The internal motion through porous chambers generated by wall-normal injection has received considerable attention in the second half of the twentieth century. This may be attributed to its relevance to a large number of phenomenological applications. In actuality, the motion of fluids driven by either wall injection or suction can be used to describe a variety of practical problems that encompass a wide range of industries and research areas. To name a few, these include: paper manufacturing (Taylor, 1956), ablation or sweat cooling (Peng & Yuan, 1965; Yuan & Finkelstein, 1958), boundary layer control (Acrivos, 1962; Libby, 1962; Libby & Pierucci, 1964), peristaltic pumping (Fung & Yih, 1968; Uchida & Aoki, 1977), gaseous diffusion or filtration, isotope separation (Berman, 1953; 1958a;b), irrigation, and the mean flow modeling of both solid (Culick, 1966; Zhou & Majdalani, 2002) and hybrid rockets (Majdalani, 2007a). Wall injected flows are initiated by the injection or suction of a fluid across the boundaries of a ducted region having an arbitrary shape and cross-sectional area. This is illustrated in Figure 1 for the special cases of porous channels and tubes. In general, one is required to solve a reduced-order form of the equations of motion for a bounded fluid in order to retrieve a meaningful solution (Terrill & Thomas, 1969). For a general three dimensional setting, this effort leads to a formidable task that is often intractable. However, when simplifying assumptions are invoked, as in the case of an incompressible stream in a channel or tube with uniform injection or suction, Berman (1953) has shown that the Navier-Stokes equations can be reduced to a fourth order nonlinear ODE that may be susceptible to both analytical and numerical treatment. Berman’s approach is based on a spatial similarity that transforms the Navier-Stokes equations to a more manageable ODE by assuming that the transverse velocity component v is axially invariant; this immediately translates into a streamfunction that varies linearly in the streamwise direction, i.e. ψ(x, y) = xF(y) (Berman, 1953; White, 2005). Then by considering the limiting case of a small suction Reynolds number, Re ∼ e, Berman employs a regular perturbation series in Re to obtain an approximate expansion for the mean flow function F(y). Berman’s Reynolds number, Re = Uwa/ν , is based on the injection speed at the wall, Uw, and the channel half height, a. As for the case of large suction, Berman (1953) first remarks that the limit of the reduced ODE cannot be used to obtain a solution owing to the reduction in order of the governing equation. Later, Sellars (1955) and Terrill (1964) invoke a procedure that permits the extraction of a closed-form analytical approximation for the large Re case by implementing a coordinate transformation that takes into account the spatial relocation of the boundary layer to the sidewall region. Internal Flows Driven by Wall-Normal Injection
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