Travelling wave solutions to Stokes’ problem for a fourth grade fluid

A non-linear partial differential equation modelling the flow of a fourth-grade fluid is derived. Both analytical and numerical travelling wave solutions admitted by the model equation are obtained.     

Stokes’ first problem for the rotating flow of a third grade fluid

This work is concerned with the unsteady rotating flow of the third grade fluid over a suddenly moving plate in its own plane. The non-linear problem governing the flow has been solved numerically. The influence of material parameter of third grade fluid and rotation upon the velocity has been discussed.     

Leray's problem for a viscous incompressible micropolar fluid

This work concerns the steady motion of a viscous incompressible micropolar fluid in unbounded domains having cylindrical outlets to infinity. We prove the existence of a solution that approaches prescribed parallel solutions along the outlets of the domain. We also study the uniqueness, the regularity and the asymptotic behavior of the solution.     

Stokes’ first problem for Sisko fluid over a porous wall

We investigate the time-dependent flow of an incompressible Sisko fluid over a wall with suction or blowing. The flow is caused by sudden motion of the wall in its own plane. The magnetodynamic nature of the fluid is taken into account by applying a variable magnetic field. The resulting nonlinear problem is solved by invoking a symmetry approach and numerical techniques. The essential features of the embedded key parameters are described. Particularly the significance of the rheological effects is studied.     

Exact solutions of Stokes’ first problem for heated generalized Burgers’ fluid in a porous half-space

This work is concerned with applying the fractional calculus approach to the fundamental Stokes’ first problem of a heated Burgers’ fluid in a porous half-space. Modified Darcy's law for a Burgers’ fluid with fractional model is introduced first time. By using the Fourier sine transform and the fractional Laplace transform, exact solutions of the velocity and temperature field are obtained. The solutions for a Navier–Stokes, second grade, Maxwell, Oldroyd-B or Burgers’ fluid appear as the limiting cases of the present analysis.     

On a problem of nonstationary two-dimensional motion of micropolar fluid when normal stress and tangential velocity are given on the boundary

The object of this paper is to investigate the solution of nonstationary motion of micropolar fluid in the half-plane when the normal stresses and tangential velocities are given on the boundary. The Laplace-Fourier transform technique is used to point out the solution by quadratures. Numerical results of the physical quantities such as tangential and normal velocities, pressure, microrotation, stresses and momentums are obtained and displayed graphically. The problem could be met in the study of the vibrations of a memberance or a plate contacting with the fluid.     

Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a local existence theorem

An initial-boundary value problem for 1-D flow of a compressible viscous heat-conducting micropolar fluid is considered; the fluid is assumed thermodynamically perfect and polytropic. The original problem is transformed into homogeneous one and studied the Faedo-Galerkin method. A local-in-time existence of generalized solution is proved.      

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.     

An exact solution of start-up flow for the fractional generalized Burgers’ fluid in a porous half-space

Modified Darcy’s law for fractional generalized Burgers’ fluid in a porous medium is introduced. The flow near a wall suddenly set in motion for a fractional generalized Burgers’ fluid in a porous half-space is investigated. The velocity of the flow is described by fractional partial differential equations. By using the Fourier sine transform and the fractional Laplace transform, an exact solution of the velocity distribution is obtained. Some previous and classical results can be recovered from our results, such as the velocity solutions of the Stokes’ first problem for viscous Newtonian, second grade, Maxwell, Oldroyd-B or Burgers’ fluids.     

Group theoretic method for unsteady free convection flow of a micropolar fluid along a vertical plate in a thermally stratified medium

The group theoretic approach is applied for solving the problem of unsteady natural convection flow of micropolar fluid along a vertical flat plate in a thermally stratified medium. The application of two-parameter transformation group reduces the number of independent variables in the governing system consisting of partial differential equations and a set of auxiliary conditions from three to only one independent variable, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. Numerical solution of the velocity, microrotation and heat transfer have been obtained. The possible forms of the ambient temperature variation with position and time are derived.     

Exponential stability for a one‐dimensional compressible viscous micropolar fluid

In this paper, we consider one‐dimensional compressible viscous and heat‐conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. The homogenous boundary conditions for velocity, microrotation, and temperature are introduced. This problem has a global solution with a priori estimates independent of time; with the help of this result, we first prove the exponential stability of solution in (H¹(0,1))⁴, and then we establish the global existence and exponential stability of solutions in (H²(0,1))⁴ under the suitable assumptions for initial data. The results in this paper improve those previously related results. Copyright © 2015 John Wiley & Sons, Ltd.     

Global attractor for heat convection problem in a micropolar fluid

The article is devoted to describe asymptotics in the heat convection problem for a micropolar fluid in two dimensions. We show the existence and the uniqueness of global in time solutions and then prove the existence of a global attractor for considered model. Next, the Hausdorff dimension of the global attractor is estimated. Copyright © 2006 John Wiley & Sons, Ltd.     

Regularization and error estimate of infinite‐time ruin probabilities for Cramer‐Lundberg model

In this article, we consider the problem of finding the ultimate ruin probability in the classical risk mode. Using Laplace transform inversion and Fourier transform, we obtain ultimate ruin probability of an insurance company. First, we show that this problem is ill‐posed in the sense of Hadamard. Then, we apply the Tikhonov and truncation methods for establishing the approximate function for the ultimate ruin probability. Furthermore, convergence of the method, together with some examples, will be given. Finally, we present a numerical example to show efficiency of the method.     

A Hausdorff‐like moment problem and the inversion of the Laplace transform

We consider the problem of finding u ∈ L ²(I ), I = (0, 1), satisfying ∫_I u (x )x dx = μ _k , where k = 0, 1, 2, …, (α _k ) is a sequence of distinct real numbers greater than –1/2, and μ = (μ _kl ) is a given bounded sequence of real numbers. This is an ill‐posed problem. We shall regularize the problem by finite moments and then, apply the result to reconstruct a function on (0, +∞) from a sequence of values of its Laplace transforms. Error estimates are given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Fourier method and an extrapolation technique for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative

In this article, we consider Stokes’ first problem for a heated generalized second grade fluid with fractional derivative (SFP-HGSGF). Implicit and explicit numerical approximation schemes for the SFP-HGSGF are presented. The stability and convergence of the numerical schemes are discussed using a Fourier method. In addition, the solvability of the implicit numerical approximation scheme is also analyzed. A Richardson extrapolation technique for improving the order of convergence of the implicit scheme is proposed. Finally, a numerical test is given. The numerical results demonstrate the good performance of our theoretical analysis.     

Rayleigh problem for a MHD Sisko fluid

The problem of unsteady unidirectional flow of an incompressible Sisko fluid bounded by a suddenly moved plate is studied. The fluid is magnetohydrodynamic (MHD) in the presence of a time-dependent magnetic field applied transversely to the flow. The non-linear flow problem arising from the laws of momentum, mass and suitable boundary and initial conditions is solved analytically using Lie symmetry method. The manner in which the various emerging parameters affect the structure of the velocity is delineated.     

On a first boundary value problem for hyperbolic equations in the plane

In the paper we study a boundary value problem for a hyperbolic equation with two independent variables; this problem is a generalization of the well-known Darboux problem. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 294–306, February, 1999.