scholarly journals The analytic solutions for the unsteady rotating flows of the generalized Maxwell fluid between coaxial cylinders

2020 ◽  
Vol 24 (6 Part B) ◽  
pp. 4041-4048
Author(s):  
Fan Wang ◽  
Wang-Cheng Shen ◽  
Jin-Ling Liu ◽  
Ping Wang
Keyword(s):  
Shear Stress ◽  
Maxwell Fluid ◽  
Analytic Solutions ◽  
Rotating Flows ◽  
Circular Cylinders ◽  
Rotating Flow ◽  
Coaxial Cylinders ◽  
Similar Solutions ◽  
Generalized Maxwell Fluid ◽  
R Functions

In this paper, we consider the unsteady rotating flow of the generalized Maxwell fluid with fractional derivative model between two infinite straight circular cylinders, where the flow is due to an infinite straight circular cylinder rotating and oscillating pressure gradient. The velocity field is determined by means of the combine of the Laplace and finite Hankel transforms. The analytic solutions of the velocity and the shear stress are presented by series form in terms of the generalized G and R functions. The similar solutions can be also obtained for ordinary Maxwell and Newtonian fluids as limiting cases.

scholarly journals Exact solutions for unsteady axial Couette flow of a fractional Maxwell fluid due to an accelerated shear

2011 ◽  
Vol 16 (2) ◽  
pp. 135-151 ◽  
Author(s):  
Muhammad Athar ◽  
Corina Fetecau ◽  
Muhammad Kamran ◽  
Ahmad Sohail ◽  
Muhammad Imran
Keyword(s):  
Shear Stress ◽  
Positive Real Number ◽  
Fractional Integration ◽  
Maxwell Fluid ◽  
Unsteady Motion ◽  
Positive Real ◽  
Special Cases ◽  
Multiple Integrals ◽  
Similar Solutions ◽  
R Functions

The velocity field and the adequate shear stress corresponding to the flow of a fractional Maxwell fluid (FMF) between two infinite coaxial cylinders, are determined by means of the Laplace and finite Hankel transforms. The motion is produced by the inner cylinder that at time t = 0+ applies a shear stress fta (a ≥ 0) to the fluid. The solutions that have been obtained, presented under series form in terms of the generalized G and R functions, satisfy all imposed initial and boundary conditions. Similar solutions for ordinary Maxwell and Newtonian fluids are obtained as special cases of general solutions. The unsteady solutions corresponding to a = 1, 2, 3, ... can be written as simple or multiple integrals of similar solutions for a = 0 and we extend this for any positive real number a expressing in fractional integration. Furthermore, for a = 0, 1 and 2, the solutions corresponding to Maxwell fluid compared graphically with the solutions obtained in [1–3], earlier by a different technique. For a = 0 and 1 the unsteady motion of a Maxwell fluid, as well as that of a Newtonian fluid ultimately becomes steady and the required time to reach the steady-state is graphically established. Finally a comparison between the motions of FMF and Maxwell fluid is underlined by graphical illustrations.

EXACT SOLUTIONS FOR THE POISEUILLE FLOW OF A GENERALIZED MAXWELL FLUID INDUCED BY TIME-DEPENDENT SHEAR STRESS

2010 ◽  
Vol 51 (4) ◽  
pp. 416-429 ◽  
Author(s):  
W. AKHTAR ◽  
CORINA FETECAU ◽  
A. U. AWAN
Keyword(s):  
Shear Stress ◽  
Velocity Field ◽  
Circular Cylinder ◽  
Poiseuille Flow ◽  
Fluid Motion ◽  
Maxwell Fluid ◽  
Time Dependent ◽  
Infinite Cylinder ◽  
Similar Solutions ◽  
Generalized Maxwell Fluid

AbstractThe Poiseuille flow of a generalized Maxwell fluid is discussed. The velocity field and shear stress corresponding to the flow in an infinite circular cylinder are obtained by means of the Laplace and Hankel transforms. The motion is caused by the infinite cylinder which is under the action of a longitudinal time-dependent shear stress. Both solutions are obtained in the form of infinite series. Similar solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases. Finally, the influence of the material and fractional parameters on the fluid motion is brought to light.

scholarly journals On analytical study of factional Oldroyd-B flow in annular region of two torsionally oscillating cylinders

2012 ◽  
Vol 16 (2) ◽  
pp. 411-421 ◽  
Author(s):  
A. Mahmood
Keyword(s):  
Shear Stress ◽  
Velocity Field ◽  
Analytical Study ◽  
Analytic Solutions ◽  
Circular Cylinders ◽  
Second Grade ◽  
General Solutions ◽  
Exact Analytic Solutions ◽  
Governing Differential Equation ◽  
R Functions

