Momentum equation, laminar boundary

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. 

The integral in the momentum equation may now be evaluated for the laminar boundary... 

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... 

With terms III-V in Eq. 4.7 deleted, Eqs. 4.5-4.7, together with the x and y momentum equations, constitute the simplified equations of motion appropriate to natural convection problems. For constant T . and 7U, the boundary conditions on these equations are 0 = 1 and u = v = w = 0 on the body and 0 = 0 far from the body. Steady-state laminar solutions to these equations are those that are obtained after setting the time partials (i.e., terms containing partial derivatives with respect to t ) in the equations equal to zero. Steady-state turbulent... 

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). 

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. 

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). 

These equations were solved utilizing a finite difference technique similar to that developed by Appeldoorn and co-workers (15,22,23). The major difference between this analysis and that presented by Appeldoorn is that in this study the inertial terms are included in the axial momentum equation since the Reynolds numbers of the flow fields considered in this study are significantly higher than those analyzed by Appeldoorn and co-workers. In conclusion, the pressure drop required for a given volumetric flow rate for laminar flow in a capillary at high-shear rates can be theoretically determined if the capillary characteristics and the appropriate physical properties of the fluid are known. As indicated in the above boundary conditions, two oases concerning the thermal conditions at the... 

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. 

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 ... 

Other boundary conditions for laminar flow are discussed in the next chapter. In this way, the velocity profile is expressed in terms of u and 5. If the wall shearing stress rw is then also related to these quantities, the momentum integral equation (2.173) will allow the variation of S with x to be found for any specified variation of the freestream velocity, u. ... 

Using these and other boundary conditions, some of which for laminar flow will be discussed in the next chapter, the temperature distribution is expressed as a function of St. If qw is then related to the wall thermal conditions and St, the energy integral can be solved, using the solution to the momentum integral equation, to give... 

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