Contraction flow

Consider flow into a circular channel or pipe with the velocity a constant at the entrance. As the fluid enters the channel, the velocity soon develops a parabolic profile. The velocity at the centerline increases from (v) (at the entrance) to 2 (v) (far downstream) while the velocity at the wall decreases suddenly from (v) to zero. The entry length is defined as the length of channel it takes before the centerline velocity is 99% of its ultimate value, 2 (v). Sometimes the entry length is defined using a 98% or 95% criterion. Atkinson et al. [2,4] have solved this problem using the hnite element method and correlated the results with the equation  

This equation is based on the Stokes flow solution (Re = 0, giving the 0.59) plus i boundary layer solution (giving the 0.056 Re). It is not actually a curve fit of the 

It is impossible to realize a constant velocity at the inlet to the circular channel unless the wall is paper thin and the flow can go past the outer wall, too. A more realistic case is where a larger channel decreases to a smaller channel (Fig. 8.3). Then a fully developed velocity profile can be used far upstream in the larger channel. The flow rearrangement occurs before the contraction (Fig. 8.8). This is important in laminar flow because vorticity is generated at the corner and diffuses upstream. As the velocity increases, this region gets pushed against the inlet and becomes smaller. Thus, for turbulent flow the effect is much shorter. 

The approach length is presented in Fig. 8.9 as a function of Reynolds number. Inside the pipe the entry length for a 4 1 contraction in a circular channel or pipe is derived from finite element calculations including the upstream segment  

The effect of Reynolds number is clearly displayed in Fig. 8.10. When the upstream region is included the entry length is much shorter (LJd = 0.1264 rather than 0.43) since the velocity has already rearranged somewhat upstream. 

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Figure 3.2 Defonnation of a fluid particle along its trajectory in a contracting flow...

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. 

Smith et al. [13] are amongst the first to employ an abrupt contraction flow to investigate chain conformation. A modified version of Smith s design was used by Merrill and Leopairat [108], and later on by the Lausanne group [109] (Fig. 26)... 

Fig. 32. Streamlines in abrupt contraction flow computed at a midtravel distance of the piston at x = — 40 mm (the origin of the abscisse is taken at the orifice entrance). Due to axial symmetry, only half of the flow tube is shown. The dimensionless stream function

Fig. 35. Velocity distribution along different streamlines in abrupt contraction flow (s is the curvilinear distance along the streamline from the entrance, as shown in Fig. 32).

First, the function e(t) computed from e(x) (Fig. 33), is divided into a number of time intervals which are sufficiently short to justify the approximation of a constant average strain-rate within each period. Only the region of space where the strain rate is significantly different from zero, i.e. from — 4r 5 x +r0 in the case of abrupt contraction flow (Fig. 33), will contribute to the degradation and needs to be considered in the calculations. The system of Eq. (87) is then solved locally using the previously mentioned matrix technique [153]. 

Figure 6.24 Flow through contracting geometries and contracting flow fields...

The techniques used in the Ml project were those of fibre spinning, open siphon, filament stretching, contraction flows, converging channel, opposing jets, climbing constants and falling drop methods. The results led to two major findings ... 

X. L. Luo, A Control Volume Approach for Integral Viscoelastic Models and Its Application to Contraction Flow of Polymer Melts, J. Non-Newt. Fluid Mech., 64, 173-189 (1996). 

FIG. 16.21 Apparent or transient extensional viscosity of the round robin test fluid Ml as a function of Hencky strain, measured in many different devices (lames and Walters, 1993). The various instruments are spin line Binding et al. Ferguson and Hudson Ngyuen et al. horizontal spin line Oliver open siphon Binding et al. stagnation flow. Laun and Hingmann Schweitzer et al. contraction flow. Binding et al. ... 

Ma, N., K.W. Koelling, and J.J. Chalmers. 2002. Fabrication and use of a transient contractional flow device to quantify the sensitivity of mammalian and insect cells to hydrodynamic forces. Biotechnol Bioeng 80 428-437. 

Yoo, J. Y, and Na, Y., A numerical study of the planar contraction flow of a viscoelastic fluid using the SIMPLER algorithm. J. Non-Newtonian Fluid Mech. 39,89 (1991). 

Figure 13. Stream tubes and stream bands for flows in a converging and a contraction flow - Local modified Hermite element.

For an axisymmetric pipe or channel, in a contracting flow, the kinetic energy change is derived as follows. Letjfl = be the area ratio (d2 < dj for contraction), and use V = Vq [1 - rjRi) ] for the inlet and outlet. Then ... 

Fig. 6 Contraction flow field. The flow direction is along the dashed line...

The streamlines of a converging flow field is shown in Fig. 6. In the contraction flow, the polymer coil may not have sufficient time to extend fully. The jet may reach very high speed to create cmitract flow. This technique was widely studied by Nguyen and Kausch [105-113]. 

In the last few decades, understanding and controlling contraction flow or entry flow behavior has been one of the classical problems of fluid mechanics. Reviews of investigations published prior to 1987 can be found in Boger [7] and White et al. [8], and a brief summary of more recent works is given in Rodd et al. [4]. 

Nigen S, Walters K (2002) Viscoelastic contractions flows comparison of axisymmetry and planar configurations. J Non-Newtonian Fluid Mech 102 343-359... 

In addition to De, which characterizes the role of elasticity in the non-Newtonian flow response, the Reynolds number Re describing the relative magnitudes of inertia and viscous stresses remains an important parameter in contraction flows. 

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