Reduced equations, boundary layers

The simpler analyses reduce the boundary layer equations to ordinary differential equations, with the distance normal to the surface as the independent variable. This results from the assumption of Couette flow, where changes of the dependent variables in the streamwise direction may be neglected. The continuity and momentum equations then become... 

For a detailed derivation, interested readers are referred to the pertinent literature (Boateng, 1998). As A/L 0 the equations reduce to boundary layer equations which, deleting the from the non-dimensional terms has the form... 

Some empirical equations to predict cyclone pressure drop have been proposed (165,166). One (166) rehably predicts pressure drop under clean air flow for a cyclone having the API model dimensions. Somewhat surprisingly, pressure drop decreases with increasing dust loading. One reasonable explanation for this phenomenon is that dust particles approaching the cyclone wall break up the boundary layer film (much like spoiler knobs on an airplane wing) and reduce drag forces. 

For contact angle <(> = 0 and 8 << R, Eq. (2-58) should be reduced to Plesset-Zwick s equation based on the thin boundary layer assumption thus, 4>e = tt/2. 

In turbulent flow, the edge effect due to the shape of the support rod is quite significant as shown in Fig. 6. The data obtained with a support rod of equal radius agree with the theoretical prediction of Eq. (52). The point of transition with this geometry occurs at Re = 40000. However, the use of a larger radius support rod arbitrarily introduces an outflowing radial stream at the equator. The radial stream reduces the stability of the boundary layer, and the transition from laminar to turbulent flow occurs earlier at Re = 15000. Thus, the turbulent mass transfer data with the larger radius support rod deviate considerably from the theoretical prediction of Eq. (52) a least square fit of the data results in a 0.092 Re0 67 dependence for... 

The thermodynamic approach does not make explicit the effects of concentration at the membrane. A good deal of the analysis of concentration polarisation given for ultrafiltration also applies to reverse osmosis. The control of the boundary layer is just as important. The main effects of concentration polarisation in this case are, however, a reduced value of solvent permeation rate as a result of an increased osmotic pressure at the membrane surface given in equation 8.37, and a decrease in solute rejection given in equation 8.38. In many applications it is usual to pretreat feeds in order to remove colloidal material before reverse osmosis. The components which must then be retained by reverse osmosis have higher diffusion coefficients than those encountered in ultrafiltration. Hence, the polarisation modulus given in equation 8.14 is lower, and the concentration of solutes at the membrane seldom results in the formation of a gel. For the case of turbulent flow the Dittus-Boelter correlation may be used, as was the case for ultrafiltration giving a polarisation modulus of ... 

Hiemenz (in 1911) first recognized that the relatively simple analysis for the inviscid flow approaching a stagnation plane could be extended to include a viscous boundary layer [429]. An essential feature of the Hiemenz analysis is that the inviscid flow is relatively unaffected by the viscous interactions near the surface. As far as the inviscid flow is concerned, the thin viscous boundary layer changes the apparent position of the surface. Other than that, the inviscid flow is essentially unperturbed. Thus knowledge of the inviscid-flow solution, which is quite simple, provides boundary conditions for the viscous boundary layer. The inviscid and viscous behavior can be knitted together in a way that reduces the Navier-Stokes equations to a system of ordinary differential equations. 

For channels that are narrow compared to their length (rs zs) and for Rer > 1, it is apparent from Eq. 7.15 that the only order-one term is the pressure gradient. Therefore we conclude that in the boundary-layer approximation the entire radial-momentum equation reduces to... 

Fig. C.l Exact solutions for the mathematical prototype for boundary-layer behavior. The solution shown is for varying values of the parameter e and for a = 0.4. Also shown is the solution to the reduced outer equation that does not satisfy the boundary condition at x=0.

For the case of two-dimensional wavy film flow, Levich (L9) has shown that Eqs. (4) and (5) reduce to the familiar form of the boundary layer equations ... 

