Surface excess momentum

Let us consider in a gas, a very large, flat surface (e.g., a piston) which is uniformly accelerated, in some fixed time taj from rest to a final velocity Vb- If we consider the state of the gas in successive increments of time (Fig. XIV.8), we see that each successive increment of motion of the surface imparts to the gas in front of it an excess momentum which is then carried into the gas with molecular velocity, i.e., the speed of sound. However, because of the essentially adiabatic compression occurring in the gas,... 

O Brien s theoretical analysis (8,10) is for a suspension of solid particles, but the evidence to date indicates that emulsion droplets behave in the same way as solid particles at the frequencies involved in the ESA effect. This is understandable on a number of counts. First, it is usually observed that surfactant-stabilized emulsion droplets in a flow field do not behave as though they were liquid. The presence of the stabilizing layer at the interface restricts the transfer of momentum across the phase boundary so that there is little or no internal motion in the drop. Also, the motions which are involved are extremely small (involving displacements of the order of fractions of a nanometer) so the perturbations are small compared to the size of the drop. Finally, O Brien has shown in some unpublished calculations that if the surface is unsaturated, so that the surfactant groups can move under the influence of the electric field, then the effect on the electroacoustic signal would depend on the quantity dy/dT, where y is the surface tension and T is the surface excess of the surfactant. We have not been able to find any evidence for such an effect, if it exists, so we will assume that the analysis for a solid particle holds also for emulsions. 

To close the system of equations for the fluid motion the tangential stress boundary condition and the force balance equation are used. The boundary condition for the balance of the surface excess linear momentum, see equations (8) and (9), takes into account the influence of the surface tension gradient, surface viscosity, and the electric part of the bulk pressure stress tensor. In the lubrication approximation the tangential stress boundary condition at the interface, using Eqs. (17) and (18), is simplified to... 

In the impulse model, the excess energy Ek is transferred to an NO molecule as the momentum p0 given only to an N atom. Here, p0 is normal to the surface and Ek = p /2m, where m is mass of the N atom. Recoil of substrate Pt atoms can be ignored, because the mass of a Pt atom is much larger than that of an N atom. After desorption the momentump0 is converted to the linear momentum of the center of mass, P, and the linear momentum of the internal coordinate, p. A relationship p0 = m dri/df is satisfied in the impulse model and it can be approximated to dr2/df = 0 at the moment of the Pt-N bond breaking, where and r2 are the position vectors of N and O atoms, respectively, in an adsorbed NO molecule. 

There will be two different streams that have fixed excess components of velocity. When a = 0 (smooth surface), there is no momentum transport and the molecules have excess velocities V/2. For a = 1 (rough surface) the excess velocities become the excess velocities of the plates, namely 0 and V, respectively. The coefficient a is known as the accommodation coefficient for momentum and must be measured experimentally. 

The thickness of the liquid sheet on the drum is independent of the pressure applied between the applicator roller and the drum. It depends largely on the viscosity and the surface tension of the liquid as well as on the ratio of rotational speeds of the drum and roller and the nip width. A stable thickness can be achieved for medium viscosity shear-thinning liquids at moderate Reynolds number. Re (the ratio between momentum and viscous forces), low capillary number, Ca (the ratio of the viscous force and surface tension), and low rotational speed ratio." A smaller Ca (<0.01) or a larger surface tension extends coating stability to higher rotation speeds. A stable sheet is established between a minimum and a maximum drum speed and at a critical ratio of rotational speeds of the applicator roller and the drum. The sheet formed beyond the eritical point is unstable, leading to ribbing and may entrain air bubbles as well especially if the liquid contains excessive surfactants." This rarely occurs in drum dryer operation because both the rotation speed and the ratio of rotation speeds are low (linear speed of drum surface rarely exceeds 0.3 m/s). 

Enhance furnace bottom temperature with many small high-velocity (high-momentum) burners, firing with constant air, variable fuel, that is, excess air as they turn to low fire, to hold the same temperatures below the load(s) as above. Install fuel meters on each zone. When the fbel flows in all zones reaeh their minimums, hold as long as necessary for the required minimum temperature differential between surface and core, as determined from time-temperature heating eurves. Then remove and process the loads. 

The inteifacial momentum balance equation (Equation 5.35) will now be written for the case of negligible siuface excess mass and momentum and two Newtonian fluids. The stress tensor in a Newtonian fluid is written following Bird et al. (2002) as pi - pt [Vv + (Vv) ], where I is the identity tensor and the superscript T represents the transpose of a tensor. When the inertial forces are considered as well, the total force on an interface exerted by the ith phase is n (Pi(v - d )v + pj - / [VVj + (Vv )T]). The dot product with n denotes that the forces act on a surface characterized by m. It is in making a force balance on the intraface that the effects of interfacial tension make themselves felt. The balance is known to be (see Chapter 5)... 

The 0(6) jump in x n results from unbalanced tangential stresses. It is proportional to the vorticity [Vx ] produced at the flame and to the rotation V x n of a surface element. We note in particular the dependence on both the heat release q, and the Prandtl number Pr. Finally, equation (13) states that a net normal momentum flux across the flame, results from unbalanced pressure gradients, flame curvature and from the momentum associated with the net excess mass given by (11). 

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 ... 

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