[Paper Review] Some exact solutions for the rotational flow of a generalized second grade fluid between two circular cylinders

Amir Mahmood, Saif Ullah+1 more|ArXiv.org|Mar 29, 2008

Fractional Differential Equations Solutions7 references4 citations

TL;DR

This paper derives exact analytical solutions for the rotational flow of a generalized second-grade fluid between two concentric circular cylinders using Laplace and Hankel transforms. The key contribution is the derivation of velocity and shear stress profiles that depend on the fractional order β, showing that lower β values enhance fluid velocity, especially near boundaries, with solutions reducing to second-grade and Newtonian fluid cases when β = 1 or α = 0.

ABSTRACT

The velocity field and the associated tangential stress corresponding to flow of a generalized second grade fluid between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time $t=0$ the fluid is at rest and at $t=0^+$ cylinders suddenly begin to rotate about their common axis with a constant angular acceleration. The solutions that have been obtained satisfy the governing differential equations and all imposed initial and boundary conditions. The similar solutions for a second grade fluid and Newtonian fluid are recovered from our general solutions. The influence of the fractional coefficient on the velocity of the fluid is also analyzed by graphical illustrations.

Motivation & Objective

• To derive exact analytical solutions for the velocity field and shear stress in rotational flow of a generalized second-grade fluid between two coaxial cylinders.
• To investigate the influence of the fractional derivative order β on fluid dynamics in viscoelastic flows.
• To recover classical second-grade and Newtonian fluid solutions as special cases when β = 1 or α = 0.
• To analyze the transient behavior of the fluid under sudden rotation with constant angular acceleration.
• To provide a framework for modeling complex rheological behavior using fractional constitutive equations.

Proposed method

• Employing Laplace and Hankel transforms to solve the time-fractional partial differential equations governing the flow.
• Using the Riemann-Liouville fractional derivative operator D_t^β to model viscoelastic memory effects in the stress-strain relationship.
• Applying the generalized G-function representation to express the solutions in closed-form integral and series forms.
• Deriving the velocity and shear stress fields by inverting the Laplace and Hankel transforms using orthogonality of Bessel functions.
• Solving the resulting eigenvalue problem to determine the roots r_n from the transcendental equation B_1(R_1 r_n) = 0.
• Validating solutions by recovering known results for second-grade and Newtonian fluids when β = 1 or α = 0.

Experimental results

Research questions

• RQ1 How does the fractional order β in the constitutive equation affect the transient velocity profile of a generalized second-grade fluid in cylindrical Couette flow?
• RQ2 What are the exact analytical expressions for the velocity field and shear stress in the rotational flow of a generalized second-grade fluid between two concentric cylinders with time-dependent angular accelerations?
• RQ3 To what extent do the solutions for the generalized second-grade fluid reduce to those of second-grade and Newtonian fluids under specific parameter limits?
• RQ4 How does the fractional parameter β influence the development of velocity profiles over time and across the radial domain?
• RQ5 What is the role of the radial position and boundary conditions in shaping the transient response of the fluid under sudden rotation?

Key findings

• The velocity field increases as the fractional order β decreases, with the most significant enhancement observed near the cylinder boundaries.
• The generalized second-grade fluid flows faster than both second-grade and Newtonian fluids under identical boundary conditions, particularly for lower β values.
• The solutions reduce to classical second-grade fluid solutions when β = 1, and to Newtonian fluid solutions when β = 1 and α = 0.
• The transient behavior of the fluid is governed by exponential decay terms involving νr_n²t / (1 + αr_n²), indicating relaxation dynamics dependent on radial mode and fluid parameters.
• Numerical illustrations confirm that the influence of β is strongest near the inner and outer cylinders, with velocity profiles showing steeper gradients at lower β.
• The derived solutions are valid for all t > 0 and satisfy all initial and boundary conditions, including the sudden start of rotation with constant angular acceleration.