Momentum equation, laminar boundary layer

Take into consideration two-dimensional, rectilinear, steady, incompressible, constant-property, laminar boundary layer flow in the x direction along a flat plate. Assume that viscous energy dissipation may be neglected. Write the continuity, momentum and energy equations. 

For a mesh with a constant rectangular grid, the incompressible laminar boundary layer equations include the momentum equation as... 

If the turbulent momentum equation is expressed in nondimensional form in the same way as was done in deriving the laminar boundary layer equations then the additional term becomes ... 

The boundary layer integral equations have been derived above without recourse to the partial differential equations for boundary layer flow. They can, however, be determined directly from these equations. Consider, for example, the laminar momentum equation (2.140). Integrating this equation across the boundary layer to some distance from the wall, i being greater than the boundary layer thickness, gives because du/dy is zero outside the boundary layer and because dp/dx is independent of y ... 

First consider the finite difference form of the momentum equation, i.e., Eq. (6.128). As with laminar boundary layer flow, the four nodal points shown in Fig. 6.9 are used in deriving the finite difference form of this equation. 

This is the momentum equation of the laminar boundary layer with constant properties. The equation may be solved exactly for many boundary conditions, and the reader is referred to the treatise by Schlichting ll] for details of the various methods employed in the solutions. In Appendix B we have included the classical method for obtaining an exact solution to Eq. (5-13) for laminar flow over a flat plate. For the development in this chapter we shall be satisfied with an approximate analysis which furnishes an easier solution without a loss in physical understanding of the processes involved. The approximate method is due to von Karman [2],... 

What is the momentum equation for the laminar boundary layer on a flat plate What assumptions are involved in the derivation of this equation ... 

We have already seen that the phenomenological laws governing heat, mass, and momentum transfer are similar. In Chap. 5 it was shown that the energy and momentum equations of a laminar boundary layer are similar, viz.. 

We wish to obtain a solution to the laminar-boundary-layer momentum and energy equations, assuming constant fluid properties and zero pressure gradient. We have ... 

In this section we derive the equation of motion that governs the natural convection flow in laminar boundary layer. The conservation of mass and energy equations derived in Chapter 6 for forced convection are also applicable for natural convection, but tlie momentum equation needs to be modified to incorporate buoyancy. 

In Chapter 5, we learned the foundations of convection. Integrating the governing equations for laminar boundary layers, we obtained expressions for the heat transfer associated with forced convection over a horizontal plate and natural convection about a vertical plate. We also found analytically, as well as by the analogy between heat and momentum, that the thermal and momentum characteristics of laminar flow over a flat plate are related by... 

Similar solutions for Prandtl numbers other than unity may be obtained from Eqs. 6.117 and 6.118 or their equivalent. A major simplification is the independence of the momentum equation (Eq. 6.117), from the energy equation (Eq. 6.118), which makes/independent of /. Also, the linear form of the energy equation in / permits handling arbitrary surface temperature distributions as in the case of the flat plate. (See the section on the two-dimensional laminar boundary layer.)... 

In a few limited situations mass-transfer coefficients can be deduced from theoretical principles. One very important case in which an analytical solution of the equations of momentum transfer, heat transfer, and mass transfer has been achieved is that for the laminar boundary layer on a flat plate in steady flow. 

Blasius steady-flow, laminar, flat-plate, boundary-layer solution is a numerical solution of his simplification of Prandtl s boundary-layer equations, which are a simplified, one-dimensional momentum balance and a mass balance. This type of solution is known in the boundary-layer literature as an exact solution. Exact solutions can be found for only a very limited number of cases. Therefore, approximate methods are available for making reasonable estimates of the behavior of laminar boundary layers (Prob. 11.8). 

Integral momentum balance for laminar boundary layer. Before we use Eq. (3.10-48) for the turbulent boundary layer,.this equation will be applied to the laminar boundary layer over a flat plate so that the results can be compared with the exact Blasius solution in Eqs. (3.10-6)-(3.10-12). 

First let us examine mass transport through this film under isothermal conditions by employing the continuity equations for mass (a mass balance) and for momentum (an energy balance). In this stagnant film, which can correspond to the laminar boundary layer that develops when a fluid passes over a flat surface, there is no motion of the fluid, hence the latter equation is irrelevant. The continuity equation for mass describes the spacial dependence of concentration in terms of the velocities parallel, u, and perpendicular, V, to the surface ... 

