Elliptic Cross-section

For the elliptic cross-section, let the center of the ellipse be at (y, z) = (0,0). The major axis of length a and the minor axis of length b are parallel to the y-axis and z-axis, respectively, as 

This trial velocity field guarantees that u y,z) satisfies the no-slip boundary condition (equation (2.36)). It also automatically satisfies the continuity equation. Substitution of the trial solution into the left-hand side of the N-S equation (2.35) yields 

To calculate the flow rate Q for the elliptic channel, we need to evaluate a 2D integral in an elliptically shaped integration region. This is accomplished by coordinate transformation. Let p, 4 ) be the polar coordinates of the unit disk, that is, the radial and azimuthal coordinates, which obey 0 p 1 and 0 (p 2n, respectively. The physical coordinates (y, z) and the velocity field u can then be expressed as functions of p, p) as 

The advantage of this coordinate transformation is that now the boundary dC can be expressed in terms of just one coordinate instead of two. Hence, at the boundary dC, we have 

The hydraulic resistance for an elliptic channel thus can be calculated as 

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The most frequently applied mechanical manometers in ventilation applications are fluid manometers, bur the following types are also used. The Bourdon tube is a small-voiume tube with an elliptic cross-section bent to the shape of a circular arc, the C-type. One end is open to the applied pressure while the other end is closed. The pressure inside the tube causes an elastic defonnation ot the tube and displaces the closed end, which is then converted, by means of a linkage mechanism, into the movement of a pointer. The Bourdon tube may be of a spiral or helical design as well. 

A mud slurry is drained from a tank through a 50 ft long plastic hose. The hose has an elliptical cross section with a major axis of 4 in. and a minor axis of 2 in. The open end of the hose is 10 ft below the level in the tank. The mud is a Bingham plastic, with a yield stress of 100 dyn/cm2, a limiting viscosity of 50 cP, and a density of 1.4g/cm3. 

It is a simple and reliable gage covering six decades of pressure (not with one gage). Below 1 torr, its sensitivity is very low. The gage is usually made up of a bended flexible metallic tube (see Fig. 1.27) of elliptic cross-section. When pressure inside the tube is different for the outside (atmospheric) pressure, the tube bends. A gear and lever system moves the needle. 

Howell, E. 2001. Dynamic surface tension measurements of liquid solder using oscillating jets of elliptical cross section. Mechanical and Industrial Engineering. University of Illinois at Chicago, Chicago, IL. pp. 75. 

Conductance values will depend not only on the pressure and the nature of the gas which is flowing, but also on the sectional shape of the conducting element (e.g. circular or elliptical cross section). Other factors are the length and whether the element is straight or curved. The result is that various equations are required to fake into account practical situations. 

If a stream of liquid be forced through a small jet of elliptical cross section it will be observed that the issuing jet is in apparent vibration and forms a series of nodes equidistant from one another. From a determination of the wave length of the vibration, a measurement which can be made with some accuracy provided that the stream of liquid issues at a constant speed, the surface tension of the liquid may be determined by means of the following equation developed by Bohr. 

In Belgium the official testing gallery was erected at Frameries under direction of Watteyne and Stassart in 1902. It was of elliptical cross section (1.85 m/1.40 m), 85 m long. 

As an illustration of this a tube of elliptical cross-section may be considered. If the major and minor semi-axes are a and b respectively... 

The Dortmund-Derne gallery is 25m long and its elliptical cross-section is 1.30m wide and 1.80m high. One end of the gallery is closed by a brick wall, while the other is open. The expln chamber is 5m long and is separated from the open section of gallery by a paper partition. The chamber can be filled with 8-9 5% methane/air and/or coal dust/air mixts. 

Finally there are dynamic methods to measure the surface tension. For example, a liquid jet is pushed out from a nozzle, which has an elliptic cross-section. The relaxation to a circular cross-section is observed. An advantage of this method is that we can measure changes of the surface tension, which might be caused by diffusion of amphiphilic substances to the surface. 

Figure 9 shows schematic depictions of the two newly discovered structures described above. A cross section of a distorted hexagonal lattice is shown in Fig. 9a, displaying lipid micelles of an elliptical cross section and DNA molecules localized in the interstitial space. The distortion of the lattice is likely caused by the asymmetry in the micellar shape. X-ray diffraction data shows that the distortion increases with 4>MV[.BisG2-... 

This is a dynamic method which enables one to measure the tensions of surfaces at very short times (c. 0.01 s) from the moment of their creation. (The methods previously described are used to measure equilibrium tensions.) A jet of liquid emerging from a nozzle of elliptical cross-section is unstable and oscillates about its preferred circular cross-section. Surface tensions can be calculated from the jet... 

If a jet of liquid is created using a nozzle of elliptical cross-section, changes in shape of the jet occur (Figure 3.11). This is because the elliptical shape is unstable and the jet will oscillate in shape (due to momentum) about the preferred shape, which is circular due to surface tension. The surface tension can be calculated from the wavelength of the oscillations, knowing the jet dimensions and flow rate, etc. 

As Fig. 12.1 indicates, the manifold cross section may be bead shaped and not circular. Thus, pressure flow in an elliptical cross-section channel may be more appropriate for the solution of the manifold flow. Such a problem, for Newtonian incompressible fluids, has been solved analytically. (J. G. Knudsen and D. L. Katz, Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958). See also, Table 12.4 and Fig. 12.51. 

Ellipsometry is useful in examining the behavior of enzymes on electrodes. Another excellent tool is the atomic force microsope (Section 7.5.18). Eppel (1993) used AFM to examine the absorption of a glycoprotein on some biosurfaces (AFM does not need conduction electrons to function), and was able to measure its elliptical cross section in the absorbed state. It was possible to map changes in the charge density of the protein (which plays a role in thrombus formation in arterial blockage) upon absorption by the change in the height of various parts of the molecule. This is an excellent example of the detail in which absorbed species can be examined. 

Badr, H.M.. Mixed Convectioh from a Straight Isothermal Tube of Elliptic Cross-Section Ini. J. Heat and Mass Jransfer, Vol. 37, p. 2343-2365. 1994. 

Using the procedure outlined in this chapter for using the boundars laser equations to find the-forced convective heat transfer rate from a circular cylinder buried in a saturated porous medium, investigate the heat transfer rate from cylinders with an elliptical cross-section with their major axes aligned with the forced flow. The surface velocity distribution should be obtained from a suitable book on fluid mechanics. 

The Bourdon110 tube gauge (range 1 mbar to 0.1 Mbar) measures the deflections, due to pressure, on a solid barrier connected to a semicircular hollow tube of elliptical cross-section, whose contraction is amplified mechanically in a needle dial. It is used for pressures in various ranges, with an accuracy of 0.1% of the calibration. 

It is however possible to design specifically NP arrays with a strong polarization dependence in order to investigate polarization effects. This has recently been reported [21] using arrays of elongated gold NPs (with elliptic cross-section), which exhibit two distinct LSP resonances, each associated with a polarization along the two main axis of the ellipse. The spectral response for... 

Find the torsion function and the torque for a viscoelastic hollow rod of elliptical cross section. 

Now we consider the hollow elliptical cross section obtained from the solid cylinder where a concentric elliptical area has been removed. The axes of the area eliminated are respectively 2ka and 2kb, where, obviously, k <. ... 

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