scholarly journals Slip Effects on Fractional Viscoelastic Fluids

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Muhammad Jamil ◽  
Najeeb Alam Khan
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Fluid Motion ◽  
Maxwell Fluid ◽  
Newtonian Fluids ◽  
Generalized Hypergeometric Functions ◽  
General Solutions ◽  
Slip Effects ◽  
Special Cases ◽  
Wright Generalized Hypergeometric Functions

Unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied, where the no-slip assumption between the wall and the fluid is no longer valid. The solutions obtained for the velocity field and shear stress, written in terms of Wright generalized hypergeometric functions , by using discrete Laplace transform of the sequential fractional derivatives, satisfy all imposed initial and boundary conditions. The no-slip contributions, that appeared in the general solutions, as expected, tend to zero when slip parameter is . Furthermore, the solutions for ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as special cases of general solutions. The solutions for fractional and ordinary Maxwell fluid for no-slip condition also obtained as limiting cases, and they are equivalent to the previously known results. Finally, the influence of the material, slip, and the fractional parameters on the fluid motion as well as a comparison among fractional Maxwell, ordinary Maxwell, and Newtonian fluids is also discussed by graphical illustrations.

scholarly journals Summation Formulas Obtained by Means of the Generalized Chain Rule for Fractional Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
S. Gaboury ◽  
R. Tremblay
Keyword(s):  
General Formula ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Mathematical Physics ◽  
Chain Rule ◽  
Generalized Hypergeometric Functions ◽  
Summation Formulas ◽  
Several Variables ◽  
Special Cases

In 1970, several interesting new summation formulas were obtained by using a generalized chain rule for fractional derivatives. The main object of this paper is to obtain a presumably new general formula. Many special cases involving special functions of mathematical physics such as the generalized hypergeometric functions, the Appell F1 function, and the Lauricella functions of several variables FD(n) are given.

scholarly journals Analytical Solutions of Upper Convected Maxwell Fluid with Exponential Dependence of Viscosity under the Influence of Pressure

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 334
Author(s):  
Constantin Fetecau ◽  
Dumitru Vieru ◽  
Tehseen Abbas ◽  
Rahmat Ellahi
Keyword(s):  
Fluid Motion ◽  
Maxwell Fluid ◽  
Newtonian Fluids ◽  
Exponential Dependence ◽  
Physical Parameters ◽  
Shear Stresses ◽  
Parallel Plates ◽  
Pressure Dependent Viscosity ◽  
Laplace Transform Technique ◽  
Dependent Viscosity

Some unsteady motions of incompressible upper-convected Maxwell (UCM) fluids with exponential dependence of viscosity on the pressure are analytically studied. The fluid motion between two infinite horizontal parallel plates is generated by the lower plate, which applies time-dependent shear stresses to the fluid. Exact expressions, in terms of standard Bessel functions, are established both for the dimensionless velocity fields and the corresponding non-trivial shear stresses using the Laplace transform technique and suitable changes of the unknown function and the spatial variable in the transform domain. They represent the first exact solutions for unsteady motions of non-Newtonian fluids with pressure-dependent viscosity. The similar solutions corresponding to the flow of the same fluids due to an exponential shear stress on the boundary as well as the solutions of ordinary UCM fluids performing the same motions are obtained as limiting cases of present results. Furthermore, known solutions for unsteady motions of the incompressible Newtonian fluids with/without pressure-dependent viscosity induced by oscillatory or constant shear stresses on the boundary are also obtained as limiting cases. Finally, the influence of physical parameters on the fluid motion is graphically illustrated and discussed. It is found that fluids with pressure-dependent viscosity flow are slower when compared to ordinary fluids.

Helices of fractionalized Maxwell fluid

2015 ◽  
Vol 4 (4) ◽  
Author(s):  
Muhammad Jamil ◽  
Kashif Ali Abro ◽  
Najeeb Alam Khan
Keyword(s):  
Angular Velocity ◽  
General Solution ◽  
Circular Cylinder ◽  
Special Function ◽  
Fluid Model ◽  
Fluid Motion ◽  
Maxwell Fluid ◽  
Initial Moment ◽  
Special Cases ◽  
In Series

AbstractIn this paper the helical flows of fractionalized Maxwell fluid model, through a circular cylinder, is studied. The motion is produced by the cylinder that at the initial moment begins to rotate around its axis with an angular velocity Omegatp, and to slide along the same axis with linear velocity Utp. The solutions that have been obtained using Laplace and finite Hankel transforms and presented in series form in terms of the newly defined special function M(z), satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for ordinary Maxwell and Newtonian fluid obtained as special cases of the present general solution. Finally, the influence of various pertinent parameters on fluid motion as well as the comparison among different fluids models is analyzed by graphical illustrations.