In laminar flow with low mass-transfer rates and constant physical properties past a solid surface, as for the two-dimensional laminar boundary layer of Fig. 3.10, the momentum balance or equation of motion (Navier-Stokes equation) for the X direction becomes [7]... 

The quasilaminar sublayer resistance / b describes the excess resistance for the transfer of matter from the atmosphere to the surfaces of the vegetation, that is, the difference between the resistance for matter and the resistance for momentum. It is primarily associated with molecular diffusion through quasi laminar boundary layers. Several parameterizations for Rb have been developed, but that employed by Brook et al. (1999), which like Equations 7.3 and 7.6 is valid for conditions of neutral atmospheric stability, is particularly easy to apply ... 

The continuity equations for mass, x-direction momentum, chemical species and energy in the plane, stationary, laminar boundary layer flow have already been given as Eqs. (7.1) to (7.4). The stream function ij/, by means of which the mass continuity equation is automatically satisfied, is defined by Eqs. (7.5). Following the approaches of Lees (1956), Fay and Riddell (1958), and Chung (1965), self-similar solutions in the stagnation region are obtained via transformations from (x, y) co-ordinates to the two new variables... 

Equation 11.12 does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer, shown in Figure 11.1, is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure 11.7). The approach is based on the assumption that the shear stress at a plane surface can be calculated from the simple power law developed by Blasius, already referred to in Chapter 3. 

Obtain the momentum equation for an element of boundary layer. If the velocity profile in the laminar region may be represented approximately by a sine function, calculate the boundary-layer thickness in terms of distance from the leading edge of the surface. 

This is termed the boundary layer momentum integral equation. As previously mentioned, it is equally applicable to laminar and turbulent flow. In laminar flow, u is the actual steady velocity while in turbulent flow it is the time averaged value. 

The way in which the momentum integral equation is applied will be discussed in detail in the next chapter. Basically, it involves assuming the form of the velocity profile, i.e., of the variation of u with y in the boundary layer. For example, in laminar flow a polynomial variation is often assumed. The unknown coefficients in this assumed form are obtained by applying the known condition on velocity at the inner and outer edges of the boundary layer. For example, the velocity must be zero at the wall while at the outer edge of the boundary layer it must become equal to the freestream velocity, u. Thus, two conditions that the assumed velocity profile must satisfy are ... 

Next we apply three fundamental laws to this fluid element Conservation of mass, conservation of momentum, and conservation of energy to obtain the continuity, momentum, and energy equations for laminar flow in boundary layers. 

In laminar flow, heat transfer occurs only by conduction, as there are no eddies to carry heat by convection across an isothermal surface. The problem is amenable to mathematical analysis based on the partial differential equations for continuity, momentum, and energy. Such treatments are beyond the scope of this book and are given in standard treatises on heat transfer, Mathematical solutions depend on the boundary conditions established to define the conditions of fluid flow and heat transfer. When the fluid approaches the heating surface, it may have an already completed hydrodynamic boundary layer or a partially developed one. Or the fluid may approach the heating surface at a uniform velocity, and both boimdary layers may be initiated at the same time. A simple flow situation where the velocity is assumed constant in all cross sections and tube lengths is called... 

Laminar Free Convection. Sparrow and Gregg [33] were the first to use the boundary layer method to study laminar, gravity-driven film condensation on a vertical plate. They improved upon the approximate analysis of Nusselt by including fluid acceleration and energy convection terms in the momentum and energy equations, respectively. Their numerical results can be expressed as ... 

Formulate the equations of continuity, momentum, energy, and continuity of component A for the corresponding boundary layers over a flat plate in laminar flow. 

When a nondeformable object is implanted in the flow field and the streamlines and equipotentials are distorted, the nature of the interface does not affect the potential flow velocity profiles. However, the results should not be used with confidence near high-shear no-slip solid-liquid interfaces because the theory neglects viscous shear stress and predicts no hydrodynamic drag force. In the absence of accurate momentum boundary layer solutions adjacent to gas-liquid interfaces, potential flow results provide a reasonable estimate for liquid-phase velocity profiles in Ihe laminar flow regime. Hence, potential flow around gas bubbles has some validity, even though an exact treatment of gas-Uquid interfaces reveals that normal viscous stress is important (i.e., see equation 8-190). Unfortunately, there are no naturally occurring zero-shear perfect-slip interfaces with cylindrical symmetry. 

