Archive of Applied Mechanics

, Volume 81, Issue 8, pp 1153–1163

# RETRACTED ARTICLE: Flow of fractional Maxwell fluid between coaxial cylinders

Original

DOI: 10.1007/s00419-011-0536-x

Fetecau, C., Fetecau, C., Jamil, M. et al. Arch Appl Mech (2011) 81: 1153. doi:10.1007/s00419-011-0536-x

## Abstract

This paper deals with the study of unsteady flow of a Maxwell fluid with fractional derivative model, between two infinite coaxial circular cylinders, using Laplace and finite Hankel transforms. The motion of the fluid is produced by the inner cylinder that, at time t = 0+, is subject to a time-dependent longitudinal shear stress. Velocity field and the adequate shear stress are presented under series form in terms of the generalized G and R functions. The solutions that have been obtained satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases of general solutions. Finally, the influence of the pertinent parameters on the fluid motion as well as a comparison between the three models is underlined by graphical illustrations.

### Keywords

Maxwell fluidFractional calculusCoaxial cylindersVelocity fieldTime-dependent shear stressLaplace and Hankel transforms

### List of symbols

V(r, t)

Velocity field

S(r, t)

Extra-stress tensor

v(r, t)

Axial component of velocity field

τ(r, t)

Non-trivial shear stress

μ

The dynamic viscosity

λ

The relaxation time

ν

Kinematic viscosity

ρ

Density of the fluid

α

The fractional parameter

$${D_{t}^{\alpha}}$$

Fractional differential operator

r, t, q

Variables

R1, R2

Radii of inner and outer cylinders

a, b, c, d

Real/complex numbers

f

Constant

Ra, b(c, t), Ga, b, c(d, t)

Generalized functions

$${\overline{v}(r,q), \overline{v}_{\rm H}(r_{n}, q)}$$

Laplace and finite Hankel transforms of v(r, t)

$${\overline{\tau}(r,q)}$$

Laplace transform of τ(r, t)

vM(r, t), τM(r, t)

Velocity component and shear stress for classical Maxwell fluid

vN(r, t), τN(r, t)

Velocity component and shear stress for classical Newtonian fluid

## Authors and Affiliations

• C. Fetecau
• 1
• 2
• Corina Fetecau
• 3
• M. Jamil
• 1
• 4
• A. Mahmood
• 1
• 5
1. 1.Abdus Salam School of Mathematical SciencesGC UniversityLahorePakistan
2. 2.Department of MathematicsTechnical University of IasiIasiRomania
3. 3.Department of Theoretical MechanicsTechnical University of IasiIasiRomania
4. 4.Department of MathematicsNED University of Engineering and TechnologyKarachiPakistan
5. 5.Department of MathematicsCOMSATS Institute of Information TechnologyLahorePakistan