This paper presents a computational technique that called Tau-collocation method for the developed solution of non-linear integro-differential equations which involves a population model. To do this, the nonlinear integro-differential equations are transformed into a system of linear algebraic equations in matrix form without interpolation of non-poly-nomial terms of equations. Then, using coll...

Usually the Stokes equations that govern a flow in a smooth thin domain (with thickness of order ε) are related to the Reynolds equation for the pressure psmooth. In this paper, we show that for a rough thin domain (with rugosities of order ε) the flow is governed by a modified Reynolds equation for a pressure prough. Moreover we find the relation prough = K psmooth where K is an explicit coeff...

Abstract In this paper, we focus on the two-dimensional linear Telegraph equation with some initial and boundary conditions. We transform the model of partial differential equation (PDE) into a system of first order, linear, ordinary differential equations (ODEs). Our method is based on finding a solution in the form of a polynomial in three variables Un(x, y, t) = ∑n i=0 ∑n j=0 ∑n k=0 U(i, j, ...

Abstract. Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse length scales for weak solutions of the three-dimensional, incompressible Navier-Stokes equations on a periodic box. The estimate for the smallest of these inverse scales coincides with the inverse Kolmogorov length, but thereafter the exponents of the Reynolds number rise rapidly for corresp...

Abstract. The asymptotic structure of laminar modulated travelling waves in two-dimensional high-Reynolds-number plane Poiseuille flow is investigated on the upper-energy branch. A finite set of independent slowly varying parameters are identified which parameterize the solution of the Navier–Stokes equations in this subset of the phase space. Our parameterization of the weakly stable modes des...

We study the asymptotic behavior of solutions of the evolution Stokes equation in a thin three-dimensional domain bounded by two moving surfaces in the limit as the distance between the surfaces approaches zero. Using only a priori estimates and compactness it is rigorously verified that the limit velocity field and pressure are governed by the time-dependent Reynolds equation.

We consider the Navier–Stokes equations for compressible isothermal flow in the steady two dimensional case and show the existence of a weak solution in the case of periodic and of mixed boundary conditions.

In this paper, we perform an input-output analysis of the linearized Navier-Stokes equations by investigating their spatio-temporal ’impulse response’. Our first motivation for this problem comes from experimental work, where transition to turbulence is observed to occur either naturally, with so-called ’turbulent spots’, or when these spots are intentionally induced in the flow by deliberate e...

The tradition in Navier-Stokes analysis of finding estimates in terms of the Grashof number Gr , whose character depends on the ratio of the forcing to the viscosity ν, means that it is difficult to make comparisons with other results expressed in terms of Reynolds number Re, whose character depends on the fluid response to the forcing. The first task of this paper is to apply the approach of D...