Exact solutions for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit

The exact solutions are obtained for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit. The fractional calculus in the constitutive relationship of a non-Newtonian fluid is introduced. We construct the solutions by means of Fourier transform and the discrete Laplace transform of the sequential derivatives and the double finite Fourier transform. The solutions for Newtonian fluid between two infinite parallel plates appear as limiting cases of our solutions.     

Analytic solution for flow of Sisko fluid through a porous medium

This paper presents the analytic solution for flow of a magnetohydrodynamic (MHD) Sisko fluid through a porous medium. The non-linear flow problem in a porous medium is formulated by introducing the modified Darcy’s law for Sisko fluid to discuss the flow in a porous medium. The analytic solutions are obtained using homotopy analysis method (HAM). The obtained analytic solutions are explicitly expressed by the recurrence relations and can give results for all the appropriate values of material parameters of the examined fluid. Moreover, the well-known solutions for a Newtonian fluid in non-porous and porous medium are the limiting cases of our solutions.     

MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium

Exact analytical solutions for magnetohydrodynamic (MHD) flows of an incompressible second grade fluid in a porous medium are developed. The modified Darcy's law for second grade fluid has been used in the flow modelling. The Hall effect is taken into account. The exact solutions for the unsteady flow induced by the time-dependent motion of a plane wall between two side walls perpendicular to the plane has been constructed by means of Fourier sine transforms. The similar solutions for a Newtonian fluid, performing the same motion, appear as limiting cases of the solutions obtained here. The influence of various parameters of interest on the velocity and shear stress at the bottom wall has been shown and discussed through several graphs. A comparison between a Newtonian and a second grade fluids is also made.     

The Rayleigh-Stokes-Problem for a fluid of Maxwellian type

The velocity fields corresponding to an incompressible fluid of Maxwellian type subjected to a linear flow on an infinite flat plate and within an infinite edge are determined by means of the Fourier sine transforms. They are in close proximity of those of a second grade fluid. The well known solutions for a Navier-Stokes fluid appear as a limiting case of our solutions.     

Self-similar solutions of the problem of formation of a hydraulic fracture in a porous medium

The problem of hydraulic fracture formation in a porous medium is investigated in the approximation of small fracture opening and inertialess incompressible Newtonian fluid fracture flow when the seepage through the fracture walls into the surrounding reservoir is asymptotically small or large. It is shown that the system of equations describing the propagation of the fracture has self-similar solutions of power-law or exponential form only. A family of self-similar solutions is constructed in order to determine the evolution of the fracture width and length, the fluid velocity in the fracture, and the length of fluid penetration into the porous medium when either the fluid flow rate or the pressure as a power-law or exponential function of time is specified at the fracture entrance. In the case of finite fluid penetration into the soil the system of equations has only a power-law self-similar solution, for example, when the fluid flow rate is specified at the fracture entrance as a quadratic function of time. The solutions of the self-similar equations are found numerically for one of the seepage regimes.     

Approximate Analytical Solutions for Flow of a Third-Grade Fluid Through a Parallel-Plate Channel Filled with a Porous Medium

The flow of a non-Newtonian fluid through a porous media in between two parallel plates at different temperatures is considered. The governing momentum equation of third-grade fluid with modified Darcy’s law and energy equation have been derived. Approximate analytical solutions of momentum and energy equations are obtained by using perturbation techniques. Constant viscosity, Reynold’s model viscosity, and Vogel’s model viscosity cases are treated separately. The criteria for validity of approximate solutions are derived. A numerical residual error analysis is performed for the solutions. Within the validity range, analytical and numerical solutions are in good agreement.     

Internal and inertial waves in a viscous rotating stratified fluid

The effects of viscosity on the propagation of a St. Andrew's cross wave which is generated by a simple-harmonic localized disturbance in a rotating stratified fluid are considered. A similarity solution of the linearised equations shows that the velocities decay and that the wave width increases away from the disturbance. Previous solutions in a stratified non-rotating fluid are recovered by letting the rotation tend to zero. The solutions are also valid in the limit of a homogeneous rotating fluid. Further solutions for waves in a realistic ocean and in an isothermal atmosphere on a rotating Earth are also included.     

Decay of a potential vortex and propagation of a heat wave in a second grade fluid

The propagation of a heat wave in an incompressible second grade fluid within the context of a potential vortex is studied. The solutions for the Newtonian fluid can be obtained from those for fluids of second grade as a limiting case.     

Eigenoscillations of a fluid in a canonical domain and functional difference equations

In this work we construct and discuss special solutions of a homogeneous problem for the Laplace equation in a domain with cone-shaped boundaries. The problem at hand is interpreted as that describing oscillatory linear wave movement of a fluid under gravity in such a domain. These solutions are found in terms of the Mellin transform and by means of the reduction to some new functional-difference equations solved in an explicit form (by quadrature). The behavior of the solutions at large distances is studied by use of the saddle point technique. The corresponding eigenoscillations of a fluid are then interpreted as generalized eigenfunctions of the continuous spectrum.     

Dual solutions of Casson fluid flows over a power-law stretching sheet

A steady boundary layer flow of a non-Newtonian Casson fluid over a power-law stretching sheet is investigated. A self-similar form of the governing equation is obtained, and numerical solutions are found for various values of the governing parameters. The solutions depend on the fluid material parameter. Dual solutions are obtained for some particular range of these parameters. The fluid velocity is found to decrease as the power-law stretching parameter β in the rheological Casson equation increases. At large values of β, the skin friction coefficient and the velocity profile across the boundary layer for the Casson fluid tend to those for the Newtonian fluid.     

