Poiseuille flow model

Abnormahty in the stractuie and dynamics of water inside the sub-2 nm CNTs are often reported. To interpret the fast water movement, there is no physical basis to apply the Poiseuille flow equation which uses the bulk viscosity in solving the equation. One way of modifying the Poiseuille flow model is to incorporate the slip length the ratio of the translational velocity at the wall, v, to the characteristic... 

Fig. 3—Comparison of velocity profiles of plane Poiseuille flow between different models.

Fig. 7. Velocity profile for Poiseuille flow and shear flow. The points are the LBM data, and the solid lines are the theoretical profiles. For the simulations, we used the D3Q18 model with 25 lattice sites across the channel.

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... 

From this, the velocities of particles flowing near the wall can be characterized. However, the absorption parameter a must be determined empirically. Sokhan et al. [48, 63] used this model in nonequilibrium molecular dynamics simulations to describe boundary conditions for fluid flow in carbon nanopores and nanotubes under Poiseuille flow. The authors found slip length of 3nm for the nanopores [48] and 4-8 nm for the nanotubes [63]. However, in the first case, a single factor [4] was used to model fluid-solid interactions, whereas in the second, a many-body potential was used, which, while it may be more accurate, is significantly more computationally intensive. 

For relatively porous nanofiltration membranes, simple pore flow models based on convective flow will be adapted to incorporate the influence of the parameters mentioned above. The Hagen-Poiseuille model and the Jonsson and Boesen model, which are commonly used for aqueous systems permeating through porous media, such as microfiltration and ultrafiltration membranes, take no interaction parameters into account, and the viscosity as the only solvent parameter. It is expected that these equations will be insufficient to describe the performance of solvent resistant nanofiltration membranes. Machado et al. [62] developed a resistance-in-series model based on convective transport of the solvent for the permeation of pure solvents and solvent mixtures ... 

It is instructive to consider steady fluid flow (sometimes called Poiseuille flow) in a thin capillary tube. This example has many purposes it provides (1) a model flow calculation, (2) an illustration of how velocity profiles arise, (3) an explanation of the nature of flow in capillary chromatography, and (4) a foundation for capillary flow models of packed beds. 

Flow between parallel plates is modeled by Couette flow Flow in tubes, pipes and blood vessels is modeled by Poiseuille flow The moving plate is the driving factor in Couette flow The pressure gradient is the driving factor in Poiseuille flow. 

Similar results can be obtained for Couette or Poiseuille flows of several fluids in parallel layers these flows are important in particular in the modelling of coextrusion experiments. Le Meur [50] has studied the existence, uniqueness and nonlinear stability with respect to one dimensional perturbations of such flows. The behaviour of each fluid is governed by an Oldroyd model such as (16)-(17), where the nondimensional numbers Re and We are defined locally in each fluid. On the rigid top or bottom walls, the velocity is given—zero on both walls for Poiseuille flow, and zero or one depending on the wall for Couette flow. The interface conditions on the given interfaces are... 

The situation for the plane Poiseuille flow for Oldroyd models is not as simple, as shown by the following result. 

The constitutive equations are the Oldroyd-B model and a modified Oldroyd-B model in which the viscosity depends on the rate of strain. In [79], Laure et al. study the spectral stability of the plane Poiseuille flow of two viscoelastic fluids obeying an Oldroyd-B law in two configurations the first one is the two layer Poiseuille flow in the second case the same fluid occupies the symmetric upper and lower layers, surrounding the central fluid. (See Figure 9.)... 

The mass transfer in the boundary layers can be described by a mass transfer coefficient. In the membrane phase, the diffusion of water vapor can be described by either of the four mechanisms, namely molecular/Knudsen diffusion model, or Poiseuille flow, or by the dusty gas model (DGM). Heat transfer coefficients are used to describe the heat transfer in the boundary layer on either side of the membrane. In the membrane heat transfer occurs through the vapor and by conduction. These aspects have been explained in detail in the following sections. 

Resistance to mass transfer comes both from the membrane stmcture and the air present within the membrane pores. The resistance by the membrane structure (in the absence of air) can be described by Poiseuille flow. In presence of air within the membrane pores either Knudsen diffusion or molecular diffusion, or a combined Knudsen-molecular diffusion flow model can be used. 

The membrane constant a is the permeability divided by the membrane thickness evaluated at the reference pressure. The physical significance of a is that it represents the proportionality constant between flux and pressure drop at the reference pressure. The parameter b indicates the extent to which Poiseuille flow contributes to the permeability and lies between 0 and 1. Equation 19.30 is know as Knudsen-Poiseuille model, where is the dimensionless pressure defined as Pavg/ ref. the Pref is chosen in such a way that P becomes close to unity for the range of application of the equation. If a different pressure is assumed, Equation 19.30 may be modified retaining its original form, but having different values for the parameters a and b with the reference pressure as a new P close to unity. In such a case the new parameters a and b can be written as [59]... 

For solids with continuous pores, a surface tension driven flow (capillary flow) may occur as a result of capillary forces caused by the interfacial tension between the water and the solid particles. In the simplest model, a modified form of the Poiseuille flow can be used in conjunction with the capillary forces equation to estimate the rate of drying. Geankoplis (1993) has shown that such a model predicts the drying rate in the falling rate period to be proportional to the free moisture content in the solid. At low solid moisture contents, however, the diffusion model may be more appropriate. 

A porous medium is modeled as made up of uniformly distributed straight circular capillaries of the same diameter. The flow through each capillary is an inertia free Poiseuille flow. By comparing the Poiseuille pressure drop and the Darcy pressure drop formulas, deduce an expression for the permeability. Discuss the difference between the result obtained and the Kozeny-Carman permeability. 

Characterizing the porous bed by means of a capillary model of the interstitial space, the physical basis of the size separation procedure can be demonstrated through examination of the convection and Brownian diffusion of the colloidal particles in a liquid flow through a circular capillary. Figure 5.7.1 shows two freely-rotating spherical Brownian particles of different size sampling a nonuniform Poiseuille flow. The center of the larger particle in its travel... 

The Pore Flow Model uses the Hagen-Poiseuille Equation to describe solvent flow through cylindrical pores of the membrane. No membrane characteristics other than pore size or pore density are accounted for, and neither limitation of flux due to friction nor diffusion is considered. Flux occurs due to convection under an applied pressure. The equation is derived from the balance between the driving force pressure and the fluid viscosity, which resists flow (Braghetta (1995), Staude (1992)). Solvent flux ( ) is described by equation (3.26) and solute flux (Js) by equation (3.27), where rp is the pore radius, np the number of pores, T the tortuosity factor. Ax the membrane thickness and ct the reflection coefficient. 

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