The tangential component of the dimensionless equation of motion is written explicitly for steady-state two-dimensional flow in rectangular coordinates. This locally flat description is valid for laminar flow around a solid sphere because it is only necessary to consider momentum transport within a thin mass transfer boundary layer at sufficiently large Schmidt numbers. The polar velocity component Vo is written as Vx parallel to the solid-liquid interface, and the x direction accounts for arc length (i.e., x = R9). The radial velocity component Vr is written... 

One concludes from (12-17a) and (12-17c) that neither 4> nor Vp is a function of the Reynolds number because Re does not appear in either equation. Consequently, dynamic pressure and its gradient in the x direction are not functions of the Reynolds number because Re does not appear in the dimensionless potential flow equation of motion, given by (12-16), from which /dx is calculated. In summary, two-dimensional momentum boundary layer problems in the laminar flow regime (1) focus on the component of the equation of motion in the primary flow direction, (2) use the equation of continuity to calculate the other velocity component transverse to the primary flow direction, (3) use potential flow theory far from a fluid-solid interface to calculate the important component of the dynamic pressure gradient, and (4) impose this pressme gradient across the momentum boundary layer. The following set of dimensionless equations must be solved for Vp, IP, u, and v in sequential order. The first three equations below are solved separately, but the last two equations are coupled ... 

This comparison focuses on the comer regions in square ducts that are nonexistent in tubes. In both configurations, the momentum boundary layer thickness is substantial (i.e., Effective/2) for fully developed laminar flow. The no-slip boundary condition for viscous flow near the walls increases the mass transfer boundary layer thickness and reduces the flux of reactants toward the catalytic surface relative to plug flow. This effect is significant in the comer regions of the channel with square cross section. Since the entire active surface in heterogeneous tubular reactors is equally accessible to reactants, one predicts larger conversion in tubes via equation (23-71) ... 

As mentioned previously, even when the flow becomes turbulent in the boundary layer, there exists a thin sub-layer close to the surface in which the flow is laminar. This layer and the fully turbulent regions are separated by a buffer layer, as shown schematically in Figure 7.1. In the simplified treatments of flow within the turbulent boundary layer, however, the existence of the buffer layer is neglected. In the laminar sub-layer, momentum transfer occurs by molecular means, whereas in the turbulent region eddy transport dominates. In principle, the methods of calculating the local values of the boundary layer thickness and shear stress acting on an immersed surface are similar to those used above for laminar flow. However, the main difficulty stems from the fact that the viscosity models, such as equations (7.13) or (7.27),... 

Recently, Hanratty presented a comprehensive review of the attempts to account for the interfacial waviness in modelling the interfacial shear stress for the stability analysis of gas-liquid two-phase flows [53]. Basically, the approach taken was to implement the models obtained for the surface stresses in air flow over a solid wavy boundary as a boundary condition for the momentum equation of the liquid layer over its it mobile wavy interface. Craik [98] adopted the interfacial stresses components which evolve from the quasi-laminar model by Benjamin [84]. Jurman and McCready [99], Jurman et al. [100], and Asali and Hanratty [101] used correlated experimental values of shear stress components (phase and amplitude) based on turbulent models which consider relaxation effects in the Van Driest mixing length. Since the characteristics of the predicted surface stresses are dependent on the wave number, Asali and Hanratty picked the phase and amplitude values which correspond to the wave lengths of the capillary ripples observed in their experiments of thin liquid layers sheared by high gas velocities [101]. It was shown that the growth of these ripples is controlled by the interfacial shear stress component in phase with the wave slope. 

In order to derive the basic equation for a laminar or turbulent boundary layer, a small control volume in the boundary layer on a flat plate is used as shown in Fig. 3.10-5. The depth in the z direction is b. Flow is only through the surfacesand dj and also from the top curved surface at 8. An overall integral momentum balance using Eq. (2.8-8) and overall integral mass balance using Eq. (2.6-6) are applied to the control volume inside the boundary layer at steady state and the final integral expression by von Karman is (B2, S3)... 

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