Stagnation flow semi-infinite domain

The steady-state stagnation-flow equations represent a boundary-value problem. The momentum, energy, and species equations are second order while the continuity equation is first order. Although the details of boundary-condition specification depend in the particular problem, there are some common characteristics. The second-order equations demand some independent information about V,W,T and Yk at both ends of the z domain. The first-order continuity equation requires information about u on one boundary. As developed in the following sections, we consider both finite and semi-infinite domains. In the case of a semi-infinite domain, the pressure term kr can be determined from an outer potential flow. In the case of a finite domain where u is known on both boundaries, Ar is determined as an eigenvalue of the problem. 

Fig. 6. 5 A stencil that illustrates the finite-difference discretization of the semi-infinite-domain axisymmetric stagnation flow problem.

Fig. 6.6 Comparison of two alternative stagnation-flow configurations. The upper illustration shows the streamlines that result from a semi-infinite potential flow and the lower illustration shows streamlines that result from a uniform inlet velocity issuing through a manifold that is parallel to the stagnation plane. Both cases are for isothermal air flow at atmospheric pressure and T = 300 K. In both cases the axial inlet velocity is u = —5 cm/s. The separation between the manifold and the substrate is 3 cm. For the outer-potential-flow case, the streamlines are plotted over the same domain, but the flow itself varies in the entire half plane above the stagnation surface. The stagnation plane is illustrated as a 10 cm radius, but the solutions are for an infinite radius.

In the foregoing sections the discussion of axisymmetric stagnation flows has concerned four subcases of the same general problem—semi-infinite or finite domains and rotation or no rotation. The intent of this section is to focus attention on the fact that with suitable choices of length and velocity scales these problems can be collapsed to a common representation. Generally speaking, the length scale is called L and the velocity scale is called U. Thus nondimensional variables are defined as... 

The pressure-gradient term Ar requires either a further boundary condition or a determination of the domain size. In the semi-infinite cases, Ar is a constant that is specified in terms of the outer potential-flow characteristics. However, the extent of the domain end must be determined in such a way that the viscous boundary layer is entirely contained within the domain. In the finite-gap cases, the inlet velocity n(zend) is specified at a specified inlet position. Since the continuity equation is first order, another degree of freedom must be introduced to accommodate the two boundary conditions on u, namely u — 0 at the stagnation surface and u specified at the inlet manifold. The value of the constant Ar is taken as a variable (an eigenvalue) that must be determined in such a way that the two velocity boundary conditions for u are satisfied. 

The discussion in this chapter has been dominated by axisymmetric flow. However, there is analogous behavior for planar stagnation flow in two-dimensional cartesian coordinates. In fact Hiemenz s original work was for planar stagnation flow in a semi-infinite region. The planar flow illustrated in Fig. 6.18 is for a finite domain. 

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