Axisymmetric

Neumann has adapted the pendant drop experiment (see Section II-7) to measure the surface pressure of insoluble monolayers [70]. By varying the droplet volume with a motor-driven syringe, they measure the surface pressure as a function of area in both expansion and compression. In tests with octadecanol monolayers, they found excellent agreement between axisymmetric drop shape analysis and a conventional film balance. Unlike the Wilhelmy plate and film balance, the pendant drop experiment can be readily adapted to studies in a pressure cell [70]. In studies of the rate dependence of the molecular area at collapse, Neumann and co-workers found more consistent and reproducible results with the actual area at collapse rather than that determined by conventional extrapolation to zero surface pressure [71]. The collapse pressure and shape of the pressure-area isotherm change with the compression rate [72]. 

The axisymmetric drop shape analysis (see Section II-7B) developed by Neumann and co-workers has been applied to the evaluation of sessile drops or bubbles to determine contact angles between 50° and 180° [98]. In two such studies, Li, Neumann, and co-workers [99, 100] deduced the line tension from the drop size dependence of the contact angle and a modified Young equation... 

In this section the governing Stokes flow equations in Cartesian, polar and axisymmetric coordinate systems are presented. The equations given in two-dimensional Cartesian coordinate systems are used to outline the derivation of the elemental stiffness equations (i.e. the working equations) of various finite element schemes. 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

In an axisymmetric flow regime there will be no variation in the circumferential (i.e. 0) direction and the second term of the integrand in Equation (4.8) can be eliminated. After integration with respect to 9 between the limits of 0 -27t Equation (4.8) yields... 

Therefore the continuity equation for an incompressible axisymmetric flow is written as... 

Similarly the components of the equation of motion for an axisymmetric Stokes flow of a generalized Newtonian fluid are written as... 

Working equations of the U-V-P scheme in axisymmetric coordinate systems... 

Using a irocedure similar to the formulation of two-dimensional forms the working equations of the U-V - P scheme in axisymmetric coordinate systems are derived on the basis of Equations (4.10) and (4.11) as... 

After the substitution of pressure via the penalty relationship the flow equations in an axisymmetric coordinate system are written as... 

Using a procedure similar to the derivation of Equation (4.53) the working equations of the continuous penalty scheme for steady-state Stokes flow in an axisymmetric coordinate system are obtained as... 

Working equations of the streamline upwind (SU) scheme for the steady-state energy equation in Cartesian, polar and axisymmetric coordinate systems... 

Similarly in an axisymmetric coordinate system the terms of stiffness and load matrices corresponding to the governing energy equation written as... 

Note that in polar and axisymmetric coordinate systems the stress term will include some lower-order terms that should be included in the formulations. 

In Chapter 4 the development of axisymmetric models in which the radial and axial components of flow field variables remain constant in the circumferential direction is discussed. In situations where deviation from such a perfect symmetry is small it may still be possible to decouple components of the equation of motion and analyse the flow regime as a combination of one- and two-dimensional systems. To provide an illustrative example for this type of approximation, in this section we consider the modelling of the flow field inside a cone-and-plate viscometer. 

It should be noted that the described axisymmetric unit cells do not represent actual repetitive sections of a material but their dimensions are related to the... 

As already mentioned, the present code corresponds to the solution of steady-state non-isothennal Navier-Stokes equations in two-dimensional Cartesian domains by the continuous penalty method. As an example, we consider modifications required to extend the program to the solution of creeping (Stokes) non-isothermal flow in axisymmetric domains ... 

Step 3 Comparing systems (7.2) and (7.3) additional terms in the members of the stiffness matrix corresponding to the axisymmetric fon-nulation are identified. Note that the measure of integration in these tenns is (r drdz). 

A similar analysis can be carried out for any axisymmetric tube to yield ... 

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

The terminal velocity of axisymmetric particles in axial motion... 

Many problems of practical interest are, indeed, two dimensional in nature. Impact and penetration problems are examples of these, where bodies of revolution impact and penetrate slabs, plates, or shells at normal incidence. Such problems are clearly axisymmetric and, therefore, accurately modeled with a two-dimensional simulation employing cylindrical coordinates. 

Compact air jets are formed by cylindrical tubes, nozzles, and square or rectangular openings with a small aspect ratio that are unshaded or shaded by perforated plates, grills, etc. Compact air jets are three-dimensional and axisymmetric at least at some distance from the diffuser opening. The maximum velocity in the cross-section of the compact jet is on the axis. 

Abramovich was the first to study axisymmetric confined jets analytically. He suggested the method based on utilizing the equations of continuity and momentum conservation. He also assumed that the width of the layer of a jet mixing with a counterflow equals the width of a free jet with a velocity distribution according to Schlichting s formula ... 

An analytical solution of the interaction in the case of isothermal main and directing jets, assume that the main stream (Fig. 7.57), supplied with initial velocity (t oi) through a nozzle that has internal diameter (Tqj), is developing within a zone ( -/q, 0) as a free jet. The momentum (/,) of the jet within the zone ( -Iq -F /, 0) remains equal to the initial momentum (/oi)> the velocity distribution in the cross-section of interaction in the plane XY remains the same within the zone (0, X ). The axisymmetric main stream within the zone (0, X j) is substituted by the linear flow with velocity profile that can be described by the formula... 

Experimental studies have shown that velocity distribution in the cross-section of the directing jet can be described by the same equation as those in the axisymmetric jet in a cross-draft,... 

Gendrikson, V. A., and Y. V. Ivanov. 1973. Some regularities of axisymmetric jets supplied at an angle to the cross-draft. In Proceedings of the Tashkent Polytechnical Institute, vol. 101, pp. 184-1.98. 

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