Square constrictions

The flow coefficient of the square constriction cannot be solved analytically. It was determined numerically by the solution of equation 3 for the geometry shown in Figure 4. The geometry in figure 4 is mapped into a rectangle and the problem is solved with a Galerkin spectral technique. The relation between the flow coefficient and interface position is shown in Figure 5 for the surfactant free system. 

Figure 6. Volume of bubbles formed from a 1 wt.% solution of Amphosol at a square constriction (5), a=500iim, r =118iim, M=lcp, 0=33 dynes/cm.

If this area is square in shape, the length of each side needs to be 6.79°5 = 2.6in. We can usually make this somewhat rectangular or odd-shaped too, so long as we preserve the total area. Note that if the area required exceeds 1 square inch, a 2-oz board should be used (as in this case). A 2-oz board reduces the thermal constriction around the power device and allows the large copper area to be more effectively used for natural convection. 

When the bung is inserted in the bottle containing 70 ml of liquid, the constricted end of the tube is kept above the surface of the liquid, and the hole in the side is below the bottom of the bung. The upper end of the tube is cut off square, and is either slightly rounded or ground smooth. 

Figure 30-17 (A) Two-dimensional map of the 260-kDa a subunit of the voltage-gated Na+ channel from the electric eel Electrophorus e/ecfns.438 441 (B) Image of the sodium channel protein obtained by cryo-electron microscopy and image analysis at 1.9 nm resolution. In this side view the protein appears to be bell-shaped with a height of 13.5 nm, a square bottom (cytoplasmic surface) 10 nm on a side, and a hemispherical top with a diameter of 6.5 nm. (C) Bottom view of the protein. (D) Axial section which cuts the bottom, as viewed in (C), approximately along a diagonal. From Sato et al.438 Notice the cavities (dark) and domain structures (light). The black arrow marks a constriction between upper (extracelllar) and lower (cytoplasmic) cavities. White lines indicate approximate position of the lipid bilayer. From Sato et al.i38 Courtesy of Chikara Sato.

Figure 7.1 Pupil diameter before (A) and after a light stimulus (S). Constriction (B) and dilation (D) velocities are determined from a least square fit of the slope. Amplitude of constriction (C) represents the maximal difference in diameter before and after the flash.

Small volumes of solution (up to 2 ml in one operation) may conveniently be filtered through a dropping (Pasteur) pipette into the constriction of which has been rammed a small piece of paper tissue (about 3 cm square). The pipette is supported vertically and the solution is added from a second Pasteur pipette. Pressure to accelerate the filtration process may then be applied from a rubber bulb attached to the top of the pipette. 

Coefficients of discharge for square-edged orifices with centered circular openings and for rotameters. (Subscript 0 indicates at orifice or at constriction and subscript 1 indicates at upstream... 

Arriola (8) and Ni (5) have observed a second mechanism for snap off in strongly constricted square capillaries. At low liquid flow rates, a bubble is trapped in the converging section of the constriction and liquid flows past the bubble. As liquid flow rate increases, waves developed in the film profile and at some critical liquid flow rate these oscillations become unstable and bubbles snap off. In these experiments, the bubble front is located upstream of the constriction neck. Therefore, no driving force for the drainage mechanism exists. Bubbles formed by this mechanism are produced at a high rate and have a radius on the order of the constriction neck. No attempt has previously been made to model snap-off rate by this mechanism in noncircular constrictions. 

Snap off by the instability mechanism may occur in the following way. A bubble in an angular constriction such as a square channel will flatten against the walls as shown in Figure 2. The radius of the circular arcs in the corners for a static bubble is about one half of the tube half width. At low liquid flow rates, a bubble trapped behind a constriction has this nonaxisymmetric shape. As liquid flow rate increases, the bubble moves farther into the constriction and the fraction of cross sectional area open to liquid flow at the front of the bubble increases until the thread becomes axisymmetric at some point near the bubble front. 

The solution for the isoflux boundary condition and with external thermal resistance was recently reexamined by Song et al. [156] and Lee et al. [157]. These researchers nondimen-sionalized the constriction resistance based on the centroid and area-average temperatures using the square root of the contact area as recommended by Chow and Yovanovich [15] and Yovanovich [132,137,144-146,150], and compared the analytical results against the numerical results reported by Nelson and Sayers [158] over the full range of the independent parameters Bi, e, and x. Nelson and Sayers [158] also chose the square root of the contact area to report their numerical results. The analytical and numerical results were reported to be in excellent agreement. 

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