Steady two-dimensional flows

There is a connection between the Lagrangian representation based on advected particles and the Eulerian representation using concentration fields. As in the case of pure advection the solution of the advection-diffusion equation can be given in terms of trajectories of fluid elements. Equation (2.6) can be generalized for the diffusive case using the Feynman-Kac formula (see e.g. Durrett (1996)) as 

In the following we consider transport processes in a range of different types of flows of increasing complexity. In each case we first consider the purely advective transport that generates the trajectories of non-diffusive particles and then the combined effects of advection and diffusion on a continuous concentration field. 

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Duct flows, like steady two-dimensional flows, are poor mixers. This class of flows is defined by the velocity field... 

This method starts with knowledge of the governing equation. The governing equation for steady two-dimensional flow with no pressure gradient is... 

Mass transfer rates in steady two-dimensional flow normal to the axis of a long cylinder have been computed numerically over a range of Re (D3, M8, W6). 

Let s use the steady, two-dimensional flow in a thin channel or a narrow gap between solid objects as schematically represented in Fig. 5.11. The channel height or gap width... 

The continuity equation is the conservation of mass equation. It is derived by a mass balance of the fluid entering and exiting a volume element taken in the flow field. In Fig. 6.1, consider a differential volume element AxAyAz. For ease of understanding, we shall consider steady, two-dimensional flow with velocity components u(x,y) and v(x,y) in the x and y directions, respectively. 

We consider steady, two-dimensional flow with velocity components u and v. The corresponding acceleration in the x direction is... 

Equations (6.20) and (6.21) are the x- and y-momentum equations for the steady, two-dimensional flow of an incompressible fluid with constant properties. 

In Eqs. (6.20) and (6.21), the terms on the left-hand side are the inertia forces, the first term on the right-hand side is the body force, the second term is the pressure force, and the last term within brackets is the viscous forces acting on the fluid element. With known body forces Bx and By, the continuity equation (6.5) and the two momentum equations (6.20) and (6.21) are three independent equations for the solution of the three unknown quantities u, v and p for the steady, two-dimensional flow of an incompressible fluid. The solution of these equations are not... 

Many practical flows can be represented with sufficient accuracy by assuming the flow to be two-dimensional, i.e., by assuming that the axes can be so positioned that one of the velocity components, here always taken as w, is effectively zero. For this reason and because the study of two-dimensional flows forms the basis of the study of more complex three-dimensional flows, most of the analyses discussed in later chapters will be for two-dimensional flows. Setting w equal to zero gives the following as the equations governing the velocity and pressure fields in steady two-dimensional flow ... 

The wave field produced in the steady, two-dimensional flow of a reacting gas past a wavy wall has been treated in [63] and [64]. Lick [65] has obtained solutions to the nonlinear, steady, two-dimensional conservation equations governing the flow of a reacting gas mixture about a blunt body. Reviews of these and other studies may be found in [1], [2], and [66]-[71]. 

This is the conservation of mass relation in differential form, which is also known as the continuity equation or mass balance for steady two-dimensional flow of a fluid with constant density. 

The energy transfers by heat and mass flow associated with a differential control Volume in the thermal boundary layer in steady two-dimensional flow. 

C Fxpress continuity equation for steady two-dimensional flow with conslani properties, and explain what each terni represents,... 

C Foi steady two-dimensional flow, what are the boundary layer appto.xiniations ... 

C For steady two-dimensional flow over an isothermal flat plate in the, v-direction, express the boundary conditions for the velocity components u and v, and the temperature T at the plale surface and at the edge of the boundary layer. 

Figure 2.2 Streamlines of a meandering jet surrounded by recirculation zones showing typical structures in steady two-dimensional flows elliptic (E) and hyperbolic (H) stagnation points, flow regions with open and closed streamlines and separatrices that connect hyperbolic points. The flow is defined by the streamfunction ip(x, y) = Cy — tanh[(y — Acosx)/(Ly/1 + A2 sinx2)].

When considering large spatial and temporal scales the transport of a concentration field by advection and molecular diffusion can be be approximately described by a diffusion equation with an effective diffusion coefficient. The main question then is to find an expression for the effective diffusivity as a function of the flow parameters and molecular diffusivity. A range of this type of problems are discussed in the review by Majda and Kramer (1999). Here we consider two simple examples of this problem in the case of steady two-dimensional flows with open and closed streamlines, respectively. 

In steady two-dimensional flows the spatial structure of the streamlines imposes strong restrictions on the advection of fluid elements. 

Equation (4.34) is the angular-momentum equation for steady two-dimensional flow. It is analogous to Eq. (4.14), the momentum equation. It is assumed in deriving Eq. (4.34) that at any given radial distance r all the fluid is moving with the same velocity, so jSj = jSj = h Applications of Eq. (4.34) are given in Chaps. 8 and 9. 

Point values of the flow velocity and direction for steady, two-dimensional flow. 

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