On exact solutions for some oscillating motions of a generalized Oldroyd-B fluid

This paper deals with exact solutions for some oscillating motions of a generalized Oldroyd-B fluid. The fractional calculus approach is used in the constitutive relationship of fluid model. Analytical expressions for the velocity field and the corresponding shear stress for flows due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient are determined using Fourier sine and Laplace transforms. The obtained solutions are presented under integral and series forms in terms of the Mittag–Leffler functions. For α = β = 1, our solutions tend to the similar solutions for ordinary Oldroyd-B fluid. A comparison between generalized and ordinary Oldroyd-B fluids is shown by means of graphical illustrations.     

On exact solutions for some oscillating motions of a generalized Oldroyd-B fluid

This paper deals with exact solutions for some oscillating motions of a generalized Oldroyd-B fluid. The fractional calculus approach is used in the constitutive relationship of fluid model. Analytical expressions for the velocity field and the corresponding shear stress for flows due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient are determined using Fourier sine and Laplace transforms. The obtained solutions are presented under integral and series forms in terms of the Mittag–Leffler functions. For α = β = 1, our solutions tend to the similar solutions for ordinary Oldroyd-B fluid. A comparison between generalized and ordinary Oldroyd-B fluids is shown by means of graphical illustrations.     

Exact solutions for the flow of an Oldroyd-B fluid due to an infinite flat plate

The unsteady flow of an Oldroyd-B fluid due to an infinite flat plate, subject to a translation motion of linear time-dependent velocity in its plane, is studied by means of the Laplace transform. The velocity field and the associated tangential stress corresponding to the flow induced by the constantly accelerating plate as well as those produced by the impulsive motion of the plate are obtained as special cases. The solutions that have been determined, in all accordance with the solutions established using the Fourier transform, reduce to those for a Newtonian fluid as a limiting case. The similar solutions for a Maxwell fluid are also obtained.     

Global existence of classical solutions for the 3D generalized compressible Oldroyd-B model

In this paper, we prove the global existence of small classical solutions to the 3D generalized compressible Oldroyd-B system. It can be seen as compressible Euler equations coupling the evolution of stress tensor τ. The result mainly shows that singularity of solutions to compressible Euler equations can be prevented by the coupling of viscoelastic stress tensor. Moreover, unlike most complex fluids containing compressible Euler equations, the irrotational condition ∇×u=0 would not be proposed here to achieve the global well-posedness.     

Unsteady helical flows of a generalized Oldroyd-B fluid with fractional derivative

This paper deals with the unsteady helical flows of a generalized Oldroyd-B fluid between two infinite coaxial cylinders and within an infinite cylinder. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions are obtained with the help of integral transforms (Laplace transform, Weber transform and finite Hankel transform). The corresponding solutions for generalized second grade and Maxwell fluids as well as those for the Newtonian and ordinary Oldroyd-B fluids are also given in limiting cases. Finally, the influence of model parameters on the velocity field is also analyzed by graphical illustrations.     

Exact solutions for the rotational flow of a generalized Maxwell fluid between two circular cylinders

Unsteady flow of an incompressible generalized Maxwell fluid between two coaxial circular cylinders is studied by means of the Laplace and finite Hankel transforms. The motion of the fluid is produced by the rotation of cylinders around their common axis. The solutions that have been obtained, written in integral and series form in terms of the generalized G_a,b,c(·, t)-functions, are presented as a sum of the Newtonian solutions and the corresponding non-Newtonian contributions. They satisfy all imposed initial and boundary conditions and for λ → 0 reduce to the solutions corresponding to the Newtonian fluids performing the same solution. Furthermore, the corresponding solutions for ordinary Maxwell fluids are also obtained for β = 1. Finally, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity field ω(r, t) have been depicted against r and t for different values of the material and fractional parameters.     

Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative

The aim of this paper is to present the analytical solutions corresponding to two types of unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative between two parallel plates. The fractional calculus approach is used in solving the problems. The velocity distributions are determined by means of discrete Laplace transform and finite Fourier sine transform. The obtained results indicate that some well known solutions for the generalized second grade fluid, the generalized Maxwell fluid as well as the ordinary Oldroyd-B fluid appear as the limiting cases of the presented results.     

Exact solutions for a nonlinear model

In this paper we show new exact solutions for a type of generalized sine-Gordon equation which is obtained by constructing a Lagrange function for a dynamical coupled system of oscillators. We convert it into a nonlinear system by perturbing the potential energy from a point of view of an approach proposed by Fermi [1].     

Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative

This paper presents an analysis for magnetohydrodynamic (MHD) flow of an incompressible generalized Oldroyd-B fluid inducing by an accelerating plate. Where the no-slip assumption between the wall and the fluid is no longer valid. The fractional calculus approach is introduced to establish the constitutive relationship of a viscoelastic fluid. Closed form solutions for velocity and shear stress are obtained in terms of Fox H-function by using the discrete Laplace transform of the sequential fractional derivatives. The solutions for no-slip condition and no magnetic field can be derived as the special cases. Furthermore, the effects of various parameters on the corresponding flow and shear stress characteristics are analyzed and discussed in detail.     

