Transient Stokes flow

In the absence of body force the equations of continuity and motion representing transient Stokes flow of a generalized Newtonian fluid in a two-dimensional Cartesian system are written, on the basis of Equations (1.1) and (1,4), as... 

Small Reynolds Number Flow, Re < 1. The slow viscous motion without interfacial mass transfer is described by the Hadamard (66)-Rybcynski (67) solution. For infinite liquid viscosity the result specializes to that of the Stokes flow over a rigid sphere. An approximate transient analysis to establish the internal motion has been performed (68), Some simplified heat and mass transfer analyses (69, 70) using the Hadamard-Rybcynski solution to describe the flow field also exist. These results are usually obtained through numerical integration since analytical solutions are usually difficult to obtain. 

Statement of the problem. Preliminary remarks. Let us consider the transient convective mass and heat transfer between a spherical drop of radius a and a translational Stokes flow where the resistance to the transfer exists only in the disperse phase. We assume that at the initial time t = 0 the concentration inside the drop is constant and equal to Co, whereas for t > 0 the concentration on the interface is maintained constant and equal to Cs. 

The transient electroosmotic flow field is solved from Navier-Stokes equations and the continuity equation... 

The first numerical study on the transient flow of a single liquid droplet impinging onto a flat surface, into a shallow or deep pool was performed by Harlow and Shannon)397 In their work, the full Navier-Stokes equations were solved numerically in cylindrical... 

This chapter illustrated how to use FEMLAB to solve the Navier-Stokes equations in a variety of situations. Some of these problems are classic, such as entry flow into a pipe and transient start-up of pipe flow. Most examples were for laminar flow in two dimensions, but one model was turbulent flow into a pipe, and another model was for a complicated three-dimensional geometry. To review, the chapter covered the following specific features ... 

Let us consider a semi-infinite fluid bounded by a by a rigid plane -oo < X < oo, Y > 0. Two exact solutions are known for the transient flow near a plate [427], These solutions correspond to rather simple flows governed by linear equations of motion. However, in these cases the Navier-Stokes equations are linearized because the nonlinear convective terms are identically zero (Vx dVx /dX = 0) rather than because we neglect these terms. The equation of motion has the form... 

Transient motion of a flat plate. One of these problems (known as Stokes first problem) describes the flow near an infinite flat plate instantaneously set in motion at a velocity Uq in the plate plane. In this case, the initial and boundary conditions for Eq. (1.9.1) are written as follows ... 

Solving the full Navier-Stokes equations in the channels requires a rigorous computational fluid dynamics (CFD) simulation. During transient operation, such as start-up and shut-down, the flow fields can have a significant effect on the concentration and temperature profiles in the system. Under normal operation, it may be desirable to assume fuUy developed laminar flow to reduce the computational time and quickly estimate flow parameters based on fluid dynamics correlations. 

Obtaining proper boundary conditions for reactive flows is very important, especially when employing recent methods for solving the transient Navier-Stokes equations (DNS andLES). For such applications, most numerical schemes need high precision. The restrictions imposed on the boundary conditions for transient simulations of high order are as follows [13] ... 

Hence the respective flow fields can be exactly calculated from the Navier Stokes equations, at least for low frequencies, i. e. laminar flows. This is not the case for linear oscillations at either low frequencies, i. e. the linear spring in gas system [3.1], or high frequencies, namely piezo-quartz systems bearing sorbent samples [5.10] or oscillating rod systems (SETARAM). Also at high frequencies the state of the sorbate phase may become transient , i. e. deviate considerably from a thermodynamic equilibrium state. 

Methods accounting for mixing are most easily illustrated for steady state or stationary reactor operation, as in Fig. 12.3-1. Because of its stochastic nature, turbulent flow is in fact only statistically iiormy. The random behavior of the variables results in rapid fluctuations of their values around mean or so-called Reynolds-averaged, "steady state" values. Nevertheless, turbulent flow is governed by deterministic equations, the Navier-Stokes equations, whose terms have been explained in Chapter 7 and in which a transient term is included to account for the fluctuations around the statistically steady state values. 

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