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Abstract
The unsteady flows of Burgers’ fluid with fractional derivatives model, through a circular cylinder, is studied by means of the Laplace and finite Hankel transforms. The motion is produced by the cylinder that at the initial moment begins to rotate around its axis with an angular velocity Ωt, and to slide along the same axis with linear velocity Ut. The solutions that have been obtained, presented in series form in terms of the generalized G _{ a,b,c }(•, t) functions, satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for fractionalized OldroydB, Maxwell and second grade fluids appear as special cases of the present results. Furthermore, the solutions for ordinary Burgers’, OldroydB, Maxwell, second grade and Newtonian performing the same motion, are also obtained as special cases of general solutions by substituting fractional parameters α = β = 1. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison among models, is shown by graphical illustrations.
The authors would like to express their sincere gratitude to the referees for their careful assessment and fruitful remarks and suggestions regarding the initial version of the manuscript.
The author Muhammad Jamil highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan; Department of Mathematics, NED University of Engineering & Technology, Karachi75270, Pakistan and also Higher Education Commission of Pakistan for generous support and facilitating this research work.
The author Najeeb Alam Khan is highly thankful and grateful to the Dean of Faculty of Sciences, University of Karachi, Karachi75270, Pakistan for supporting and facilitating this research work.
I. INTRODUCTION
II. DEVELOPMENT OF THE GOVERNING EQUATIONS
III. HELICAL FLOWS OF FRACTIONALIZED BURGERS’ FLUID
A. Calculation of the velocity field
B. Calculation of shear stresses
IV. THE SPECIAL CASES
A. Helical flows of ordinary Burgers’ fluid
B. Helical flows of fractionalized OldroydB fluid
C. Helical flows of ordinary OldroydB fluid
D. Helical flows of fractionalized Maxwell fluid
E. Helical flows of ordinary Maxwell fluid
F. Helical flows of fractionalized second grade fluid
G. Helical flows of ordinary second grade fluid
H. Helical flows of Newtonian fluid
V. NUMERICAL RESULTS AND CONCLUSIONS
Key Topics
 Maxwell equations
 19.0
 Viscoelasticity
 12.0
 Kinematics
 7.0
 Boundary value problems
 5.0
 Fluid equations
 5.0
Figures
Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 3, λ_{2} = 2, λ_{3} = 2, α = 0.6, β = 0.5 and different values of t.
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Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 3, λ_{2} = 2, λ_{3} = 2, α = 0.6, β = 0.5 and different values of t.
Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{2} = 0.1, λ_{3} = 0.2, α = 0.6, β = 0.5, t = 5s and different values of λ_{1}.
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Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{2} = 0.1, λ_{3} = 0.2, α = 0.6, β = 0.5, t = 5s and different values of λ_{1}.
Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 3, λ_{3} = 2, α = 0.6, β = 0.5, t = 5s and different values of λ_{2}.
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Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 3, λ_{3} = 2, α = 0.6, β = 0.5, t = 5s and different values of λ_{2}.
Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 7, λ_{2} = 0.5, α = 0.6, β = 0.5, t = 5s and different values of λ_{3}.
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Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 7, λ_{2} = 0.5, α = 0.6, β = 0.5, t = 5s and different values of λ_{3}.
Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ρ = 895.002, λ_{1} = 5, λ_{2} = 2, λ_{3} = 3, α = 0.6, β = 0.5 and different values of ν.
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Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ρ = 895.002, λ_{1} = 5, λ_{2} = 2, λ_{3} = 3, α = 0.6, β = 0.5 and different values of ν.
Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 2, λ_{2} = 2, λ_{3} = 1.5, α = 0.6, β = 0.5 and different values of r.
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Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 2, λ_{2} = 2, λ_{3} = 1.5, α = 0.6, β = 0.5 and different values of r.
Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 2, λ_{2} = 0.25, λ_{3} = 1.5, β = 0.1, t = 5s and different values of α.
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Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 2, λ_{2} = 0.25, λ_{3} = 1.5, β = 0.1, t = 5s and different values of α.
Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 1.5, λ_{2} = 0.25, λ_{3} = 1, α = 0.8, t = 5s and different values of β.
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Profiles of the velocity components w(r, t) and v(r, t) given by Eqs. (37) and (38), for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 1.5, λ_{2} = 0.25, λ_{3} = 1, α = 0.8, t = 5s and different values of β.
Profiles of the velocity components w(r, t) and v(r, t) for fractionalized Burgers’, fractionalized OldroydB, fractionalized Maxwell, fractionalized second grade and Newtonian fluids, for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 8, λ_{2} = 2, λ_{3} = 2, α = 0.6, β = 0.5 and t = 5s.
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Profiles of the velocity components w(r, t) and v(r, t) for fractionalized Burgers’, fractionalized OldroydB, fractionalized Maxwell, fractionalized second grade and Newtonian fluids, for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 8, λ_{2} = 2, λ_{3} = 2, α = 0.6, β = 0.5 and t = 5s.
Profiles of the velocity components w(r, t) and v(r, t) for ordinary Burgers’, ordinary OldroydB, ordinary Maxwell, ordinary second grade and Newtonian fluids, for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 8, λ_{2} = 2, λ_{3} = 2, α = 1, β = 1 and t = 5s.
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Profiles of the velocity components w(r, t) and v(r, t) for ordinary Burgers’, ordinary OldroydB, ordinary Maxwell, ordinary second grade and Newtonian fluids, for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 8, λ_{2} = 2, λ_{3} = 2, α = 1, β = 1 and t = 5s.
Profiles of the velocity components w(r, t) and v(r, t) for fractionalized Burgers’, fractionalized OldroydB, fractionalized Maxwell, fractionalized second grade and Newtonian fluids, for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 8, λ_{2} = 2, λ_{3} = 2, α = 0.6, β = 0.5 and large time t = 15 or 200s.
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Profiles of the velocity components w(r, t) and v(r, t) for fractionalized Burgers’, fractionalized OldroydB, fractionalized Maxwell, fractionalized second grade and Newtonian fluids, for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 8, λ_{2} = 2, λ_{3} = 2, α = 0.6, β = 0.5 and large time t = 15 or 200s.
Profiles of the velocity components w(r, t) and v(r, t) for ordinary Burgers’, ordinary OldroydB, ordinary Maxwell, ordinary second grade and Newtonian fluids, for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 8, λ_{2} = 2, λ_{3} = 2, α = 1, β = 1 and large time t = 13 or 45s.
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Profiles of the velocity components w(r, t) and v(r, t) for ordinary Burgers’, ordinary OldroydB, ordinary Maxwell, ordinary second grade and Newtonian fluids, for R = 1, Ω = 1, U = 1, ν = 0.0357541, μ = 32, λ_{1} = 8, λ_{2} = 2, λ_{3} = 2, α = 1, β = 1 and large time t = 13 or 45s.
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Abstract
The unsteady flows of Burgers’ fluid with fractional derivatives model, through a circular cylinder, is studied by means of the Laplace and finite Hankel transforms. The motion is produced by the cylinder that at the initial moment begins to rotate around its axis with an angular velocity Ωt, and to slide along the same axis with linear velocity Ut. The solutions that have been obtained, presented in series form in terms of the generalized G _{ a,b,c }(•, t) functions, satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for fractionalized OldroydB, Maxwell and second grade fluids appear as special cases of the present results. Furthermore, the solutions for ordinary Burgers’, OldroydB, Maxwell, second grade and Newtonian performing the same motion, are also obtained as special cases of general solutions by substituting fractional parameters α = β = 1. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison among models, is shown by graphical illustrations.
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