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.

Multi-parametric mittag-leffler functions and their extension

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Anatoly Kilbas ◽  
Anna Koroleva ◽  
Sergei Rogosin
Keyword(s):  
Fractional Calculus ◽  
Historical Development ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Integral Representations ◽  
Generalized Hypergeometric Functions ◽  
Late Professor ◽  
Wright Generalized Hypergeometric Functions

AbstractThis paper surveys one of the last contributions by the late Professor Anatoly Kilbas (1948–2010) and research made under his advisorship. We briefly describe the historical development of the theory of the discussed multi-parametric Mittag-Leffler functions as a class of the Wright generalized hypergeometric functions. The method of the Mellin-Barnes integral representations allows us to extend the considered functions to the case of arbitrary values of parameters. Thus, the extended Mittag-Leffler-type functions appear. The properties of these special functions and their relations to the fractional calculus are considered. Our results are based mainly on the properties of the Fox H-functions, as one of the widest class of special functions.

scholarly journals Certain Unified Integrals Associated with Product of M-Series and Incomplete H-functions

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1191 ◽  
Author(s):  
Manish Kumar Bansal ◽  
Devendra Kumar ◽  
Ilyas Khan ◽  
Jagdev Singh ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Special Cases ◽  
Fox’S H Function

In this paper, we established some interesting integrals associated with the product of M-series and incomplete H-functions, which are expressed in terms of incomplete H-functions. Next, we give some special cases by specializing the parameters of M-series and incomplete H-functions (for example, Fox’s H-Function, Incomplete Fox Wright functions, Fox Wright functions and Incomplete generalized hypergeometric functions) and also listed few known results. The results obtained in this work are general in nature and very useful in science, engineering and finance.

scholarly journals On a New Class of Laplace-Type Integrals Involving Generalized Hypergeometric Functions

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 87 ◽  
Author(s):  
Wolfram Koepf ◽  
Insuk Kim ◽  
Arjun K. Rathie
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
New Class ◽  
Special Cases ◽  
Laplace Type

In the theory of generalized hypergeometric functions, classical summation theorems for the series 2 F 1 , 3 F 2 , 4 F 3 , 5 F 4 and 7 F 6 play a key role. Very recently, Masjed-Jamei and Koepf established generalizations of the above-mentioned summation theorems. Inspired by their work, the main objective of the paper is to provide a new class of Laplace-type integrals involving generalized hypergeometric functions p F p for p = 2 , 3 , 4 , 5 and 7 in the most general forms. Several new and known cases have also been obtained as special cases of our main findings.

General Solutions for Magnetohydrodynamic Natural Convection Flow with Radiative Heat Transfer and Slip Condition over a Moving Plate

2013 ◽  
Vol 68 (10-11) ◽  
pp. 659-667 ◽  
Author(s):  
Constantin Fetecau ◽  
Dumitru Vieru ◽  
Corina Fetecau ◽  
Shahraz Akhter
Keyword(s):  
Natural Convection ◽  
Radiative Heat ◽  
Fluid Motion ◽  
Slip Condition ◽  
Convection Flow ◽  
Natural Convection Flow ◽  
Moving Plate ◽  
General Solutions ◽  
Special Cases ◽  
Laplace Transform Technique

General solutions for the magnetohydrodynamic (MHD) natural convection flow of an incompressible viscous fluid over a moving plate are established when thermal radiation, porous effects, and slip condition are taken into consideration. These solutions, obtained in closed-form by Laplace transform technique, depend on the slip coefficient and the three essential parameters Gr, Preff, and Keff. They satisfy all imposed initial and boundary conditions and can generate a large class of exact solutions corresponding to different fluid motions with technical relevance. For illustration, two special cases are considered and some interesting results from the literature are recovered as limiting cases. The influence of pertinent parameters on the fluid motion is graphically underlined.

Hypergeometric Mellin transforms

1955 ◽  
Vol 51 (4) ◽  
pp. 577-589 ◽  
Author(s):  
L. J. Slater
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Mellin Transforms ◽  
Special Cases

The Mellin transforms of generalized hypergeometric functions are discussed in this paper, and it is shown how some of the most general integrals of the Mellin type can be deduced from them. Four general theorems are considered and a number of special cases are given in detail.

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