 39-55.  doi: 10.1615/CritRevBiomedEng.v36.i1.40.

[26]

V. Mathur and K. Khandelwal, Exact solution for the flow of Oldroyd-B fluid between coaxial cylinders, Int. J. Eng. Res. Technol. (IJERT), 3 (2014), 949-954. 

[27] N. W. McLachlan, Bessel Functions for Engineers, Oxford Univeristy Press, London, 1955. 
[28]

G. Nagaraju and J. V. Ramana Murthy, MHD flow of longitudinal and torsional oscillations of a circular cylinder with suction in a couple stress fluid, Int. J. Appl. Mech. Eng., 18 (2013), 1099-1114.  doi: 10.2478/ijame-2013-0069.

[29]

J. G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc. Lond. A., 200 (1950), 523-541.  doi: 10.1098/rspa.1950.0035.

[30]

J. G. Oldroyd, The motion of an elastico-viscous liquid contained between coaxial cylinders, Q. J. Mech. Appl. Math., 4 (1951), 271-282.  doi: 10.1093/qjmam/4.3.271.

[31] I. Podlubny, Fractional Differential Equations, Academic Press, San Diege, 1999. 
[32]

K. R. Rajagopal and P. K. Bhatnagar, Exact solutions for some simple flows of an Oldroyd-B fluid, Acta Mech., 113 (1995), 233-239.  doi: 10.1007/BF01212645.

[33]

A. RaufA. A. Zafar and I. A. Mirza, Unsteady rotational flows of an Oldroyd-B fluid due to tension on the boundary, Alexandria Eng J, 54 (2015), 973-979.  doi: 10.1016/j.aej.2015.09.001.

[34]

N. Raza, Unsteady rotational flow of a second grade fluid with non-integer caputo time fractional derivative, Punjab University Journal of Mathematics, 49 (2017), 15-25. 

[35]

N. RazaM. AbdullahA. Rashid ButtA. U. Awan and E. U. Haque, Flow of a second grade fluid with fractional derivatives due to a quadratic time dependent shear stress, Alexandria Eng. J., 57 (2018), 1963-1969.  doi: 10.1016/j.aej.2017.04.004.

[36]

M. Renardy, Inflow boundary condition forsteady flow of viscoelastic fluids with differential constitutive laws, Rocky Mount. J. Math., 18 (1998), 445-453.  doi: 10.1216/RMJ-1988-18-2-445.

[37]

M. Renardy, An alternative approach to inflow boundary conditions for Maxwell fluids in three space dimensions, J. Non-Newtonian Fluid Mech., 36 (1990), 419-425.  doi: 10.1016/0377-0257(90)85022-Q.

[38]

M. B. RiazM. I. Asjad and K. Shabbir, Analytic solutions of fractional Oldroyd-B fluid in a circular duct that applies a constant couple, Alexandria Eng. J., 55 (2016), 3267-3275. 

[39]

H. RudolfY. Luchko and Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal, 12 (2009), 299-318. 

[40]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048.

[41]

B. Singh and N. K. Varshney, Effect of MHD visco-elastic fluid (Oldroyd) and porous medium through a circular cylinder bounded by a permeable bed, Int. J. Math. Arch., 3 (2012), 2912-2917. 

[42]

R. Talhouk, Unsteady flows of viscoelastic fluids with inflow and outflow boundary conditions, Appl. Math. Lett., 9 (1996), 93-98.  doi: 10.1016/0893-9659(96)00080-8.

[43]

D. Vieru and I. Siddique, Axial flow of several non-Newtonian fluids through a circular cylinder, Int. J. Appl.Mech., 2 (2010), 543-556.  doi: 10.1142/S1758825110000640.

[44]

N. D. Waters and M. J. King, Unsteady flow of an elastico-viscous liquid, Rheol. Acta., 93 (1970), 345-355. 

[45]

A. A. Zafar, N. A. Shah and Niat Nigar, On some rotational flows of non-integer order rate type fluids with shear stress on the boundary, Ain Shams Eng J, 9 (2018), 1865-1876, https://www.sciencedirect.com/science/article/pii/S2090447917300138 doi: 10.1016/j.asej.2016.08.018.

[46]

A. A. ZafarM. B. Riaz and M. I. Asjad, Unsteady Rotational Flow of fractional Maxwell Fluid in a cylinder subject to shear stress on the boundary, Punjab University Journal of Mathematics, 50 (2018), 21-32. 

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