Integral transform solution of developing laminar duct flow in Navier‐Stokes formulation

Abstract

The generalized integral transform technique is employed in the hybrid numerical-analytical solution of the Navier-Stokes equations in streamfunction-only formulation, which govern the incompressible laminar flow of a Newtonian fluid within a parallel plate channel. Owing to the analytic nature of this approach, the outflow boundary condition for an infinite duct is handled exactly, and the error involved in considering finite duct lengths is investigated. The present error-controlled solutions are used to inspect the relative accuracy of previously reported purely numerical schemes and to compare Navier-Stokes and boundary layer formulations for various combinations of inlet conditions and Reynolds number.

Numerical Model of Electrokinetic Flow for Capillary Electrophoresis

“…A typical CPU time for a converged solution is about 150 s. Figure 5 compares the center line velocities computed by various researchers. It is shown that the present solution agrees well with the solutions due to an integrated transform technique (24) and a finite element method (25). …”

Integral transform solution of internal flow problems based on Navier–Stokes equations and primitive variables formulation

“…Our concern at this point is to confirm that the present solution path under the primitive variables formulation and with the Poisson equation for the pressure field with adequate boundary conditions, results in a sufficiently fast convergence behaviour and reproduces previously obtained benchmark results [18]. Those reference results on the other hand, were provided by the GITT approach itself but under the streamfunction-only formulation, which brings up a fourth-order partial differential equation and the associated biharmonic-type eigenvalue problem [17].…”

“…The ODE system was numerically solved by making use of subroutine DBVPFD for boundary value problems, available in the IMSL library [31], with a relative error target of 10 −4 . Two versions of the code were prepared for comparison purposes, one involving truncation of the channel at a certain prescribed length, and the other with the exact transformation of the infinite channel as presented in Reference [18], that maps the longitudinal domain in the interval (0, 1). Numerical results were then obtained for the velocity components, pressure field and friction factor along the channel length for various values of the Reynolds number, and a selected set is thus reported below for illustration purposes.…”

“…The classical test problem of developing laminar flow is studied for the incompressible situation inside a smooth parallel-plates channel [18,28]. The fluid enters the channel under uniform and irrotational flow, and a fully-developed state is reached at the channel outlet.…”

“…Its application in the solution of the boundary layer equations for internal flows is well documented in different contributions [9][10][11][12][13][14][15][16]. In parallel, and also successfully, the integral transform approach was employed to solve the full Navier-Stokes equations, for both cavity and channel flows [17][18][19][20][21][22][23][24][25][26].…”