Parabolic Poiseuille solution

For a uniform pressure gradient (AP) along a tube we get the parabolic Poiseuille solution... 

We have just discussed several variations of the flow in ducts, assuming that there are no axial variations. In fact there well may be axial variations, especially in the entry regions of a duct. Consider the situation illustrated in Fig. 4.8, where a square velocity profile enters a circular duct. After a certain hydrodynamic entry length, the flow must eventually come to the parabolic velocity profile specified by the Hagen-Poiseuille solution. 

Figure 13 shows the axial velocity profiles computed from equation 124. One can observe the variation of the axial velocity with the normalized bed radius, DF1/2/2. When the normalized bed radius is zero, the axial velocity displays a parabolic profile that corresponds to the Hagen-Poiseuille solution. As the normalized bed radius increases, the axial velocity profile flattens. When the normalized bed radius is infinite, the axial velocity corresponds to a unidirectional flow (flat) profile. 

For infinite diluted solutions, Sc 1000, therefore Sd O.ldu. Consequently, Sd Su, and the distribution of velocities in the diffusion boundary layer may be determined independently of the appropriate hydrodynamic problem. As an example, consider stationary laminar flow of viscous incompressible liquid in a flat channel. It is known that at some distance from the channel entrance, the velocity profile changes to a parabolic Poiseuille profile (Fig. 6.2) [5]. 

Several geometries can be used to create velocity gradients in fluids for the computation of tj. The Navier-Stokes equation provides the basis for finding the relationship between tj, the geometry and applied forces. One of the most common arrangements is the capillary viscometer, for which the Poiseuille solution to the Navier-Stokes equation, which has a parabolic flow profile, is used ... 

Solution of Equation (8.63) for the case of constant viscosity gives the parabolic velocity profile. Equation (8.1), and Poiseuille s equation for pressure drop. Equation (3.14). In the more general case of /r = /r(r), the velocity profile and pressure drop are determined numerically. 

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.

The three boundary conditions still apply. The nearly trivial solution reveals that the velocity profile must take a parabolic form, which is the expected result for plane Poiseuille flow. 

Consider the flow of an incompressible fluid in the entry region of a circular duct. Assuming the inlet velocity profile is flat, determine the length needed to achieve the parabolic Hagen-Poiseuille profile. Recast the momentum equation in nondimensional form, where the Reynolds number is based on channel diameter and inlet velocity emerges as a parameter. Based on solutions at different Reynolds numbers, develop a correlation for the entry length as a function of inlet Reynolds number. 

Velocity Profiles In laminar flow, the solution of the Navier-Stokes equation, corresponding to the Hagen-Poiseuille equation, gives the velocity v as a function of radial position r in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity twice the average velocity, is shown in Fig. 6-10. 

Equation 9.11 is usually referred to as Poiseuille s law and sometimes as the Hagen-Poiseuille law. It assumes that the fluid in the cylinder moves in layers, or laminae, with each layer gliding over the adjacent one (Fig. 9-14). Such laminar movement occurs only if the flow is slow enough to meet a criterion deduced by Osborne Reynolds in 1883. Specifically, the Reynolds number Re, which equals vd/v (Eq. 7.19), must be less than 2000 (the mean velocity of fluid movement v equals JV, d is the cylinder diameter, and v is the kinematic viscosity). Otherwise, a transition to turbulent flow occurs, and Equation 9.11 is no longer valid. Due to frictional interactions, the fluid in Poiseuille (laminar) flow is stationary at the wall of the cylinder (Fig. 9-14). The speed of solution flow increases in a parabolic fashion to a maximum value in the center of the tube, where it is twice the average speed, Jv. Thus the flows in Equation 9.11 are actually the mean flows averaged over the entire cross section of cylinders of radius r (Fig. 9-14). 

The solution of this equation under the no-slip condition on the surface of a tube of radius a (V = 0 for Tl = a) describes an axisymmetric Poiseuille flow with parabolic velocity profile ... 

The sixth reactor design criterion requires that the pressure drop at the minimum residence time be less than 100 psi. For a small diameter channel, the flow through that channel wiU be laminar for all flow rates of interest for this particular applicahon. Neglecting end effects, the solutions to the equations of continuity and of motion for steady-state laminar flow of an incompressible Newtonian fluid are well-known, yielding a parabolic velocity distribution and the Hagen-Poiseuille equahon for pressure drop, as given in Eqs. (9) and (10) ... 

The velocity profile described by this solution has the familiar parabolic form known as Poiseuille flow (Figure 7.1). The velocity at the wall (r = a) is clearly zero, as required by the no-sHp condition, while as expected on physical grounds, the maximum velocity occurs on the axis of the tube (r = 0) where Mjjjax = Kfl /4/u. At any position between the wall and the tube axis, the velocity varies smoothly with r, with no step change at any point. 

In long cylindrical pores, a pressure drop sets up a Poiseuille flow (parabolic velocity profile) and particles suspended in the solution are transported through the pore with transit times depending on the radial positions at which the particles enter the pore. In Poiseuille flow, solid spheres migrate to an off-axis... 

Initial collection of acetophenone and of various coloured dye tracers indicated that when using a stainless steel cylinder (10 mm x 250 mm, 20 mL volume) as the collection vessel the total volume of the cylinder was not available for the solute. The dye experiments indicated that about half the total volume was available for collection of the solute. The reasons for this are as follows. The Hagen-Poiseuille equation, which describes the velocity profile of a liquid flowing through a tube under laminar flow conditions predicts that the maximum velocity of the head of the parabolic flow profile will be twice the mean velocity of the liquid [23]. Breakthrough of the dye tracer would therefore be expected in about half the volume of the tube. Fluid mechanics predicts that if the flow is turbulent the difference between the mean and maximum flow velocities is much less than in laminar flow [23]. This means that un-... 

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