Generalized Steady Axisymmetric Stagnation Flow

Deriving the axisymmetric stagnation-flow equations begins with the steady-state three-dimensional Navier-Stokes equations (Eqs. 3.58, 3.60, and 3.60), but considering flow only in the z-r plane. In general, there may be a circumferential velocity component ui, but there cannot be variations of any variable in the circumferential direction 0. The derivation depends on two principal conjectures. First, the velocity field is presumed to be described in terms of a streamfunction that has the separable form 

From the definition of the axisymmetric stream function (Section 3.1.2), it can be seen that 

Notice that radial density derivatives have dropped out because the density is assumed to be a function of z alone. The divergence of velocity also is reduced according to the assumptions, leaving 

By substituting the relationships above into the Navier-Stokes equations, eliminating the terms that involve radial derivatives of density or U, and simplifying what remains, the following ordinary differential equations emerge  

These equations are written to isolate the pressure-gradient terms on the left-hand side to emphasize the point that the right-hand sides are functions of z alone. If there is no circumferential velocity (i.e., w = 0), then it is apparent that the right-hand sides depend only on z. If there are circumferential velocities, then the further assumption is made that 

---

Consider the steady flow inside a cylindrical channel, which is described by the two-dimensional axisymmetric continuity and Navier-Stokes equations (as summarized in Section 3.12.2). Assume the Stokes hypothesis to relate the two viscosities, low-speed flow, a perfect gas, and no body forces. The boundary-layer derivation begins at the same starting point as with axisymmetric stagnation flow, Section 6.2. Assuming no circumferential velocity component, the following is a general statement of the Navier-Stokes equations ... 

---