On the generalized Navier–Stokes equations

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Abstract

In this paper, we present a general model of the classical Navier–Stokes equations. With the help of Laplace, Fourier Sine transforms, finite Fourier Sine transforms, and finite Hankel transforms. An exact solutions for three different special cases have been obtained.

Introduction

The Navier–Stokes and continuity equations given by:vt+(v·)v=−1ρp+ν2v,·v=0,where t is the time, v is the velocity vector, p is the pressure, ν is the kinematics viscosity and ρ is the density. The model based on fractional derivatives has been shown to be one of the most effective approaches [1], [4], [6], [7]. If the fractional derivative model is used to represent the time derivative term, the equation of motion (1) assumes the form:Dtαv+(v·)v=−1ρp+ν2v.The fractional operator Dtα based on Caputo’s definition is defined as [1]:Dtαf(t)=1Γ(n−α)0tf(n)(x)(t−x)α−n+1dx,n−1<α<n,where f(n) represents the nth ordinary derivative of f(t) with respect to t.

Since the Navier–Stokes’ equations are nonlinear in character, there is no known general method to solve these equations. There are very few cases in which the exact analytic solution of the Navier–Stokes equations can be obtained, where have to make certain assumptions about the state of the fluid and a simple configuration for the flow pattern is to be considered. In the present paper we shall concentrate ourselves only on those configurations of fluid motion in which, besides time as one of the independent variable, the velocity field is a function of only one space coordinate.

Section snippets

Required integral transforms and special functions

If f(r) satisfies Dirichlet conditions in closed interval (0,1) and if its finite Hankel transform [5, p. 82] is defined to be:f̄n)=∫0Rrf(r)J0(rλn)dr,where λn are the roots of the equation J0(r)=0. Then at each point of the interval at which f(r) is continuous:f(r)=2R2n=1f̄n)J0nr)J12nR),where the sum is taken over all positive roots of J0(r)=0, J0 and J1 are Bessel functions of first kind.

In application of the finite Hankel transforms to physical problems, it is useful to have the

Starting flow in a pipe

Let us consider a long circular pipe in which the fluid is initially at rest. A constant pressure gradient along the axis of the pipe is suddenly imposed, which will set the fluid in motion. Let the axis of the pipe be taken as z-axis along which the flow takes place and r denotes the radial direction outward from the z-axis. We consider the flow as axially symmetric and fully developed. The equation governing the fluid motion in the present case is [2, p. 116]:ρut=−pz2ur2+1rur,where ρ

Flow due to a plane wall suddenly set in motion

The simplest unsteady flow is that which results due to the impulsive motion of a flat plate in its own plane in an infinite mass of fluid which is otherwise at rest. This flow was first studied by Stokes and in the literature it is generally known as Stokes’ first problem [2, p. 109].

Let x-axis be taken in the direction of motion of the wall, which is suddenly accelerated from rest and moves with a constant velocity U0. Let y-axis be perpendicular to the plane of the wall. Then Eq. (3) takes

Starting flow in plane Coutte motion

If the fluid is not infinite but bounded another stationary parallel wall at distance h, then Eq. (3) takes the formαutα2uy2,the initial and boundary conditions are:u(y,0)=0,u(y,t)=U0,aty=0,u(y,t)=0,aty=h.The solution of Eq. (33) is obtained by the consecutive use of the Laplace and finite Fourier Sine transformsu(y,t)=2νU0htαn=1Eα,α+1−nπ2h2tαsinnπyhwhen α→1, using Eq. (22) we obtainu(y,t)=2νhU0π2n=11n1−e−nπ2n2tsinnπyh.

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