Run up flow of an incompressible micropolar fluid between parallel plates – A state space approach

In this paper, we study the run up flow of an incompressible micropolar fluid between two horizontal infinitely long parallel plates. Initially a flow of the fluid is induced by a constant pressure gradient until steady state is reached. After the steady state is reached, the pressure gradient is suddenly withdrawn while the two plates are impulsively started with different velocities in their own plane. Using the Laplace transform technique and adopting the state space approach, we obtain the velocity and microrotation components in Laplace transform domain. A standard numerical inversion procedure is used to find the velocity and microrotation in space-time domain for various values of time, distance, material parameters and pressure gradient. The variation of velocity and microrotation components is studied and the results are illustrated through graphs. It is observed that the micropolarity parameter has a decreasing effect on velocity component. It is also found that as the gyration parameter increases there is a decrease in microrotation component and an increase in velocity component.     

On the falling of objects in non-Newtonian fluids

The falling of a lamina in between two parallel plates containing a fluid of second grade is studied. The velocity of the lamina and the fluid are determined by solving the mixed initial—boundary value problem using Laplace transform. Explicit exact solutions are obtained for the velocity of the lamina and the fluid. Next, the falling of a cylinder in a tube containing a fluid of second grade is analyzed using Laplace transform, and once again exact solutions are found.

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Sunto Si studia la caduta di una lamina fra due piastre parallele contenenti un fluido di secondo grado. La velocità della lamina e del fluido sono determinate risolvendo un problema misto al contorno—a valori iniziali per mezzo della trasformata di Laplace—.Si studia poi la caduta di un cilindro in un tubo contenente un fluido di secondo grado utilizzando ancora la trasformata di Laplace e anche in questo caso si determina la soluzione esatta.

     

Exact solutions for the unsteady axial flow of Non-Newtonian fluids through a circular cylinder

The velocity field and the shear stress corresponding to the motion of a generalized Oldroyd-B fluid due to an infinite circular cylinder subject to a longitudinal time-dependent shear stress are established by means of the Laplace and finite Hankel transforms. The exact solutions, written under series form, can be easily specialized to give the similar solutions for generalized Maxwell and generalized second grade fluids as well as for ordinary Oldroyd-B, Maxwell, second grade and Newtonian fluids performing the same motion. Finally, some characteristics of the motion as well as the influence of the material parameters on the behavior of the fluid are shown by graphical illustrations.     

Peristaltic transport of a Newtonian fluid in a vertical asymmetric channel with heat transfer and porous medium

The problem of peristaltic flow of a Newtonian fluid with heat transfer in a vertical asymmetric channel through porous medium is studied under long-wavelength and low-Reynolds number assumptions. The flow is examined in a wave frame of reference moving with the velocity of the wave. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase. The analytical solution has been obtained in the form of temperature from which an axial velocity, stream function and pressure gradient have been derived. The effects of permeability parameter, Grashof number, heat source/sink parameter, phase difference, varying channel width and wave amplitudes on the pressure gradient, velocity, pressure drop, the phenomenon of trapping and shear stress are discussed numerically and explained graphically.     

Effects of stenosis on arterial rheology through a mathematical model

The paper presents an analytical study of blood flow through a stenosed artery using a suitable mathematical model. The artery is modelled as an anisotropic viscoelastic cylindrical tube containing a non-Newtonian viscous incompressible fluid representing blood. The blood flow is assumed to be characterized by the Herschel–Bulkley model. The effect of the surrounding connective tissues on the motion of the arterial wall has been incorporated. Initially, the relevant solutions of the boundary value problem are obtained in the Laplace transform space, through the use of a suitable finite difference technique. Laplace inversion is carried out by employing suitable numerical techniques. Finally, the variations of the vascular wall displacements, the velocity distribution of the blood flow, the flux, the resistance to flow and the wall shear stress in the stenotic region are quantified through numerical computations and presented graphically.     

3D flow of a generalized Oldroyd-B fluid induced by a constant pressure gradient between two side walls perpendicular to a plate

This paper deals with the 3D flow of a generalized Oldroyd-B fluid due to a constant pressure gradient between two side walls perpendicular to a plate. The fractional calculus approach is used to establish the constitutive relationship of the non-Newtonian fluid model. Exact analytic solutions for the velocity and stress fields, in terms of the Fox H-function, are established by means of the finite Fourier sine transform and the Laplace transform. Solutions similar to those for ordinary Oldroyd-B fluid as well as those for Maxwell and second-grade fluids are also obtained as limiting cases of the results presented. Furthermore, 3D figures for velocity and shear stress fields are presented for the first time for certain values of the parameters, and the associated transport characteristics are analyzed and discussed.     

New exact solutions corresponding to the second problem of Stokes for second grade fluids

New exact solutions for the velocity field corresponding to the second problem of Stokes, for second grade fluids, have been established by the Laplace transform method. These solutions, presented as a sum of the steady-state and transient solutions, are in accordance with the previous solutions obtained by a different technique. The required time to reach the steady state is determined by graphical illustrations. This time decreases if the frequency of the velocity increases. The effects of the material parameters on the decay of the transients are also investigated by graphs.     

Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus

This paper deals with the unsteady axial Couette flow of fractional second grade fluid (FSGF) and fractional Maxwell fluid (FMF) between two infinitely long concentric circular cylinders. With the help of integral transforms (Laplace transform and Weber transform), generalized Mittag–Leffler function and H-Fox function, we get the analytical solutions of the models. Then we discuss the exact solutions and find some results which have been known as special cases of our solutions. Finally, we analyze the effects of the fractional derivative on the models by using the numerical results and find that the oscillation exists in the velocity field of FMF.     