Nonorthogonal stagnation-point flow over a lubricated surface

An attempt is made to study a steady two-dimensional flow of a viscous incompressible fluid incident at some angle onto a plate lubricated with a thin layer of a power-law fluid. Similar and nonsimilar solutions of the governing partial differential equations are obtained numerically by imposing the continuity of velocity and shear stress at the interface layer between the fluid and the lubricant. The Keller box method is applied to obtain the solutions. The limiting cases for full and no-slip conditions are compared.     

Unsteady flows with a constant total pressure,described by the equations of ideal magnetohydrodynamics

Exact solutions of the equations of ideal magnetohydrodynamics describing the class of unsteady flows of an electrically conducting fluid with a constant total pressure are constructed. The solutions are written in the Lagrange coordinate system; arbitrariness in its choice was used to parameterize magnetic field lines. The wide functional arbitrariness the solutions provide a significant variation in the picture of the described fluid motions. An example of unsteady flow of an ideal electrically conducting fluid in a cylindrical channel with fixed magnetic tubes is given.     

Magnetohydrodynamic transient flows of a non-Newtonian fluid

Some exact solutions of the time-dependent partial differential equations are discussed for flows of an Oldroyd-B fluid. The fluid is electrically conducting and incompressible. The flows are generated by the impulsive motion of a boundary or by application of a constant pressure gradient. The method of Laplace transform is applied to obtain exact solutions. It is observed from the analysis that the governing differential equation for steady flow in an Oldroyd-B fluid is identical to that of the viscous fluid. Several results of interest are obtained as special cases of the presented analysis.     

Scattering coefficients for a sphere in a visco-acoustic medium for arbitrary partial wave order

Analytical solutions are reported for the scattering coefficients of a solid elastic sphere suspended in a viscous fluid for arbitrary partial wave order. Expressions are derived for incident compressional and shear wave modes, taking into account the viscosity of the surrounding fluid and resultant wave mode conversion. The long compressional wavelength limit is employed to simplify the derivation, whereas no restriction is placed on the shear wavelength in the fluid compared to the particle dimension. The analytical approximations are compared with numerical results obtained from matrix inversion of the boundary equations and agree within the validity domain of the solutions.     

Stokes＇ first problem for a viscoelastic fluid with the generalized Oldroyd-B model

The flow near a wall suddenly set in motion for a viscoelastic fluid with the generalized Oldroyd-B model is studied. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox H-function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions. The project supported by the National Natural Science Foundation of China (10272067), the Doctoral Program Foundation of the Education Ministry of China (20030422046), the Natural Science Foundation of Shandong Province, China (Y2006A14) and the Research Foundation of Shandong University at Weihai. The English text was polished by Keren Wang.     

Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary

Based on a modified Darcy's law, Stokes’ first problem was investigated for a second grade fluid in a porous half-space with a heated flat plate. Exact solutions of the velocity and temperature fields were obtained using Fourier sine transforms. In contrast to the classical Stokes’ first problem, there is a steady-state solution for the second grade fluid in the porous half-space, which is a damping exponential function with respect to the distance from the flat plate. The well-known solutions for Newtonian fluids in non-porous or porous half-space appear in limiting cases of our solutions.     

Instability of steady shearing flows in a non-linear viscoelastic fluid

This paper proves the non-existence of global smooth solutions to an equation for a viscoelastic fluid shearing flow. The non-existence of smooth solutions is interpreted physically as the formation of a vortex sheet and an instability in the fluid motion.Dedicated to Clifford Truesdell on the occasion of his sixtieth birthday     

Exact solutions of a dipolar fluid flow

Exact solutions for three canonical flow problems of a dipolar fluid are obtained: (i) The flow of a dipolar fluid due to a suddenly accelerated plate, (ii) The flow generated by periodic oscillation of a plate, (iii) The flow due to plate oscillation in the presence of a transverse magnetic field. The solutions of some interesting flows caused by an arbitrary velocity of the plate and of certain special oscillations are also obtained.     

A note on sinusoidal motion of a viscoelastic non-Newtonian fluid

In this note, the exact solutions of velocity field and associated shear stress corresponding to the flow of second-grade fluid in a cylindrical pipe, subject to a sinusoidal shear stress, are determined by means of Laplace and finite Hankel transform. These solutions are written as sum of steady-state and transient solutions, and they satisfy governing equations and all imposed initial and boundary conditions. The corresponding solutions for the Newtonian fluid, performing the same motion, can be obtained from our general solutions. At the end of this note, the effects of different parameters are presented and discussed by showing flow profiles graphically.     

Natural oscillations of a rotating spherical fluid layer of variable depth

Small harmonic oscillations of the free surface of a thin fluid layer covering a rotating sphere are considered. The fluid is in the central field of sphere gravity and is exposed to the centrifugal and Coriolis forces. It is assumed that the fluid layer depth is independent of the longitude. In this formulation the problem is governed by a differential equation with singular coefficients that generalizes the Laplace tidal equation. The method of local separation of singularities is applied to integrate this equation. The solutions obtained are compared with the corresponding modes of the Laplace tidal equation, that is, the solutions of the problem for a fluid layer of constant depth.