Singular pressure drops

Write out the head variation between the furnace outlet and the inlet of shop n. Justify that the singular pressure drop coefficient, defined by AH = pg U / 2, should decrease as the secondary pipe cortsidered gets further away from the furnace. 

Flow of trains of surfactant-laden gas bubbles through capillaries is an important ingredient of foam transport in porous media. To understand the role of surfactants in bubble flow, we present a regular perturbation expansion in large adsorption rates within the low capillary-number, singular perturbation hydrodynamic theory of Bretherton. Upon addition of soluble surfactant to the continuous liquid phase, the pressure drop across the bubble increases with the elasticity number while the deposited thin film thickness decreases slightly with the elasticity number. Both pressure drop and thin film thickness retain their 2/3 power dependence on the capillary number found by Bretherton for surfactant-free bubbles. Comparison of the proposed theory to available and new experimental... 

In most experimental devices, the main problem is to eliminate the different sources of error. For pressure drop measurements, the pressure sensors must not be intrusive and interfere with the physical phenomenon. In most pubhshed works, the pressure sensors are added to the circuit and the fitting itself can create a singular pressure loss. Two experiments are presented. The first one has a rectangular channel whose hydraulic diameter varies from 100 pm to 1 mm with pressure sensors on either side of the test channel and includes entrance effects. The second one whose hydrauhc diameter is 7.1 pm has the pressure taps far from the inlet and outlet to eliminate entrance and exit effects. 

Inside the drop, we require that the velocity and pressure fields be bounded at the origin [which is a singular point for the spherical coordinate system that we will use to solve (7 199)]. Finally, at the drop surface, we must apply the general boundary conditions at a fluid interface from Section L of Chap. 2. However, a complication in using these boundary conditions is that the drop shape is actually unknown (and, thus, so too are the unit normal and tangent vectors n and t and the interface curvature V n). As already noted, we can expect to solve this problem analytically only in circumstances when the shape of the drop is approximately (or exactly) spherical, and, in this case, we can use the method of domain perturbations that was first introduced in Chap. 4. In this procedure, we assume that the shape is nearly spherical, and develop an asymptotic solution that has the solution for a sphere as the first approximation. An obvious question in this case is this When may we expect the shape to actually be approximately spherical ... 

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