Circumferential momentum

Since every term in the radial and circumferential momentum equations that involves velocity is zero, it follows that... 

The circumferential-momentum equation is a parabolic partial differential that requires solution for the radial dependence of the circumferential velocity. With the velocity profiles in hand, the radial-momentum equation can be used to determine the resulting radial pressure dependence. 

If there is a circumferential velocity component, the circumferential momentum equation follows from substitutions into Eq. 6.185,... 

The governing equations for the rotating disk must include a circumferential momentum equation, and the circumferential velocity becomes a dependent variable. Also the circumferential velocity contributes to the radial-momentum equation. As simplified by the general equations of Section 6.2, the nonreacting, constant-property equations are summarized... 

Inclusion of the circumferential-momentum equation demands two new boundary conditions on the scaled circumferential velocity W = w/r. At the rotating surface z = 0, W = 2, where Q is the rotation rate (rad/s). Usually the outer flow is considered to have no circumferential velocity, W = 0. In general, there can be a swirl component in the outer flow. However, as discussed in Section 6.7.3, an inlet-swirl velocity can destabilize the flow. For the pure rotating disk situation with a semi-infinite outer environment the pressure-curvature eigenvalue will vanish. However, the eigenvalue is retained in the analysis because it will be needed for the analysis of fixed-gap rotating-disk situations. 

The columns of cells below row 16 contain the values of the dependent variables at the node points. They will all be iterated until a final solution is achieved. The formula in each cell represents an appropriate form of the difference equations. Each column represents an equation. Column B represents the continuity equation, column C represents the radial momentum equation, column D represents the circumferential momentum equation, and column E represents the thermal energy equation. Column F represents the perfect-gas equation of state, from which the nondimensional density is evaluated. The difference equations involve interactions within a column and between columns. Within a column the finite-difference formulas involve the relationships with nearest-neighbor cells. For example, the temperature in some cell j depends on the temperatures in cells j — 1 and j + 1, that is, the cells one row above and one row below the target cell. Also, because the system is coupled, there is interaction with other columns. For example, the density, column F, appears in all other equations. The axial velocity, column B, also appears in all other equations. 

D18 - D47 In cell Cl9, enter the difference formula for the circumferential momentum equation... 

Consider a long cylindrical shell whose interior is filled with an incompressible fluid. If the fluid is initially at rest when the cylinder begins to rotate, a boundary layer develops as the momentum diffuses inward toward the center of the cylinder. The fluid s circumferential velocity vu comes to the cylinder-wall velocity immediately, owing to the no-slip condition. At very early time, however, the interior fluid will be only weakly affected by the rotation, with the influence increasing as the boundary layer diffuses inward. If the shell continues to rotate at a constant angular velocity, the fluid inside will eventually come to rotate as a solid body. 

The circumferential ((j>) momentum equation is a partial differential equation. Identify some of its basic properties. Is it elliptic, parabolic, or hyperbolic Is it linear of nonlinear What is its order ... 

Based on a differential cylindrical control volume, derive steady-state momentum balances for the axial and circumferential directions, i.e., the Navier-Stokes equations. 

Written in this form, these equations clearly show that the pressure-gradient terms are both functions of r alone as long as the circumferential velocity w is either zero or a function of r alone. The axial momentum equation has been divided by z, so the term... 

Whereas the circumferential variations of the local wall shear stress (i.e., the momentum flux) in itself are not of interest in the study of the BSR, the analogous variations in mass flux or surface concentration are indeed. In Ref. 15 a graph is presented of the local heat flux relative to the circumferential average, for the constant-temperature boundary condition, as a function of a and s/dp. These data are based on a semianalytical solution of the governing PDE, following the procedure described by Ref. 8 (see Section II.B.2). At a relative pitch of 1.2 the local flux at a = 0 is ca. 64% lower than the circumferential average at a relative pitch of 1.5 the flux at a = 0 is still ca. 20% lower than the circumferential average. In the case of a constant surface temperature, the local heat fluxes are directly proportional to the local Nusselt (or Sherwood) numbers. 

Fig. 33.7. In this figure the atomizing gas enters fi om the top while the liquid enters firom a circumferential slot. As both fluids reach the core opening, the liquid is pushed toward the nozzle exit by the gas pressure. At an arbitrary time (tj), the liquid flow is redirected by the gas pressure and a thin film is formed at the nozzle wall. The hquid partially blocks the gas flow, building a pressure. As the pressure builds to a critical value, a hquid chunk is removed. This process causes an oscillatory spray formation. The frequency of this oscillation depends on the liquid and gas flow rates. The frequency increases with increasing the velocity of the liquid or the gas. Two separate variables are important for the pulsation (a) shear stresses at the liquid/gas interface, and (b) fluid momentum.

In a planar turbine configuration, the flow expands radially through circumferential rows of blades. Stator blade rows are fixed and tend to deviate the flow in the tangential direction. The swirling flow then enters the next blade row that is attached to the disk, the rotor. The curved rotor blades turn the flow in the opposite direction, which imparts a reaction force on the rotor. With respect to the center of the disk, this force is a moment, or torque T, that sets the rotor in motion. This process of fluid-to-mechanical energy conversion is best described by the conservation of angular momentum applied to a control volume enclosing a blade row ... 

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