2D flow

Al-Rashed etal. (1996) and Al-Rashed and Jones (1999a,b) presented a CFD-based model to prediet the effeets of mixing during bateh-wise gas-liquid reaetive preeipitation in the flat interfaee eell used by Waehi and Jones, 1991b. A 2D flow simulation was developed for the ehemieal reaetion with... 

If V v = 0, e,s(l/2)m in two-dimensional (2D) flows and (2/3)1/2 in three-dimensional (3D) flows, where i = A,r/. The efficiency can be thought of as the specific rate of stretching of material elements normalized by a factor proportional to the square root of the energy dissipated locally. 

Flows that produce an exponential increase in length with time are referred to as strong flows, and this behavior results if the symmetric part of the velocity gradient tensor (D) has at least one positive eigenvalue. For example, 2D flows with K > 0 and uniaxial extensional flow are strong flows simple shear flow (K = 0) and all 2D flows with K < 0 are weak flows. 

Fig. 39. (a) The growth of average cluster size for clusters with a constant capture radius in various 2D flows, (b) Variation of polydispersity with average cluster size, (c) Scaled distribution of the cluster sizes at different times. The regular flow is the journal bearing flow with only the inner cylinder rotating. One time unit is equivalent to the total displacement of the boundaries equal to the circumference of the outer cylinder (Hansen and Ottino, 1996b). 

D flows and time-dependent 2D flows, and mixes the fluid by continuously stretching different volumes of the fluid and folding them into one another. In a qualitative sense, the path taken by a given fluid element in the flow depends in a sensitive way on its encounters with a series of weak secondary flows or eddies, present even at low Re in the comers of channels, which transport the element across the flow [167],... 

In a 2 1/2D flow case we must also include the z-component... 

The main features of the stream-tube method in 2D and 3D flows, discussed more extensively elsewhere [40,53,55], are now summarized for two-dimensional situations. Flows with open streamlines are considered. The main flow region D of the physical domain D (D 2 D ) is mapped into a domain D such that the transformed streamlines are straight and parallel to an assumed main direction Oz of the flow. An example corresponding to 2D flows is illustrated in Fig 11. 

Y. Tsujikawa, S. Fuji, H. Sadamori, S. Ito, and S. Katsura, Numerical simulation of 2D flow of catalytic combustor, Proc. 2nd Int. Workshop Catalytic Combustion, Tokyo, 18-20 April, (H. Arai. ed.). Catalysis Society of Japan, Tokyo, 1994, p. 96. 

We obtain the final thickness of ponded plume material following Huppert (1982) for four geometrically ideal cases ponding of radially symmetric plume material beneath a very wide craton, ponding beneath a craton of limited area, 2D ponding where material flows outward along a channel of constant width, and 2D flow from a craton of limited width. The objective is to obtain the dimensional dependence of the thickness of plume material on physical parameters. 

TABLE 10.1. Using FEMLAB to Solve Transport Problems (this example is for a 2D flow problem)... 

In this case, we must therefore begin with the full Navier Stokes and continuity equations for a 2D flow, (2 91) and (2 20). In terms of the Cartesian coordinate system described in Fig. 4-8, these are... 

We note, for future reference, that the function f plays a role in the present problem at 0(a/R) that is analogous (in the plane) to the streamfunction for a general 2D flow. The concept of a streamfunction will be discussed in detail in Chap. 7. 

In general, for 2D flows, 3 can be identified with a Cartesian variable z, orthogonal to the plane of motion, and /13 = 1. However, for axisymmetric flows, c/3 represents the azimuthal angle

spherical coordinates, for example,... 

However, for 2D flows, the third direction corresponds to a Cartesian coordinate direction and... 

The solution of these equations by means of standard eigenfunction expansions can be carried out for any curvilinear, orthogonal coordinate system for which the Laplacian operator V2 is separable. Of course, the most appropriate coordinate system for a particular application will depend on the boundary geometry. In this section we briefly consider the most common cases for 2D flows of Cartesian and circular cylindrical coordinates. 

Figure 7-5. 2D flow in a sharp comer, caused by motion of the bottom surface (at = 0) with the velocity U. The plot shows streamlines, i/r = t/// U, calculated from Eq. (7-79) for a = jr/3. Contour values range from 0 at the walls in increments of 0.02. 

The general solution (7-71) can be applied to examine 2D flows in the region between two plane boundaries that intersect at a sharp corner. This class of creeping motion problems was considered in a classic paper by Moffatt,11 and our discussion is similar to that given by Moffatt. A typical configuration is shown in Fig. 7-5 for the case in which one boundary at 6 = 0 is moving with constant velocity U in its own plane and the other at 6 = a is stationary. 

Figure 7-8. A sketch of the 2D flow near a sharp comer that is induced by an arbitrary stirring flow at large distances from the comer (a) antisymmetric, (b) symmetric.

Figure 9-15. A sketch of streamlines for the class of linear 2D flows for — 1 < A < 1.

A general discussion of flow types and particle shapes for which these conditions are satisfied is beyond the scope of this book. As an alternative, at least a qualitative sense may be imparted for the issues involved by consideration of the special case of a spherical particle and the class of linear 2D flows described by... 

The question at hand is whether circumstances exist for this rather simple situation in which the conditions (1) and (2) are satisfied so that boundary-layer analysis can be applied. So far as the first condition is concerned, the only flows of (9-266) that have open streamlines are those with X > 0 (which includes simple shear flow). On the other hand, there is a nonzero hydrodynamic torque on the sphere that causes it to rotate for all flows in this subgroup except X = 1. Thus, for a sphere in the general linear 2D flow, given by (9-266), there are only two cases that satisfy the conditions for applicability of boundary-layer theory ... 

Similarly, if we consider the case of a circular cylinder with its axis at x = 0 and oriented normal to the plane of the 2D flow, (9-266), the requirement of open-streamline flow is satisfied only if X = 1 or if the cylinder is constrained from rotation and 0 < X < 1. 

Streamline implies the existence of a streamfunction, which we have seen to he true only for axisymmetric and 2D flows. In three dimensions, recirculating flows are associated with regions of closed stream surfaces, or pathlines. 

The main value of the vorticity transport equation, in the present context, is that a direct analogy exists for 2D motions between this equation and the thermal energy equation of Chap. 9. Specifically, for a 2D flow,... 

Problem 10-14. Two-Dimensional Jet. Consider the 2D flow created when fluid is force with high velocity through a very narrow slit in a wall as pictured in the figure (this flow is referred to as a 2D symmetric jet). 

Problem 10-16. 2D Flow Near a Stagnation Point. Consider a flow impinging on a horizontal flat plate, as illustrated in the figure. Assume that far away from the plate the flow is given by... 

The introduction o(V/> is equivalent to introducing a stream function for this 2D flow problem. If we introduce these expressions for u and w into (12-307a), we see that... 

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