Some Exact Solutions for Helical Flows of Maxwell Fluid in an Annular Pipe due to Accelerated Shear Stresses

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Muhammad Jamil ◽  
Constantin Fetecau ◽  
Najeeb Alam Khan ◽  
Amir Mahmood
Keyword(s):  
Exact Solutions ◽  
Large Time ◽  
Maxwell Fluid ◽  
Hankel Transform ◽  
Circular Cylinders ◽  
Shear Stresses ◽  
Material Parameters ◽  
In Series ◽  
Finite Hankel Transform ◽  
Annular Pipe

Some exact solutions corresponding to helical flows of Maxwell fluid between two infinite coaxial circular cylinders are determined by means of finite Hankel transform, presented in series form, satisfied all imposed initial and boundary conditions. The motion is produced by the inner cylinder that applies the torsional and longitudinal time-dependent accelerated shear stresses to the fluid. The corresponding solutions for Newtonian fluid and large-time solutions are also obtained as limiting cases and effect of material parameters on the large-time solutions are discussed. Finally, the influence of pertinent parameters as well as a comparison between Maxwell and Newtonian fluids on the velocity components and shear stresses profiles is also examined by graphical illustrations.

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.

Couette Flow of a Generalized Oldroyd-B Fluid due to Constantly Accelerated Shear Stress Coupled with the Pressure Gradient

2011 ◽  
Vol 354-355 ◽  
pp. 179-182
Author(s):  
Chun Rui Li ◽  
Lian Cun Zheng ◽  
Xin Xin Zhang ◽  
Jia Jia Niu
Keyword(s):  
Shear Stress ◽  
Laplace Transform ◽  
Pressure Gradient ◽  
Exact Solutions ◽  
Couette Flow ◽  
Fractional Derivatives ◽  
Hankel Transform ◽  
Time Dependent ◽  
Infinite Cylinder ◽  
Finite Hankel Transform

This paper presented an analysis for the couette flow of a generalized Oldroyd-B fluid within an infinite cylinder subject to a time-dependent shear stress with the influence of the internal constantly decelerated pressure gradient. The exact solutions are established by means of the combine of the sequential fractional derivatives Laplace transform and finite Hankel transform and presented by integral and series form in terms of the Mittag-Leffler function. Moreover, the effects of various parameters are analyzed in detail by graphical illustrations.

scholarly journals An Accurate MHD Flux Solutions of a Viscose Fluid and Generalized Burgers' Model fluxwithin an Annular Pipe Under Sinusoidal Pressure

2018 ◽  
Vol 31 (1) ◽  
pp. 203
Author(s):  
Hanan F. Qasim
Keyword(s):  
Laplace Transform ◽  
Hankel Transform ◽  
Burgers Model ◽  
Analytical Studies ◽  
Finite Hankel Transform ◽  
Annular Pipe

The aim of this work presents the analytical studies of both the magnetohydrodynamic (MHD) flux and flow of the non-magnetohydro dynamic (MHD) for a fluid of generalized Burgers’ (GB) withinan annular pipe submitted under Sinusoidal  Pressure (SP)gradient. Closed beginning velocity's' solutions are taken by performing the finite Hankel transform (FHT) and Laplace transform (LT) of the successivefraction derivatives. Lastly, the figures were planned to exhibition the transformations effects of different fractional parameters (DFP) on the profile of velocity of both flows.

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 MHD Flows of Second Grade Fluid Through the Moving Porous Cylindrical Domain

2019 ◽  
Vol 12 (3) ◽  
pp. 1149-1175 ◽  
Author(s):  
Muhammad Jamil ◽  
Muhammad Zafarullah
Keyword(s):  
Outer Cylinder ◽  
Hankel Transform ◽  
Newtonian Fluids ◽  
Circular Cylinders ◽  
Second Grade ◽  
Convolution Product ◽  
Shear Stresses ◽  
Cylindrical Domain ◽  
Second Grade Fluid ◽  
Grade Fluid

The flows of Magnetohydrodynamics(MHD) second grade fluid between two infinite porous coaxial circular cylinders are studied. At time t=0^+, the inner cylinder begins to rotate around its axis and to slide along the same axis due to torsional and longitudinal time dependent shear stresses and the outer cylinder is also rotate around its axis and to slide along the same axis with acceleration. The exact solutions obtained with the help of discrete Laplace and finite Hankel transform, satisfy all imposed initial and boundary conditions. The solution presented in convolution product of Laplace transform . The corresponding solutions for second grade and Newtonian fluids are also obtained as limiting cases with and without MHD effect. Finally, the influence of pertinent parameters on the velocity components and shear stresses, as well as a comparison among, second grade and Newtonian fluids with and without MHD  is also analyzed by graphical illustrations.

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.

Laminar Flow in an Annulus With Arbitrary Time-Varying Pressure Gradient and Arbitrary Initial Velocity

1969 ◽  
Vol 36 (2) ◽  
pp. 309-311 ◽  
Author(s):  
Satya Prakash
Keyword(s):  
Pressure Gradient ◽  
Initial Velocity ◽  
Hankel Transform ◽  
Initial Distribution ◽  
Circular Cylinders ◽  
Time Varying ◽  
Annular Space ◽  
Arbitrary Time ◽  
Constant Quantity ◽  
Finite Hankel Transform

In this paper, the problem of incompressible laminar viscous flow in the annular space bounded by two coaxial infinite circular cylinders with an arbitrary time-varying pressure gradient and with an arbitrary initial distribution of velocity has been studied. The present problem generalizes the several earlier works in which the pressure gradient and the initial distribution of velocity have been taken in special forms. The analysis has been made by the use of finite Hankel transform. The case of steady flow when the pressure gradient is constant has been deduced by taking the pressure gradient to be a constant quantity and then letting the time since the start of the motion be infinite. This result has been shown in agreement with the already well-established result.

scholarly journals Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows

1975 ◽  
Vol 67 (4) ◽  
pp. 787-815 ◽  
Author(s):  
Allen T. Chwang ◽  
T. Yao-Tsu Wu
Keyword(s):  
Exact Solutions ◽  
Stokes Flow ◽  
Fundamental Solutions ◽  
The Body ◽  
Circular Cylinders ◽  
Geometrical Parameters ◽  
Flow Problems ◽  
Singularity Method ◽  
Wide Range ◽  
Body Shapes

The present study further explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyper-bolic profiles), while the body shapes cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.

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