Exact solutions for some oscillating flows of a second grade fluid with a fractional derivative model

In the present work, the exact analytic solutions for some oscillating flows of a generalized second grade fluid are investigated using Fourier sine and Laplace transforms. A more appropriate model is presented for fluid material between viscous and elastic to introduce the fractional calculus approach into the constitutive relationship. This paper employs the fractional calculus approach to study second grade fluid flows. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method has been used. Similar solutions for second grade fluid appear as the limiting cases of our solutions. The influence of pertinent parameters on the flows is delineated and appropriate conclusions are drawn.     

Unsteady longitudinal flow of a generalized Oldroyd-B fluid in cylindrical domains

Considering a fractional derivative model the unsteady flow of an Oldroyd-B fluid between two infinite coaxial circular cylinders is studied by using finite Hankel and Laplace transforms. The motion is produced by the inner cylinder which is subject to a time dependent longitudinal shear stress at time t = 0⁺. The solution obtained under series form in terms of generalized G and R functions, satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, generalized and ordinary Maxwell, and Newtonian fluids are obtained as limiting cases of our general solutions. The influence of pertinent parameters on the fluid motion as well as a comparison between models is illustrated graphically.     

Exact solutions of the generalized Dugdale model of two-dimensional decagonal quasicrystals

The basic properties of the generalized Dugdale model of quasicrystals have significance in the theory of fracture mechanics of this solids. This study suggests a conformal mapping which transforms the complicated region in physical plane onto the exterior of the unit circle in the ζ plane, then in terms of Cauchy integral theory, the complicated boundary value problem is solved, and the solutions are expressed in close form exactly. The crack tip opening displacement (CTOD), cohesive force zone and the most important physical quantity, stress intensity factor, are obtained with exact analytic form, which are useful in materials science and engineering.     

New exact solutions for magnetohydrodynamic flows of an Oldroyd-B fluid

This paper presents the new exact analytical solutions for magnetohydrodynamic (MHD) flows of an Oldroyd-B fluid. The explicit expressions for the velocity field and the associated tangential stress are established by using the Laplace transform method. Three characteristic examples: (i) flow due to impulsive motion of plate, (ii) flow due to uniformly accelerated plate, and (iii) flow due to non-uniformly accelerated plate are considered. The solutions for the hydrodynamic flows are special cases of the presented solutions. Moreover, the similar solutions corresponding to Maxwell and Newtonian fluids in the presence as well as absence of a magnetic field appear as the limiting cases of our solutions. The influences of the exerted magnetic field on the flow are also graphically presented and discussed. In particular, graphical results for the Oldroyd-B fluid are compared with those of a Newtonian fluid.     

Exact solutions for the flow of a viscoelastic fluid induced by a circular cylinder subject to a time dependent shear stress

The velocity field and the adequate shear stress corresponding to the flow of a Maxwell fluid with fractional derivative model, between two infinite coaxial cylinders, are determined by means of the Laplace and finite Hankel transforms. The motion is due to the inner cylinder that applies a longitudinal time dependent shear to the fluid. The solutions that have been obtained, presented under integral and series form in terms of the generalized G and R functions, satisfy all imposed initial and boundary conditions. They can be easy particularizes to give the similar solutions for ordinary Maxwell and Newtonian fluids. Finally, the influence of the relaxation time and the fractional parameter, as well as a comparison between models, is shown by graphical illustrations.     

Starting solutions for oscillating motions of an Oldroyd-B fluid over a plane wall

In this paper, we establish the starting solutions for oscillating motions of an Oldroyd-B fluid between two side walls perpendicular to a plane wall. The expressions for the velocity field and the associated tangential stress at the bottom wall are obtained, presented under integral and series form. These satisfy all imposed initial and boundary conditions. The obtained solutions are graphically analyzed for the variations of interesting flow parameters. In the absence of side walls, all solutions that have been obtained reduce to those corresponding to the motion over an infinite plate. Moreover, the obtained solutions can be specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motions.     

Exact solutions for generalized Maxwell fluid flow due to oscillatory and constantly accelerating plate

This paper deals with the analytical solutions for generalized Maxwell fluid flow due to oscillatory and constantly accelerating plate. The fractional calculus approach is used to establish the constitutive relationship of fluid model. Exact solutions are presented for the velocity field and the corresponding shear stress in series forms in terms of generalized G and R functions by using the discrete inverse Laplace transform method.     

Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a generalized Maxwell fluid, 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 solutions 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 ordinary Maxwell fluid are also obtained as limiting cases of our general solutions. At the end, flows corresponding to the ordinary Maxwell and generalized Maxwell fluids are shown and compared graphically by plotting velocity profiles at different values of time and some important results are remarked.