Momentum integrals

As will be shown, the steady solution for the detonation velocity does not involve any knowledge of the structure of the wave. The Hugoniot plot discussed in Chapter 4 established that detonation is a large Mach number phenomenon. It is apparent, then, that the integrated momentum equation is included in obtaining a solution for the detonation velocity. However, it was also noted that there are four integrated conservation and state equations and five unknowns. Thus, other... 

Dimensionless integration momentum in (3.33) was measured in electron mass. We return here to dimensionful integration momenta, which results in an extra factor in the numerators in (6.10), (6.11) and (6.12) in comparison with the factor in the skeleton integral (3.33). Notice also the minus sign before the momentum in the arguments of form factors it arises because in the equations below k = fc. ... 

The entrainment flow rate is from Sakiadis, who used an integral momentum approximation rather than the exact similarity solution. 

As discussed in Chapters 2 and 3, in the integral method it is assumed that the boundary layer has a definite thickness and the overall or integrated momentum and thermal energy balances across the boundary layer are considered. In the case of flow over a body in a porous medium, if the Darcy assumptions are used, there is, as discussed before, no velocity boundary layer, the velocity parallel to the surface near the surface being essentially equal to the surface velocity given by the potential flow solution. For flow over a body in a porous medium, therefore, only the energy integral equation need be considered. This equation was shown in Chapter 2 to be ... 

Fig. 5-5 Elemental control volume for integral momentum analysis of laminar boundary layer.

This is the integral momentum equation of the boundary layer. If the pressure is constant throughout the flow,... 

To determine the turbulent-boundary-layer thickness we employ Eq. (5-17) for the integral momentum relation and evaluate the wall shear stress from the empirical relations for skin friction presented previously. According to Eq. (5-52),... 

Now, using the integral momentum equation for zero pressure gradient [Eq. (5-17)] along with the velocity profile and wall shear stress, we obtain... 

For the free-convection system, the integral momentum equation becomes... 

The equivalent integral momentum equation for zero pressure gradient is... 

One can similarly define average integral, momentum-transfer and total cross sections. 

Problem 2-12. From Cauchy s equation of motion for the steady flow of an incompressible fluid in the absence of body forces, derive the integral momentum balances for the hydrodynamic force F and torque L on an arbitrary body Sb immersed in the fluid ... 

Expressions based on Ree are found through the use of the integral momentum equation for a flat plate,... 

This expression, (equation (7.7)), known as the integral momentum equation, is vahd for both laminar and turbulent flow, and no assmnption has also been made about the nature of the fluid or its behaviour. In order to integrate equation (7.7), the relation between Vx and y must be known. 

However, to take account of the yield stress acting at the edge of the boundary layer, the integral momentum balance equation (7.7) must be modified as ... 

We seek information about characteristics of the traveling waves that develop when the interface is deformed. Because of the initial velocity, a direct stability analysis based on the differential equations of change is more complex mathematically than the analyses presented above, although it has been carried out (Benjamin, 1957 Krantz and Goren, 1970 Yih, 1963). We present here a simpler but shghtly less accurate analysis based on the integral momentum equation. It follows for the most part the procedure of Kapitsa (1948) which was described by Levich (1962). 

With this expression and with Equation 5.140 for and its derivatives, the integral momentum equation becomes, to first order in perturbation quantities. 

Starting with Newton s second law we will develop the integral momentum-balance equation for linear momentum. Angular momentum will not be considered here. Newton s law may be stated The time rate of change of momentum of a system is equal to the summation of all forces acting on the system and takes place in the direction of the net force. 

H Integral Momentum Balance for Boundary-Layer Analysis... 

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)... 

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). 

Integral momentum analysis for turbulent boundary layer. The procedures used for the integral momentum analysis for laminar boundary layer can be applied to the turbulent boundary layer on a flat plate. A simple empirical velocity distribution for pipe flow which is valid up to a Reynolds number of 10 can be adapted for the boundary... 

In a similar fashion, the integral momentum analysis method used for the turbulent hydrodynamic boundary layer in Section 3.10 can be used for the thermal boundary layer in turbulent flow. Again, the Blasius 7-power law is used for the temperature distribution. These give results that are quite similar to the experimental equations as given in Section 4.6. 

Using the velocity profile expression and integrating momentum equation, the boundary layer thickness delta is given as... 

The top face of the control volume can be considered the free surface and therefore the shear stress is zero (plane A-A in Figure 4.9). The normal stress at this surface is due to the forces exerted by the freeboard gas. if there is no significant saltation of particles on the free surface, then the normal stress at this plane is also zero and equating the net momentum to the net force yields the integral momentum equation... 

Using the above simplification, the integral momentum equation (7.73) gives... 

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