Poiseuille-Stokes equation

Note that the Poiseuille-Stokes equation for an isotropic liquid (9.28b) or for a nematic would give us... 

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. 

See also -> convection, -> Grashof number, - Hagen-Poiseuille, -> hydrodynamic electrodes, -> laminar flow, - turbulent flow, -> Navier-Stokes equation, -> Nusselt number, -> Peclet number, -> Prandtl boundary layer, - Reynolds number, -> Stokes-Einstein equation, -> wall jet electrode. 

Our analysis shows that there is a steady, unidirectional flow solution of the Navier-Stokes equations for the Poiseuille flow problem for all flow rates. However, it says nothing about the stability of this solution. To examine this question, further analysis is necessary.7... 

In the present book, we focus our attention on the solution of the Navier-Stokes equations for laminar flows, frequently without any attempt to analyze the stability (or experimental realizability) of the resulting solutions. In using these solutions, it is therefore quite apparent that we must always reserve judgment as to the range of parameter values where they will exist in practice. We have already noted that experimental observation shows that Poiseuille flow exists for Reynolds numbers only less than a critical value. A general introduction to hydrodynamic stability theory is given in Chap. 12. It should be noted, however, that the stability of Poiseiulle flow is a very difficult problem, and only a short introductory section that is relevant to this problem is provided.7... 

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. 

Conveniently, when one considers the flow of simple fluids through channels, the Stokes equation - that describes the velocity field as a function of the pressure field - can be integrated to yield a very simple relation (Hagen-Poiseuille Law) between the difference (Ap) of pressures at inlet and outlet of the tube (channel) and the volumetric rate of flow (0 of the simple fluid ... 

This is the Stokes equation that has analytic solutions for certain types of simple laminar flows, for example the laminar flow in a straight channel known as Poiseuille flow (Fig. 1.2). Assuming that the flow is unidirectional along the x axis (vy = vz = 0) it follows from incompressibility that the velocity profile vx y) is uniform along the channel. When a pressure difference 5p is applied along a channel of length L and width d the Stokes equation reduces to... 

Note that in fact the plane Poiseuille flow (1.17) is also an exact solution of the full Navier-Stokes equation. However, it was shown by linear stability analysis that this becomes unstable to small perturbations at a critical Reynolds number of 5772. In fact, the transition to turbulence is observed experimentally at even lower values of Re around 1000. 

Figure 6.221 and the friction factor plot (Fig. 6.10) show marked similarities. In both at low Reynolds numbers there is a region where/or Q is proportional to. 0t or 1/, that is, a straight line of slope —45 on log paper. For pipes this is Poiseuille s equation, which can be written /=16/ for spheres (straight line on Fig. 6.22) it is Stokes law which can be written... 

The cylindrical coordinate form of the Navier-Stokes equations is shown in many textbooks, e.g., Bird et al. [8]. Starting with that form, derive the Poiseuille equation. 

To calculate the Poiseuille coefficient Gp in the slip flow regime (5 < 8 < 100) the Navier-Stokes equation is solved with the velocity slip boundary conditions (8). Then for the channel the flow rate reads... 

Problem By writing the Stokes equation (Navier-Stokes equation without the inertial term) in cylindrical coordinates, show that the velocity profile in the tube is parabolic.Designate by G the pressure gradient responsible for the transport of fluid (in the present problem, G is generated by the Laplace underpressure existing at the upstream interface), and set the velocity at the solid/liquid interface equal to zero. Under these conditions, deduce the average velocity in the tube (Poiseuille s law) and, from there, the viscous force F that opposes the progress of the fluid [equation (5.40)]. 

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 ... 

In Figure 1.7, the modifications brought about by gravity in the case of a plane Poiseuille flow ate depicted. The pipe is inclined within the gravity field. Navier-Stokes equations in directions Oy and now reduce to ... 

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). 

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 ... 

Far from the entrance range of the pipe, the fully developed flow is called Poiseuille flow, which is characterized by a parabolic velocity profile. To illustrate this, we consider the flow in x-direction of a viscous fluid in a channel. The channel has a width in the y-direction of a, length f (z a) in the z-direction, and a length f (x a) in the x-direction. There is a pressure drop Ap along f, so that the pressure gradient is constant (such a pressure gradient could be supplied by gravity, for instance). We assume the flow is steady ( do/dt), and the speed depends only on the y-direction v(f) = Vx(y). In this case, the nonlinear term in the Navier-Stokes equation (B.13) vanishes, and we are left with a simple equation for... 

Shortly before Stokes was putting the finishing touches on the Navier-Stokes equations, experiments were done that would prove to be exceptionally valuable both from the point of view of providing an experimental verification of the Navier-Stokes equations and from a purely empirical point of view. In 1839, Hagen described a series of experiments on laminar flow in capillary tubes, and in 1841, Poiseuille published his own independent studies of this problem. Poiseuille s more accurate results could be expressed as (Rouse and Ince, page 160, 1957)... 

Here, Q represents the volumetric flow rate, D the tube diameter, and L the tube length. Rouse and Ince (1957) suggest that Stokes was not aware of the data of Hagen and Poiseuille and it remained for Franz Neumann and Eduard Hagenback to independently derive Eq. 1-70 from the Navier-Stokes equations in the years 1858-1860. 

Doolittle s torsional viscosimeter was essentially a damped, oscillating Couette viscometer, and Doolittle chose as the unit of viscosity the "number of degrees of retardation between the first and second complete arcs." There is no mention In this paper of the Hagen-Poiseuille law, the Navier-Stokes equations, the treatise of Lamb (1879) or the work of Basset (1888), and it would appear that Doolittle was unaware of the studies of fluid mechanics carried out in the previous century and half. Nearly two decades later, Gillet (1909) began his paper entitled "Analysis and Friction Tests of Lubricating Greases" with the comment ... 

The flow of liquid through confined channels is given by the Stokes equation, which for cylindrical microchannels is described by the Hagen-Poiseuille equation ... 

Most of the microfluidic devices consist of microchannel of different dimensions interconnected with each other. In the case of two straight channels of different dimensions connected to form one long channel, the translation invariance, that is, fully developed flow assumption, is broken. Hence, the expressions for the ideal Poiseuille flow no longer apply. However, it is expected that the ideal description is approximately correct if the Reynolds number (Re) of the flow is sufficiently small. This is because a very small value of Re corresponds to a vanishing small contribution from the nonlinear term (V - V)F in the Navier-Stokes equation (N-S equation), a term that is strictly zero in ideal Poiseuille flows due to translation invariance. 

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