Numerical solutions boundary layer

Specification of initial conditions completes the statement of the problem. We emphasize that little can be done analytically (we will refer to what can be done later), and in general numerical solutions are required. The problems between the fluid dynamics and the surfacr tant concentration are coupled nonlinearly and we describe numerical methods and results later. In addition to the nonlinear nature of the problem, certain dimensionless groups are large and make the equations stiff, in the sense that either small time-steps are needed or boundary layer structures need to be resolved. For example, the Peclet numbers tend to be large in applications (see earlier) and solute boundary layers develop near the surface. Also, the bulk concentration constant k which appears in the mass balance equation (39) can be of order 10 - making that equation stiff. These issues are taken up in Section . 

Thus, a velocity boundary layer and a thermal boundary layer may develop simultaneously. If the physical properties of the fluid do not change significantly over the temperature range to which the fluid is subjected, the velocity boundary layer will not be affected by die heat transfer process. If physical properties are altered, there will be an interactive effect between the momentum and heat transfer processes, leading to a comparatively complex situation in which numerical methods of solution will be necessary. 

The temperature profiles along the x-axis at various times are shown in Figure 4. These values should be compared with the theoretical solution T - erfc [ (l-x)/(2jc t) ]. Some numerical oscillations are noted at the heated boundary at short times due to the inability of the rather coarse mesh and time Increment to capture the thermal boundary layer which forms there. However, this can easily be avoided if desired by using a finer mesh in that region, and also by stepping with shorter time increments initially. 

Fig. 3. Numerical values of A and a for the solution of turbulent flow boundary layer on a rotating hemisphere. The value of meridional angle, 9, is given in degrees. From [22].

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

Sections IX,A-C have been devoted to expressions for m(z), (z), and Kyy based on atmospheric boundary layer theory. Because of the rather complicated dependence of u and K onz, Eq. (9.36) must generally be solved numerically (see, for example, Nieuwstadt and van Ulden, 1978 van Ulden, 1978). However, if they can be found, analytical solutions are advantageous for studying the behavior of the predicted mean concentration. 

Analytic solutions for flow around and transfer from rigid and fluid spheres are effectively limited to Re < 1 as discussed in Chapter 3. Phenomena occurring at Reynolds numbers beyond this range are discussed in the present chapter. In the absence of analytic results, sources of information include experimental observations, numerical solutions, and boundary-layer approximations. At intermediate Reynolds numbers when flow is steady and axisym-metric, numerical solutions give more information than can be obtained experimentally. Once flow becomes unsteady, complete calculation of the flow field and of the resistance to heat and mass transfer is no longer feasible. Description is then based primarily on experimental results, with additional information from boundary layer theory. 

Here we consider three theoretical approaches. As for rigid spheres, numerical solutions of the complete Navier-Stokes and transfer equations provide useful quantitative and qualitative information at intermediate Reynolds numbers (typically Re < 300). More limited success has been achieved with approximate techniques based on Galerkin s method. Boundary layer solutions have also been devised for Re > 50. Numerical solutions give the most complete and... 

Attempts to obtain theoretical solutions for deformed bubbles and drops are limited, while no numerical solutions have been reported. A simplifying assumption adopted is that the bubble or drop is perfectly spheroidal. SalTman (SI) considered flow at the front of a spheroidal bubble in spiral or zig-zag motion. Results are in fair agreement with experiment. Harper (H4) tabulated energy dissipation values for potential flow past a true spheroid. Moore (Mil) applied a boundary layer approach to a spheroidal bubble analogous to that for spherical bubbles described in Chapter 5. The interface is again assumed to be completely free of contaminants. The drag is given by... 

The boundary layer thickness, 5, is the thickness, z, where u/U = 0.99. From the numerical solution, this occurs at= 4.85. Then, the thickness of the boundary layer, 5, is found from a rearrangement of equation (E4.4.2) and is provided for different distances in equation (E4.4.7) and Table E4.4.1. 

Based on the numerical solutions, prepare a fit of the boundary-layer thickness as a function of the strain rate, 8(a). 

The overall mass-transfer rates on both sides of the membrane can only be calculated when we know the convective velocity through the membrane layer. For this, Equation 14.2 should be solved. Its solution for constant parameters and for first-order and zero-order reaction have been given by Nagy [68]. The differential equation 14.26 with the boundary conditions (14.28a) to (14.28c) can only be solved numerically. The boundary condition (14.28c) can cause strong nonlinearity because of the space coordinate and/or concentration-dependent diffusion coefficient [40, 57, 58] and transverse convective velocity [11]. In the case of an enzyme membrane reactor, the radial convective velocity can often be neglected. Qin and Cabral [58] and Nagy and Hadik [57] discussed the concentration distribution in the lumen at different mass-transport parameters and at different Dm(c) functions in the case of nL = 0, that is, without transverse convective velocity (not discussed here in detail). 

If the mass transfer of a gaseous reaction partner into the liquid is accompanied by a chemical reaction, the following case can occur depending on the reaction rate and the mobility of the reaction partners The concentration of A is not only reduced to zero in the solution in addition, the reaction front shifts from the liquid bulk to the liquid-side boundary layer. As a result, the liquid-side boundary is apparently reduced and finally eliminated in a chemical way ( chemisorption ), see Fig. 86. This process increases the mass transfer coefficient by the enhancement factor E as compared to its numerical value for purely physical absorption. 

After the Burgers equation, the numerical analysis of the incompressible boundary layer equations for convection heat transfer are discussed. A few important numerical schemes are discussed. The classic solution for flow in a laminar boundary layer is then presented in the example. 

While the boundary layer equations that were derived in the preceding sections are much simpler than the general equations from which they were derived, they still form a complex set of simultaneous partial differential equations. Analytical solutions to this set of equations have been obtained in a few important cases. For the majority of flows, however, a numerical solution procedure must be adopted. Such solutions are readily obtained today using modest modem computing facilities. This was, however, not always so. For this reason, approximate solutions to the boundary layer equations have in the past been quite widely used. While such methods of solution are less important today, they are still used to some extent. One such approach will, therefore, be considered in the present text. 

NUMERICAL SOLUTION OF THE LAMINAR BOUNDARY LAYER EQUATIONS... 

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