Hagen/Poiseuille flow

Hafnium metal, analysis of, 13 87 Hafnium nitride, 13 89—90, 93 Hafnium oxide, 13 89, 93—94 reduction of, 73 84 Hafnium sulfides, 73 94 Hafnium tetrabromide, 73 93 Hafnium tetrachloride, 73 92 26 631 vapor reduction of, 73 84-85 Hafnium tetrafluoride, 73 90, 91 Hafnium tetrahydridoborate, 73 90 Hagen-Poiseuille expression/law, 27 726, 729 Hagen-Poiseuille flow, in microfluidics, 26 961... 

Fig. 4.6 Force balance on a differential control volume for Hagen-Poiseuille flow in a circular tube.

From the governing equation itself, it is clear that the sinusoidal contribution to the pressure gradient averages to zero over a full cycle, leaving the average Hagen-Poiseuille flow. 

Fig. 4.10 Instantaneous nondimensional velocity profiles in a circular duct with an oscillating pressure gradient. The mean velocity, averaged over one full period, shows that the parabolic velocity profile or the Hagen-Poiseuille flow. These solutions were computed in a spreadsheet with an explicit finite-volume method using 16 equally spaced radial nodes and 200 time steps per period. The plotted solution is that obtained after 10 periods of oscillation.

It is interesting to note several special cases of the system. First, it may be recognized that system approaches the parallel-plate Hagen-Poiseuille flow in the limit that V - 0 and A V -> 0. In this case both Reynolds number terms are eliminated from the momentum equation. Furthermore, since W — C/, only the U velocity remains in the pressure-gradient eigenvalue K. The momentum equation is simplified to... 

It follows that net wall shear stress for the parallel-plate Hagen-Poiseuille flow is characterized as... 

Figure 5.16 shows the product Rej/ / as a function of Rev. For Rev less than approximately 2, the wall-injection has very little effect. In this case the wall friction approaches that of the Hagen-Poiseuille flow (i.e., Reyf = 24). For Rev greater than approximately 2, the V velocity serves to skew the axial velocity profile and thus affect the wall stress. For Rev greater than approximately 20, Re / approaches a linear relationship as... 

Show that the limiting case of flow in a planar wedge for a < C 1 (Section 5.2.4) degenerates into the parallel-plate Hagen-Poiseuille flow. Show that the parabolic velocity profile is recovered and that the pressure gradient approaches a constant. 

Tube flow is encountered in several polymer processes, such in extrusion dies and sprue and runner systems inside injection molds. When deriving the equations for pressure driven flow in tubes, also known as Hagen-Poiseuille flow, we assume that the flow is steady, fully developed, with no entrance effects and axis-symmetric (see Fig.5.13). 

Using the nomenclature presented in Fig. 6.12 and assuming a land thickness of h we can assume the land length to be described by slit flow and the manifold by the Hagen-Poiseuille flow with a variable radius as... 

Using eqn. (6.219) and the analytical solution for Newtonian Hagen-Poiseuille flow given by... 

Assuming a Hagen Poiseuille flow, we can write the following equations for the 4 runner system sections... 

Hydrodynamic, fully developed, tubular flows were first studied by Hagen and Poiseuille. The flow is therefore also called Hagen-Poiseuille flow. We can see from (3.223), that the velocity at the tube axis is twice the mean velocity, wx(r = 0) = 2w m. From (3.222) we get the following for the pressure drop... 

This problem was first dealt with by Graetz (1850-1891) in 1883 [3.27], later in 1910 by Nusselt (1882-1957) [3.28] and by many other authors. It is also known as a Graetz or Graetz-Nusselt problem. It is described by the energy equation (3.232), in which, according to the suppositions made, the radial velocity component disappears, wr = 0, and the axial velocity is that of a Hagen-Poiseuille flow (3.223). With that the energy equation becomes... 

N.A. Mortensen, F. Okkels, and H. Bruns, Reexamination of Hagen-Poiseuille flow Shape dependence of the hydraulic resistance in microcharmels, Physical Review E, 71, (2005). 

D. K. Hennecke, Heat Transfer by Hagen-Poiseuille Flow in the Thermal Development Region with Axial Conduction, Warme-Stoffiibertrag., (1) 177-184,1968. 

In the limiting case of F = 0, it can be shown that equation 118 reduces to the Hagen-Poiseuille flow equation, that is,... 

Calculate the residence-time distribution (RTD) for a tubular reactor undergoing steady, laminar flow (Hagen-Poiseuille flow). The velocity profile for Hagen-Poiseuille flow is 2, p. 51]... 

This parabolic dependence of local velocity on radical position (Figure 6.4a) is characteristic of Hagen-Poiseuille flow. ... 

The more or less laminar flow condition in column chromatography causes a Hagen-Poiseuille flow profile, resulting in peak broadening (see Fig. 35-1 a). The quest is to limit the development of this flow profile. 

You can inject guanidine or urea with your sample. In this case, the high viscosity of the plug injected in front of the sample limits the development of a Hagen-Poiseuille flow profile. The sample will elute in a smaller peak volume, peak concentration increases and the peak height increases too (see Fig. 35-lb). In a reversed-phase system, urea or guanidine elute with the front and do not disturb the separation. 

Rajagopal, K.R., Saccomandi, G., Veigori, L. Linear stability of Hagen-Poiseuille flow in a chemically reacting fluid. Comput. Math. Appl. 61(2), 460-469 (2011)... 

The simplest model of steady laminar flow in a uniform circular cylinder is known as the Hagen-Poiseuille flow. For axisymmetric flow in a circular tube of internal radius Rq and length I, the boundary conditions are... 

Now, if in addition to the potential difference a pressure difference Ap = (p at z = L) - (p at z = 0) is also applied, then a term corresponding to Hagen-Poiseuille flow of - Apa /8pL is added to the right-hand side in Equation V.lO.vi. For zero flow rate, what should be the pressure drop Show that with decreasing values of radius a, the pressure drop increases more rapidly than the concomitant potential drop. Often it is Ap that is imposed and AK called the streaming potential, is measured. 

The mean free path of molecules in air at atmospheric pressure is /free — 1 /(Niiyg), where Nl 2.69 10 cm is the number density of gas molecules and cTg 10 " cm is the cross section for elastic collisions of molecules. These numbers result in /free — 3.7 10 cm, or 37 nm. The mean pore radius of the GDL is in the order of 10 pm, which means that the flow in the GDL pores occurs in a continuum regime. Thus, pressure-driven oxygen transport in a dry porous GDL can be modeled as a viscous Hagen-Poiseuille flow in an equivalent duct. However, determination of the equivalent duct radius and the dependence of this radius on the GDL porosity is a nontrivial task (Tamayol et al., 2012). Much workhas recently been done to develop statistical models of porous GDLs and to calculate viscous gas flows in these systems using Navier-Stokes equations (Thiedmann et al., 2012). 

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