Poiseuille flow in a pipe

Fig. 5.3.5 Jo int spatial-velocity images of xenon undergoing Poiseuille flow in a pipe (id = 4 mm, DXe = 4.5 mm2 s-1, Vave =...

Fig. 5.3.6 j oint spatial-velocity images of xenon undergoing Poiseuille flow in a pipe (id = 4 mm, Vave = 27 mm s 1, D = 8 mm2 s 1) at 0.7 atm recorded with a protocol shown in Figure 5.3.4(A). Only particles at walls are selected by the edge enhancement filter . A modified imaging gradient time duration... 

The first test is to compare a turbulent channel flow studied in the previous section and a laminar flow. A three dimensional Poiseuille flow in a pipe geometry was used as test case. The flow is laminar and the Reynolds number based on the bulk velocity and diameter is approximately 500. The bound-... 

An important problem is to analyze the stability of fluid flows. With the exception of the Taylor-Couette and Saffman Taylor problems, this chapter has focused on stability questions when the base state of the system was one with no motion (or rigid-body motion), so that instability addresses the conditions for spontaneous onset of flow. An equally valid question is whether a particular flow, such as Poiseuille flow in a pipe (or any of the other flows that we have analyzed in previous chapters of this book), is stable, especially to infinitesimal perturbations as linear instability determines whether the particular flow is actually realizable in experiments. This question was first mentioned back in Chapter 3 when we analyzed simple unidirectional flow problems and noted that solutions such as Poiseuille s solution for flow through a tube was a valid solution of the Navier-Stokes equations for all Reynolds numbers, even though common experience tells us that beyond some critical Reynolds number there is a transition to turbulent flow in the tube. 

In the limit where the yield stress = 0, the velocity profile reduces to that for the Poiseuille flow in a pipe of circular cross-section (Eq. 4.2.14). 

Poiseuille flow also occurs in a simple shearing situation, but it presumes that there is a pressure gradient that drives the flow and that the solid boundaries are fixed. Flow in a pipe or tube is an example of Poiseuille flow. It is a straightforward matter to combine these Couette and Poiseuille effects, and still find an exact analytic solution. 

The molecules of these other species get in the way of the molecules of species 1 (say) and, in effect, exert a drag on them in much the same way that a pipe exerts a frictional drag on the fluid flowing through it. The analogy with pipe-flow does not end here an analysis of diffusion may be carried out in essentially the same way that we may derive, for example, Poiseuille s equation for the rate of fluid flow in a pipe—through the application of Newton s second law. 

Prom the introductory courses in fluid flow one recalls that the simple parabolic profile for laminar flow in a pipe, the Hagen-Poiseuille law, is derived by integration of a sufficiently simplified form of the generalized momentum equation (see e.g., [13], Example 3.6-1) ... 

There is really not much point in having a curve for laminar flow on a friction factor plot , since laminar flow in a pipe can be solved analytically. Poiseuille s equation (Eq. 6.8) can be rewritten (Prob. 6.12) as... 

Poiseuille flow n. Laminar flow in a pipe or tube of circular cross-section under a constant pressure gradient. If the flowing fluid is Newtonian, the flow rate will be given by the Hagen-Poiseuille equation. 

This equation expresses simple Poiseuille flow through a pipe but corrects it for jet-driven motion in the outer hquid. 

FIGURE 7.1 Parabolic velocity profile characteristic of Poiseuille flow in a round pipe of radius o. x, r-coordinate system with origin on the pipe centerline. 

The Poiseuille flow in a circular cylindrical pipe is determined in the same fashion as detailed above. The main difference is the need to represent the equations in a cylindrical coordinate system, since the boundary conditions are most optimum in that coordinate system. The equations are given in Table 1.2, and Figure 1.3 indicates the way in which a point M in space is located through the distance r to an axis Oa its abscissa z along that axis, and the angle 0. At that point, the flow velocity is determined by the components ur,ue, and Uz), represented on the basis of the three vectors er,ee, indicated in the figure. Therefore, the velocity vector is ... 