Wilson (79, 80) pointed out that A is not the dimensionless thickness of the diffusion boundary layer scaled with D/Vg, as originally suggested by Burton et al. (74), except in the limit at which the velocity field in the layer is dominated by the bulk flow, that is, X >> 1. In this case, the analysis reduces to the one first presented by Levich (81), and the integral in equation 25 is approximated as follows ... 

The amount of effort required to solve the full governing equations can be considerably reduced if certain assumptions can be introduced that simplify these equations. The most commonly used assumptions are that the flow in a duct is fully developed or that the flow has a boundary layer character. 

The problem of determining the velocity profile in the boundary layer has, therefore, been reduced to that of solving the ordinary differential equation given in Eq. 

In the preceding sections, the solution for boundary layer flow over a flat plate wav obtained by reducing the governing set of partial differential equations to a pair of ordinary differential equations. This was possible because the velocity and temperature profiles were similar in the sense that at all values of x, (u u ) and (Tw - T)f(Tw - T > were functions of a single variable, 17, alone. Now, for flow over a flat plate, the freestream velocity, u, is independent of x. The present section is concerned with a discussion of whether there are any flow situations in which the freestream velocity, u 1, varies with Jr and for which similarity solutions can still be found [1],[10]. 

Similarity and integral equation methods for solving the boundary layer equations have been discussed in the previous sections. In the similarity method, it will be recalled, the governing partial differential equations are reduced to a set of ordinary differential equations by means of a suitable transformation. Such solutions can only be obtained for a very limited range of problems. The integral equation method can, basically, be applied to any flow situation. However, the approximations inherent in the method give rise to errors of uncertain magnitude. Many attempts have been made to reduce these errors but this can only be done at the expense of a considerable increase in complexity, and, therefore, in the computational effort required to obtain the solution. 

Neglecting transport due to convection, or the existence of concentration gradients (these are valid assumptions for the bulk of the solution, far from surface boundary layers where concentrations might vary), the equations describing current flow in an electrolyte containing cations and anions reduce to the familiar Ohm s law. Unidirectional current between two parallel plates can be described by (1)... 

Using the slug-flow model, show that the boundary-layer energy equation reduces to the same form as the transient-conduction equation for the semi-infinite solid of Sec. 4-3. Solve this equation and compare the solution with the integral analysis of Sec. 6-5. 

In either case, if the membrane completely rejects the solute and the concentration polarization is negligible, then the concentration across the boundary layer is constant and small. Equation (17.38) consequently reduces to... 

Ry i wj, and Rdp/dx - (dp/dx) reduces the axisymmetric equations to equations for planar boundary layers in the new variables. The indicated transformation concerning the pressure gradient means that the axisymmetric flow for a given pressure gradient corresponds to a two-dimensional flow under a different pressure gradient. If the subscript oo identifies external-flow variables just outside the boundary layer, then from equations (15), (16), and (17) we find that... 

To obtain the flux of reactant A through the stagnant boundary layer surrounding a catalyst particle, one solves Equation (6.2. ) with the appropriate boundary conditions. If the thickness of the boimdary layer S is small compared to the radius of curvature of the catalyst particle, then the problem can be solved in one dimension as depicted in Figure 6.2.1. In this case. Pick s Law reduces to ... 

It is interesting to note that for Pr - 1, this equation reduces to Eq. 6-49 when 6 is replaced by dfidr), which is equivalent to ulV (see Eq. 6-46). The boundary conditions for Q and dfldr) are also identical. Thus we conclude that (he velocity and thenrial boundary layers coincide, and the nondimensional velocity and temperature profiles (u/Pand 6) aic identical for steady, incompressible, laminar flow of a fluid with constant properties and Pr = 1 over an isothermal flat plate (Fig, 6-30), The value of the temperature gradient at the surface (y = 0 or T) = 0) infthis case is, from Table 3, dOldr) d fldr) = 0.332. 

---