Two-dimensional linear flow

Fig. 13. Streamlines and velocity profiles for two-dimensional linear flows with varying vorticity. (a) K = -1 pure rotation, (b) K = 0 simple shear flow, (c) K = 1 hyperbolic extensional flow. 

Fig. 14. Critical capillary number (Cac ) as a function of the viscosity ratio (p) for two-dimensional linear flows with varying vorticity (Bentley and Leal, 1986).

B.J. Bentley and L.G. Leal An Experimental Investigation of Drop Deformation and Breakup in Steady Two-Dimensional Linear Flows. J. Fluid. Mech. 167, 241 (1986). 

The velocity gradient in a general two-dimensional linear flow, such as that produced near the stagnation point of a four-roll mill, can be represented as... 

Bentley, B.J. and Leal, L.G., An experimental investigation of drop deformation and breakup in steady, two-dimensional linear flows. 

Bentley, B. J., and Leal, L. G., A computer-controlled four-roll mill for investigations of particle and drop dynamics in two-dimensional linear shear flows. J. Fluid Mech. 167, 219-240 (1986). 

We consider flows in ducts with aspect ratio (AR = wlh = width/height) of 1, 2, and 4. The data are obtained by linearized Boltzmann solution in ducts with the corresponding aspect ratios. Our previous analysis was valid for the two-dimensional channels, where we reported flowrate per channel width. For duct flows, three-dimensionality of the flow field (due to the side walls of the duct) must be considered. In continuum duct flows, the flowrate formula developed for two-dimensional channel flows is corrected in order to include the blockage effects of the side walls. According to this, the volumetric flowrate in a duct with aspect ratio AR for no-slip flows is (see... 

Nam et al. (2000) have shown that the theoretical framework introduced in the previous sections for the description of passively ad-vected decaying scalar field also applies to the description of the small scale structure of the vorticity field in a two-dimensional turbulent flow with linear damping. The vorticity dynamics in this case is described by the Navier-Stokes equation... 

Axisymmetric Stream Function in Spherical Coordinates. It is necessary to understand the stream function in sufficient depth because additional boundary conditions are required to solve linear fourth-order PDFs relative to the typical second-order differential equations that are characteristic of most fluid dynamics problems. Consider the following two-dimensional axisymmetric flow problem in which there is no dependence on the azimnthal angle 4> in spherical coordinates ... 

Levich (8 ) has discussed capillary motion in two-dimensional creeping flows in which the surface was flat. Yih (17) pointed out inconsistencies in Levich s analysis which were associated with the assumptions of a linear distribution of surface tension with distance along the interface, and with the deflection of the surface which inevitably occurs when capillary flow exists. He noted that under certain circumstances steady flows may not exist. Ostrach (18, 19) has discussed scaling problems in capillary flows. 

Recent mathematical work suggests that—especially for nonlinear phenomena—certain geometric properties can be as important as accuracy and (linear) stability. It has long been known that the flows of Hamiltonian systems posess invariants and symmetries which describe the behavior of groups of nearby trajectories. Consider, for example, a two-dimensional Hamiltonian system such as the planar pendulum H = — cos(g)) or the... 

Linear air jets are formed by slots or rectangular openings with a large aspect ratio. The jet flow s are approximately two-dimensional. Air velocities are symmetric in the plane at which air velocities in the cross-section are maximum. At some distance from the diffuser, linear air jets tend to transform info compact jets. 

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. 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

An experimental arrangement is illustrated in Fig 1. on p 381 of Ref 93. A metal plate thickness , is bent thru an angle deton wave, velocity DQ travelling thru a layer of explosive. When the plate was deflected, it hit at an angle of incidence i a block of expl, density p.Q The thicknesses were sufficiently small compared to the other quantities so that the flow could be considered as plane two-dimensional and stationary. The reference system R had its origin at the point of impact I and was under uhiform linear motion. Theoretical and experimental studies of the flow were carried ont in the vicinity of the noinr nf impact... 

In porous media the flow of water and the transport of solutes is complex and three-dimensional on all scales (Fig. 25.1). A one-dimensional description needs an empirical correction that takes account of the three-dimensional structure of the flow. Due to the different length and irregular shape of the individual pore channels, the flow time between two (macroscopically separated) locations varies from one channel to another. As discussed for rivers (Section 24.2), this causes dispersion, the so-called interpore dispersion. In addition, the nonuniform velocity distribution within individual channels is responsible for intrapore dispersion. Finally, molecular diffusion along the direction of the main flow also contributes to the longitudinal dispersion/ diffusion process. For simplicity, transversal diffusion (as discussed for rivers) is not considered here. The discussion is limited to the one-dimensional linear case for which simple calculations without sophisticated computer programs are possible. 

In this section, we will proceed to develop a finite element formulation for the two-dimensional Poisson s equation using a linear displacement, constant strain triangle. Poisson s equation has many applications in polymer processing, such as injection and compression mold filling, die flow, potential problems, heat transfer, etc. The general form of Poisson s equation in two-dimensions is... 

In this chapter, we have derived the two-dimensional finite element penalty formulation for creeping flows where the pressure was eliminated by assuming a compressible flow. Here, we will use a mixed formulation, where the pressure is included among the unknown variables. In the mixed formulation, we use different order of approximation for the pressure as we will for the velocity. For instance, if tetrahedral elements are used, we can use a quadratic representation for the velocity (10 nodes) and a linear representation for the pressure (4 nodes). Hence, we must use different shape functions for the velocity and pressure. For such a formulation we can write... 

The first case considered is solute desorption during unconfined compression. We consider a two dimensional plane strain problem, see Fig. 1. A sinusoidal strain between 0 and 15 % is applied at 0.001 Hz, 0.01 Hz, 0.1 Hz and 1 Hz. To account for microscopic solute spreading due to fluid flow a dispersion parameter is introduced. Against the background of the release of newly synthesized matrix molecules the diffusion parameter is set to the value for chondroitin sulfate in dilute solution Dcs = 4 x 10 7 cm2 s-1 [4] The dispersion parameter Dd is varied in the range from 0 mm to 1 x 10 1 mm. The fluid volume fraction is set to v = 0.9, the bulk modulus k = 8.1 kPa, the shear modulus G = 8.9 kPa and the permeability K = lx 10-13m4 N-1 s-1 [14], The initial concentration is normalized to 1 and the evolution of the concentration is followed for a total time period of 4000 s. for the displacement and linear discontinuous. For displacement and fluid velocity a 9 noded quadrilateral is used, the pressure is taken linear discontinuous. 

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