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a fractional Oldroyd-B fluid, also called generalized Oldroyd-B fluid (GOF), between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and after some time both cylinders suddenly begin to oscillate around their common axis with different angular frequencies of their velocities. The exact analytic solutions of the velocity field and associated shear stress, that have been obtained, are presented under integral and series forms in terms of generalized G and R functions. Moreover, these solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of classical Oldroyd-B, generalized Maxwell, classical Maxwell, generalized second grade, classical second grade and Newtonian fluids are also obtained as limiting cases of our general solutions.

scholarly journals Influence of Inclined MHD on Unsteady Flow of Generalized Maxwell Fluid with Fractional Derivative between Two Inclined Coaxial Cylinders through a Porous Medium

2020 ◽  
pp. 1705-1714
Author(s):  
Sundos Bader ◽  
Suad Naji Kadhim ◽  
Ahmed. M. Abdulhadi
Keyword(s):  
Porous Medium ◽  
Velocity Field ◽  
Fractional Derivative ◽  
Maxwell Fluid ◽  
Hankel Transform ◽  
Analytic Solutions ◽  
Circular Cylinders ◽  
Physical Parameters ◽  
The Laplace Transform ◽  
Generalized Maxwell Fluid

"This paper presents a study of inclined magnetic field on the unsteady rotating flow of a generalized Maxwell fluid with fractional derivative between two inclined infinite circular cylinders through a porous medium. The analytic solutions for velocity field and shear stress are derived by using the Laplace transform and finite Hankel transform in terms of the generalized G functions. The effect of the physical parameters of the problem on the velocity field is discussed and illustrated graphically.

Exact Solutions for the Flow of Fractional Maxwell Fluid in Pipe-Like Domains

2016 ◽  
Vol 8 (5) ◽  
pp. 784-794 ◽  
Author(s):  
Vatsala Mathur ◽  
Kavita Khandelwal
Keyword(s):  
Shear Stress ◽  
Velocity Field ◽  
Newtonian Fluid ◽  
Outer Cylinder ◽  
Fluid Motion ◽  
Maxwell Fluid ◽  
Circular Cylinders ◽  
Generalized Solutions ◽  
Special Cases ◽  
Graphical Illustration

AbstractThis paper presents an analysis of unsteady flow of incompressible fractional Maxwell fluid filled in the annular region between two infinite coaxial circular cylinders. The fluid motion is created by the inner cylinder that applies a longitudinal time-dependent shear stress and the outer cylinder that is moving at a constant velocity. The velocity field and shear stress are determined using the Laplace and finite Hankel transforms. Obtained solutions are presented in terms of the generalized G and R functions. We also obtain the solutions for ordinary Maxwell fluid and Newtonian fluid as special cases of generalized solutions. The influence of different parameters on the velocity field and shear stress are also presented using graphical illustration. Finally, a comparison is drawn between motions of fractional Maxwell fluid, ordinary Maxwell fluid and Newtonian fluid.

scholarly journals The First Solution for the Helical Flow of a Generalized Maxwell Fluid within Annulus of Cylinders by New Definition of Transcendental Function BNrrn

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Fang Wang ◽  
Jinling Liu
Keyword(s):  
Maxwell Fluid ◽  
Hankel Transform ◽  
Helical Flow ◽  
Transcendental Function ◽  
Coaxial Cylinders ◽  
Derivative Formulas ◽  
Finite Hankel Transform ◽  
Axis Of Symmetry ◽  
Definition Of ◽  
Generalized Maxwell Fluid

Most articles choose the transcendental function B1rrn to define the finite Hankel transform, and very few articles choose B0rrn. The derivations of B0rrn and B1rrn are also considered the same. In this paper, we find that the derivative formulas for the transcend function BNrrn are different and prove the derivative formulas for B0rrn and B1rrn. Based on the exact formulas of B0rrn and B1rrn, we keep on studying the helical flow of a generalized Maxwell fluid between two boundless coaxial cylinders. In this case, the inner and outer cylinders start to rotate around their axis of symmetry at different angular frequencies and slide at different linear velocities at time t=0+. We deduced the velocity field and shear stress via Laplace transform and finite Hankel transform and their inverse transforms. According to generalized G and R functions, the solutions we obtained are given in the form of integrals and series. The solution of ordinary Maxwell fluid has been also obtained by solving the limit of the general solution of fractional Maxwell fluid.

scholarly journals Exact Solutions for the Axial Couette Flow of a Fractional Maxwell Fluid in an Annulus

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
M. Imran ◽  
A. U. Awan ◽  
Mehwish Rana ◽  
M. Athar ◽  
M. Kamran
Keyword(s):  
Shear Stress ◽  
Velocity Field ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Couette Flow ◽  
Maxwell Fluid ◽  
Circular Cylinders ◽  
Rotational Flow ◽  
Material Parameters

The velocity field and the adequate shear stress corresponding to the rotational flow of a fractional Maxwell fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace and finite Hankel transforms. The solutions that have been obtained are presented in terms of generalized Ga,b,c(·,t) and Ra,b(·,t) functions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases of our general solutions. Finally, the influence of the material parameters on the velocity and shear stress of the fluid is analyzed by graphical illustrations.

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