Plane Poiseuille flow

Application of the momentum equation to ejectors of other types is discussed in Lapple (Fluid and Particle Dynamics, University of Delaware, Newark, 1951) and in Sec. 10 of the Handbook. 

Example 3 Venturi Flowmeter An incompressible fluid flows through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can be solved using the mechanical energy balance. In a well-made venturi, viscous losses are negligible, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. 

With control surfaces at 1 and 2 as shown in the figure, Eq. (6-17) in the absence of losses and shaft work gives 

The continuity equation gives V2 = ViAla, and Vi = Q/A. The pressure drop measured by the manometer is p —pz = (pm — p)gAz. Substituting these relations into the energy balance and rearranging, the desired expression for the flow rate is found. 

Example 4 Plane Poiseuille Flow An incompressible Newtonian fluid flows at a steady rate in the x direction between two very large flat plates, as shown in Fig. 6-8. The flow is laminar. The velocity profile is to be found. This example is found in most fluid mechanics textbooks the solution presented here closely follows Denn. 

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Velocity profiles of Plane Poiseuille Flow Continuum-------First-order slip Second-order slip - -... 

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

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. 

G.H. Evans and S. Paolucci. The Thermoconvective Instability of Plane Poiseuille Flow Heated from Below A Proposed Benchmark Solution for Open Boundary Flows. Int. J. Num. Meth. Fluids, 11 1001-1013,1990. 

K.S. Gage and W.H. Reid. The Stability of Thermally Stratified Plane Poiseuille Flow. J. FluidMech., 33(l) 21-32,1968. 

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. 

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.)... 

P. Laure, H. Le Meur, Y. Demay and J.-C. Saut, Linear stability of multilayer plane Poiseuille flows of Oldroyd-B fluids, Prepublication 96-12, Universite Paris-Sud, Mathematiques, submitted. 

This dimensional form of the solution emphasizes the fact that the flow is a linear combination of plane Poiseuille flow and linear shear flow, with the relative magnitude determined by the ratio of Gd2/yu to U. 

Formula (1.5.6) describes the parabolic velocity field in a plane Poiseuille flow symmetric with respect to the midplane X = h of the channel. 

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. 

When the velocity profile is known, the effective diffusivity can be calculated explicitly. For example, in a plane Poiseuille flow with a parabolic velocity profile u(y) = 4 Uy(L — y)/L2 the diffusivity is... 

K. C. Cheng, and R. S. Wu, Viscous Dissipation Effects on Convective Instability and Heat Transfer in Plane Poiseuille Flow Heated from Below, Appl. Set Res., (32), 327-346,1976. 

Fig. 3.6.9. (a) Ratio of transverse to longitudinal pressure gradient versus director orientation in plane Poiseuille flow of nematic MBBA. Sample thickness d = 200/m. Length of cell L = 4cm and lateral width 1=4cm. (b) E> ection f of flow lines with respect to the y axis versus in wide cells (L/l ). Full line represents theoretical variation. (From Pieranski and Guyon. )... 

In spiral, rectangular channels, cells and particles experience Dean drag forces in addition to inertial lift forces. The first conclusive research to analyze flow in curved channels was done by Dean in 1927 [10]. He showed that in curved channels/pipes, the plane Poiseuille flow is disturbed by the presence of centrifugal force (Fcf), and in this condition, the maximum point of velocity distribution shifts from the center of the channel toward the concave wall of the channel (Fig. 2a). This shift causes a sharp velocity gradient to develop near the concave wall between the point of maximum velocity and the outer concave channel wall where the velocity is zero. This causes decrease in the centrifugal force on the fluid near the concave wall which leads to... 

FENE potentials. A treatment of a plane Poiseuille flow similar to the plane Couette flow considered here is feasible. The influence of confining... 

Derivation of basic equations. Acta Astronautica 4, 1177 Sivashinsky, G. I. (1979) On self-turbulization of a laminar flame. Acta Astronautica 6, 569 Stewartson, K., Stuart, J. T. (1971) A non-linear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529... 

Stratonovich, R. L. (1967) Topics in the Theory of Random Noise (Gordon and Breach, New York) Stuart, J. T. (1960) On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part I The basic behavior in plane Poiseuille flow. J. Fluid Mech. 9, 353 Suzuki, R. (1976) Electrochemical neuron model. Adv. Biophys. 9, 115... 

Fluid Mechanics for Chemical Engineering 1.3. The plane Poiseuille flow... 

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