Method boundary layer theory

The thin concentration boundary layer approximation, Eq. (3-51), has also been solved for bubbles k = 0) using surface velocities from the Galerkin method (B3) and from boundary layer theory (El5, W8). The Galerkin method agrees with the numerical calculations only over a small range of Re (L7). Boundary layer theory yields... 

M7. Meksyn, D., New Methods in Laminar Boundary Layer Theory. Pergamon, New York, 1961. 

Boundary-layer theory has been applied to solve the heat-transfer problem in forced convection laminar flow along a heated plate. The method is described in detail in numerous textbooks (El, G5, S3). Some exact solutions and approximate solutions are also obtained (B2, S3). 

Laminar Boundary Layer Theory - The Integral Method... 

For large Reynolds numbers, the function K depends on the particular shape of the obstacle and can be calculated by standard methods of boundary layer theory (Schlichting, 1979, Chapter IX). Then, the equation of convective diffusion is, in the boundary layer approximation. 

But before the virtues of the results and the approach are extolled, the method must be described in detail. Let us therefore return to a systematic development of the ideas necessary to solve transport (heat or mass transfer) problems (and ultimately also fluid flow problems) in the strong-convection limit. To do this, we begin again with the already-familiar problem of heat transfer from a solid sphere in a uniform streaming flow at sufficiently low Reynolds number that the velocity field in the domain of interest can be approximated adequately by Stokes solution of the creeping-flow problem. In the present case we consider the limit Pe I. The resulting analysis will introduce us to the main ideas of thermal (or mass transfer) boundary-layer theory. 

A very readable account of higher order approximations in boundary-layer theory may be found in the textbook, M. Van Dyke, Perturbation Methods in Fluid Mechanics (annotated edition) (Parabolic Press, Stanford, CA, 1975) or in a review paper by the same author M. Van Dyke, Higher-order boundary-layer theory, Annu. Rev. Fluid. Mech. 1, 265-92 (1969). 

Problem 11-2. Specified Heat Flux. We have considered the development of thermal boundary-layer theory for a 2D body with a constant surface temperature 0q. We have also discussed a method to determine the surface temperature when the heat flux is specified. In this problem, we wish to solve for the leading-order approximation to the temperature distribution in the fluid for an arbitrary 2D body when the heat flux is specified as a constant. 

Boundary layer approximation. The Landau problem, which was described above, is an example of an exact solution of the Navier-Stokes equations. Schlichting [427] proposed another approach to the jet-source problem, which gives an approximate solution and is based on the boundary layer theory (see Section 1.7). The main idea of this method is to neglect the gradients of normal stresses in the equations of motion. In the cylindrical coordinates (71, ip, Z), with regard to the axial symmetry (Vv = 0) and in the absence of rotational motion in the flow (d/dip = 0), the system of boundary layer equations has the form... 

The flow in the wake behind a body moving in an unbounded liquid possesses all the properties of free jet flows and can be calculated by methods of the boundary layer theory [427]. Note that the wake behind a moving body is almost invariably turbulent even if the boundary layer on the body surface is laminar. This is due to the fact that there are points of inflection on all velocity profiles of the wake such velocity distributions are known to be particularly unstable. 

For Re > 1500, the corresponding hydrodynamic problems can be solved by methods of the boundary layer theory [427], However, because of the flow... 

The drop may conserve its spherical form until Re = 300 [94]. Since usually the boundary layer on a drop or a bubble is considerably thinner than on a solid sphere, one can use methods based on the boundary layer theory even for 50 < Re < 300. By using these methods, the following formula was obtained in [94] for the drag coefficient for Re 1 ... 

The integral boundary layer method for determining the transfer coefficients is explained in this section. The basic boundary layer theory has been summarized in many textbooks in chemical engineering, among them the books of Hines and Maddox [52], Incropera De Witt [53], Cussler [27], Middleman [87] and Bird et al. [15] are considered very informative. 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundary-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, "separation of variables (not the classical concept) and the use of "continuous transformation groups. The basic theory is available in Ames (1965). 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

A reciprocal proportionality exists between the square root of the characteristic flow rate, t/A, and the thickness of the effective hydrodynamic boundary layer, <5Hl- Moreover, f)HL depends on the diffusion coefficient D, characteristic length L, and kinematic viscosity v of the fluid. Based on Levich s convective diffusion theory the combination model ( Kombi-nations-Modell ) was derived to describe the dissolution of particles and solid formulations exposed to agitated systems [(10), Chapter 5.2]. In contrast to the rotating disc method, the combination model is intended to serve as an approximation describing the dissolution in hydrodynamic systems where the solid solvendum is not necessarily fixed but is likely to move within the dissolution medium. Introducing the term... 

In addition to the interphase potential difference V there exists another potential difference of fundamental importance in the theory of the electrical properties of colloids namely the electro-kinetic potential, of Freundlich. As we shall note in subsequent sections the electrokinetic potential is a calculated value based upon certain assumptions for the potential difference between the aqueous bulk phase and some apparently immobile part of the boundary layer at the interface. Thus represents a part of V but there is no method yet available for determining how far we must penetrate into the boundary layer before the potential has risen to the value of the electrokinetic potential whether in fact f represents part of, all or more than the diffuse boundary layer. It is clear from the above diagram that bears no relation to V, the former may be in fact either of the same or opposite sign, a conclusion experimentally verified by Freundlich and Rona. 

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