A note on the unsteady flow of a generalized second-grade fluid through a circular cylinder subject to a time dependent shear stress

The unsteady flow of a generalized second-grade fluid through an infinite straight circular cylinder is considered. The flow of the fluid is due to the longitudinal time dependent shear stress that is prescribed on the boundary of the cylinder. The fractional calculus approach in the governing equation corresponding to a second-grade fluid is introduced. The velocity field and the resulting shear stress are obtained by means of the finite Hankel and Laplace transforms. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method is used. The corresponding solutions for ordinary second-grade and Newtonian fluids, performing the same motion, are obtained as limiting cases of our general solutions. Finally, the influence of the material constants and of the fractional parameter on the velocity and shear stress variations is underlined by graphical illustrations.     

Unsteady MHD Couette flow in a rotating system

The unsteady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system has been considered. An exact solution of the governing equations has been obtained by using a Laplace transform. Solutions for the velocity distributions as well as shear stresses have been obtained for small times as well as for large times. It is found that for large times the primary velocity decreases with increase in the rotation parameter K² while it increases with increase in the magnetic parameter M². It is also found that with increase in K², the secondary velocity v₁ decreases near the stationary plate while it increases near the moving plate. On the other hand, the secondary velocity decreases with increase in the magnetic parameter.     

Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels

Transient electro-osmotic flow of viscoelastic fluids in rectangular micro-channels is investigated. The general twofold series solution for the velocity distribution of electro-osmotic flow of viscoelastic fluids with generalized fractional Oldroyd-B constitutive model is obtained by using finite Fourier and Laplace transforms. Under three limiting cases, the generalized Oldroyd-B model simplifies to Newtonian model, fractional Maxwell model and generalized second grade model, where all the explicit exact solutions for velocity distribution are found through the discrete Laplace transform of the sequential fractional derivatives. These exact solutions may be able to predict the flow behavior of viscoelastic biological fluids in BioMEMS and Lab-on-a-chip devices and thus could benefit the design of these devices.     

Exact solutions of accelerated flows for a Burgers’ fluid II. The cases γ = λ2/4 and γ > λ2/4

The purpose of this study is to provide the exact analytic solutions of accelerated flows for a Burgers’ fluid when the relaxation times satisfy the conditions γ = λ²/4 and γ > λ²/4. The velocity field and the adequate tangential stress that is induced by the flow due to constantly accelerating plate and flow due to variable accelerating plate are determined by means of Laplace transform. All the solutions that have been obtained are presented in the form of simple or multiple integrals in terms of Bessel functions. A comparison between Burgers’ and Newtonian fluids for the velocity and the shear stress is also made through several graphs.     

Axial Couette flow of a generalized Oldroyd-B fluid due to a longitudinal time-dependent shear stress

Abstract

The velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized Oldroyd-B fluid in an infinite circular cylinder are determined by means of Hankel and Laplace transforms. The solutions that have been obtained, written in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The similar solutions for generalized Maxwell fluids as well as those for ordinary fluids are obtained as limiting cases of our general solutions.     

Pulsatile flow of an unsteady dusty fluid through rectangular channel

The geometry of laminar flow of an unsteady viscous fluid with uniform distribution of dust particles through a rectangular channel under the influence of pulsatile pressure gradient has been considered. Initially the fluid and dust particles are at rest. The analytical expressions for velocities of fluid and dust particles is obtained by solving the partial differential equations using variable separable method and Laplace transform technique. The skin friction at the boundary plates are also calculated. Finally the changes in the velocity profiles with s and n are shown graphically.     

Unsteady flow past a flat plate with suction

A solution is given for the transient response for laminar boundary layer flow past a flat plate to a step-function change in suction velocity. An arbitrary but constant suction velocity normal to the plate is allowed prior to step-change. Using the Laplace transform technique the solutions for the unsteady velocity profile and shear stress are obtained and are graphically sketched when the suction velocity doubles in the stepchange. The results show clear evidence of boundary-layer contraction when suction velocity is increased.     

Helical flows of a heated generalized Oldroyd-B fluid subject to a time-dependent shear stress in porous medium

This paper presents an analysis for helical flows of a heated generalized Oldroyd-B fluid subject to a time-dependent shear stress in porous medium, where the motion is due to the longitudinal time-dependent shear stress and the oscillating velocity in boundary. The exact solutions are established by using the sequential fractional derivatives Laplace transform coupled with finite Hankel transforms in terms of generalized G function. Moreover, the effects of various parameters (relaxation time, fractional parameter, permeability and porosity) on the flow and heat transfer are analyzed in detail by graphical illustrations.     

Generalized thermoelastic infinite medium with spherical cavity subjected to moving heat source

This paper deals with a problem of thermoelastic interactions in an isotropic unbounded medium with spherical cavity due to the presence of moving heat sources in the context of the linear theory of generalized thermoelasticity with one relaxation time. The governing equations are expressed in the Laplace transform domain and solved in that domain. The inversion of the Laplace transform is done numerically using the Riemann-sum approximation method. The numerical estimates of the displacement, temperature, stress, and strain are obtained for a hypothetical material. The results obtained are presented graphically to show the effect of the heat source velocity and the relaxation time parameters on displacement, temperature, stress, and strain.     

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.