The Poiseuille flow in a cylindrical circular pipe is the solution of the set of equations surrounded by a double bar in Table 1.2. With equation [1.19], the Navier-Stokes equations are simplified into ... 

This is also calculated for the cylindrical coordinate system using equation [1.2] and the stress tensor (Table 1.2) for the velocity field [1.23] of the Poiseuille flow in a circular pipe. 

We assume that the friction stress r exerted by the flow is uniform on the side wall. This assumption is questionable in the case of a complex cross-sectional geometry, but is wholly valid in the case of the plane Poiseuille flow or for the Poiseuille flow in a drcnlar pipe. Therefore, it is written as ... 

The smooth laminar regime. For < 2000, we obtain 1 = 64 / Re. This result is accurate for the Poiseuille flow in a circular pipe (Chapter 1, section 1.4), as... 

We have determined in this section the hydrodynamic conditions on a turbulent flow that lead to the RTD laws of a CSTR or a tubular reactor with axial dispersion. The reverse analysis, for deriving turbulence characteristics of a flow from the measurement of a RTD, should be used with caution if it caimot be positively asserted that the flow is turbulent. A laminar flow (e.g. a Poiseuille flow) in a tubular reactor also produces an axial dispersion measured by a RTD, because the product injected on the pipe axis is carried faster than that injected near the walls. Clearly, it would be meaningless to derive turbulence characteristics from the measured RTD law." ... 

Two common types of one-dimensional flow regimes examined in interfacial studies Poiseuille and Couette flow [37]. Poiseuille flow is a pressure-driven process commonly used to model flow through pipes. It involves the flow of an incompressible fluid between two infinite stationary plates, where the pressure gradient, Sp/Sx, is constant. At steady state, ignoring gravitational effects, we have... 

EXAMPLE 4.3 Plagan Poiseuille flow - laminar, steady incompressible flow in a long pipe with a linear pressure gradient (first-order, nonlinear solution to an ordinary differential equation)... 

Linear stability theory results match quite well with controlled laboratory experiment for thermal and centrifugal instabilities. But, instabilities dictated by shear force do not match so well, e.g. linear stability theory applied to plane Poiseuille flow gives a critical Reynolds number of 5772, while experimentally such flows have been observed to become turbulent even at Re = 1000- as shown in Davies and White (1928). Couette and pipe flows are also found to be linearly stable for all Reynolds numbers, the former was found to suffer transition in a computational exercise at Re = 350 (Lundbladh Johansson, 1991) and the latter found to be unstable in experiments for Re > 1950. Interestingly, according to Trefethen et al. (1993) the other example for which linear analysis fails include to a lesser degree, Blasius boundary layer flow. This is the flow which many cite as the success story of linear stability theory. 

This section considers the case of flow in a channel of diameter d at low Re. After the flow has been in the pipe for a distance much longer than the entry length, the fluid velocity only varies with radial position. In the case of a cylindrical channel with flow along the axis, the velocity distribution is a simple quadratic, known as Hagen-Poiseuille or simply Poiseuille flow. The pressure drop is given by the following relation ... 

The most basic state of motion for fluid in a pipe is one in which the motion occurs at a constant rate, independent of time. The pressure flow relation for laminar, steady flow in round tubes is called Poiseuille s Law, after J.L.M. Poiseuille, the French physiologist who first derived the relation in 1840 [12]. Accordingly, steady flow through a pipe or channel that is driven by a pressure difference between the pipe ends of just sufficient magnitude to overcome the tendency of the fluid to dissipate energy through the action of viscosity is called Poiseuille flow. 

Table 1.1 contains two additional rate processes, which are driven by gradients. The first is Poiseuille s law, which applies to viscous flow in a circular pipe, and a similar expression, D Arcy s law, which describes viscous flow in a porous medium. Both processes are driven by pressme gradients and both vary inversely with viscosity, which is to be expected. 

Illustration of stationary flow types, (a) Plug flow at the entrance region of a pipe and (b) the Poiseuille flow in the fuUy developed flow region (c) Stokes flow around a sphere. 

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