Flow axisymmetric

Sphere in Linear Flows Axisymmetric Extensional Flow and Simple Shear... 

Hinze also discussed various well-defined flow forms and the types of droplet deformation associated with them. The flow patterns described are parallel flow, plane hyperbolic flow, rotating flow, axisymmetric hyperbolic flow, Couette flow, and irregular flow (turbulent). 

Since frequently the growth furnace has an axisymmetric conflguration, it is desirable to restrict 3D computations to the melt domain. Almost everybody fails to resist this temptation. This leads, however, to the additional artiflcial constraints on the flow - axisymmetric thermal boundary conditions, be they taken from the experiment or from the 2D global heat-transfer computations. Moreover, these conditions usually are the stationary ones. Extension of the 3D domain by adding at least the crystal, quartz and massive graphite crucibles results in attenuation of the azimuthal nonuniformities caused by the turbulent flow and reduces the errors of assuming the axisymmetric conditions at the boundary dividing 2D and 3D domains. 

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... 

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... 

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. 

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 ... 

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 ... 

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... 

Wygnanski, 1. 1964. The flow induced by two-dimensional and axisymmetric turbulent jets issuing normally from an infinite plane surface. Aeronautical Quarterly, November, pp. 373-580. 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

Flow Past a Point Sink A simple potential flow model for an unflanged or flanged exhaust hood in a uniform airflow can be obtained by combining the velocity fields of a point sink with a uniform flow. The resulting flow is an axially symmetric flow, where the resulting velocity components are obtained by adding the velocities of a point sink and a uniform flow. The stream function for this axisymmetric flow is, in spherical coordinates. 

Flow in the (r, 0) Plane The experimental data of KelsalU gave strong evidence that fluid flow in the cyclone may be assumed to be axisymmetrical. 

D. S. Noh, Y. Koh, I. S. Kang. Numerical solutions for shape evolution of a particle growing in axisymmetric flows of supersaturated solution. J Cryst Growth 183 427, 1998. 

In deriving the equation for this case, Johnson et al. (J4) assume steady-state conditions with viscous, incompressible axisymmetric flow around single... 

Flow through a convergent channel, either two-dimensional or axisymmetric, are probably the most investigated geometries (due to their connection with the technical die entry flow problem). Most of the other convergent flows could be derived from the abrupt contraction flow which is depicted schematically in Fig. 25. 

By using a tapered inlet instead of an abrupt constriction, different variations of convergent flow could be envisioned axisymmetric flow in a conical channel... 

Consider axisymmetric flow in a circular tube so that Vg = 0. Two additional assumptions are needed to treat the variable-viscosity problem in its simplest form ... 

Y. -C. Chen and R. W. Bilger, Stabilization mechanism of lifted laminar flames in axisymmetric jet flows. Combust. Flame 122 377-399, 2000. 

Viscometric flow theories describe how to extract material properties from macroscopic measurements, which are integrated quantities such as the torque or volume flow rate. For example, in pipe flow, the standard measurements are the volume flow rate and the pressure drop. The fundamental difference with spatially resolved measurements is that the local characteristics of the flows are exploited. Here we focus on one such example, steady, pressure driven flow through a tube of circular cross section. The standard assumptions are made, namely, that the flow is uni-directional and axisymmetric, with the axial component of velocity depending on the radius only. The conservation of mass is satisfied exactly and the z component of the conservation of linear momentum reduces to... 

In the context of the preceding model, a drop is said to break when it undergoes infinite extension and surface tension forces are unable to balance the viscous stresses. Consider breakup in flows with D mm constant in time (for example, an axisymmetric extensional flow with the drop axis initially coincident with the maximum direction of stretching). Rearranging Eq. (26) and defining a characteristic length Rip113, we obtain the condition, for a drop in equilibrium,... 

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