Unidirectional flow governing equation

Equation (3-10), which we have derived from the Navier-Stokes equations, governs the unknown scalar velocity function for all unidirectional flows, i.e., for any flow of the form (3-1). However, instead of Cartesian coordinates (x, y, z), it is evident that we could have derived (3-10) by using any cylindrical coordinate system (q, 1/2, z) with the direction of motion coincident with the axial coordinate z. In this case,... 

We have shown that the Navier-Stokes and continuity equations reduce to a governing equation for u of the form (3-12) for any problem in which u can be expressed in the form (3-1). We will see that the pressure-gradient function G(t) can always be specified and is considered to be a known function of time. To solve a unidirectional flow problem, we must therefore solve Eq. (3-12), with Git ) specified, subject to boundary conditions and initial conditions on u. It is clear from the governing equation that u(q, q2, t) will be nonzero only if either G(l) is nonzero or the value of u is nonzero on one (or more) of the boundaries of the flow domain. 

We assume that the flow will still remain unidirectional, so that u is a function ofy only. However, we anticipate that the temperature in the fluid will no longer be a constant, but will also be a function of y because of viscous dissipation. Hence the viscosity // will also depend on y because it is a function of temperature, and the governing equation for the velocity will then be in the form... 

For all of the flows that can be classified as unidirectional, the analysis of Section A shows that the governing equations reduce to solving a single heat equation for the magnitude of the scalar velocity component in the flow direction. For heat transfer applications, mathematically analogous problems involve heat transport by pure conduction. As noted earlier, there are excellent comprehensive books devoted exclusively to the solution of this class of problems.4 Here, we consider a related problem, which is chosen because it addresses the physically important coupling of heat transfer in the presence of phase change and also because it is another ID problem that exhibits a self-similar solution. 

In the previous transient unidirectional flow problems that we have considered, the dimensionless governing equations and boundary/initial conditions were completely free of all dimensional parameters. In these problems, however, the only relevant time scale was the diffusion time, i2c/v. Here, in contrast, the flow is characterized by a second imposed time scale that is due to the oscillatory pressure gradient, and the form of the resulting flow is predicted by (3-354) to depend upon the ratio of the diffusion time R2/v to the imposed time scale, 1 / >. The dimensionless ratio of time scales, R can also be considered to be a dimensionless frequency for the flow, and in that context is sometimes called the Strouhal number. 

The rhythmic contractions of the heart produce a pressure distribution in the arterial tree that includes both a steady component, P, and a purely oscillatory component, Pqso as does the velocity field. In contrast, flow in the trachea and bronchi has no steady component, and thus is purely oscillatory. It is common practice to refer to these components of pressure and flow as steady and oscillatory, respectively, and to use the term pulsatile to refer to the superposition of the two. A very useful feature of these flows, when they occur in rigid tubes, is that the governing equation (Equation 7.10) is linear, since the flow field is unidirectional and independent of axial position. The steady and oscillatory components can therefore be decoupled from each other, and analyzed separately. This gives... 

Using the method of moments formulation proposed by Aris, it is possible to show that the solutal spreading in a unidirectional pressure-driven flow can be calculated by solving two 2-dimensional problems. First, the velocity field u(x, y) for the analyte molecules is determined under steady-state conditions based on the fluid flow in the system. For a straight channel whose axis is aligned along the z-direction, this quantity is governed by the 2-dimensional Poisson equation... 

---