Circular tube Poiseuille flow

For laminar flow in a circular tube of radius R, the pressure gradient is given by a differential form of the Poiseuille equation ... 

The key analysis of hydrodynamic dispersion of a solute flowing through a tube was performed by Taylor [149] and Aris [150]. They assumed a Poiseuille flow profile in a tube of circular cross-section and were able to show that for long enough times the dispersion of a solute is governed by a one-dimensional convection-diffusion equation ... 

Laminar flow In circular tubes with parabolic velocity distribution Is known as Poiseuille flow. This special case is found frequently in vacuum technology. Viscous flow will generally be found where the molecules mean free path is considerably shorter than the diameter of the pipe X d. 

Next we consider a fluid flowing through a circular tube with material at the wall diffusing into the moving fluid. This situation is met with in the analysis of the mass transfer to the upward-moving gas stream in wetted-wall-tower experiments. Just as in the discussion of absorption in falling films, we consider mass transfer to a fluid moving with a constant velocity profile and also flow with a parabolic (Poiseuille) profile (see Fig. 5). 

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

Flow of a Power Law Fluid in a Straight Circular Tube (Hagen-Poiseuille Equation)... 

Poiseuille flow of a Newtonian fluid in a circular tube. For a pressure driven flow of a Newtonian fluid in a circular tube, we can obtain an analytical solution as we already did in Chapter 5. Ignoring the entrance effects, the solution for the velocity field as a function of the radial direction (see Fig. 10.18) is as follows... 

Figure 3-4. A pictorial representation of the force balance on the fluid within an arbitrarily chosen section of a circular tube in steady Poiseuille flow. Pressure forces act on the two cross-sectional surfaces at z = 0 and z = L, while a viscous stress acts at the cylindrical boundary and exactly balances the net pressure force.

In this section we consider a second problem involving heat transfer in Poiseuille flow through a circular tube. In this case, we assume that the fluid in the region — 8 < z < 8 is initially heated to a temperature 0, while the temperature elsewhere, i.e., z > 8, is held at a constant temperature 60. The wall of the tube is insulated for all z so that there is no heat loss or gain to the surroundings. Precisely the same problem could be formulated as a mass transfer problem for the redistribution of a solute in a solvent with an initial solute concentration C for —5 < z < 5 and concentration Co for z > 8, with tube walls that are impermeable to the solute. The only difference is that the thermal diffusivity k is replaced with the species diffusion coefficient/). However, to make the discussion as straightforward as possible, the analysis in this section is presented as a heat transfer problem. 

As a starting point, we recall that the limit a/R = 0 corresponds to a straight circular tube, with the flow described by the Poiseuille flow solution w = (1 — r2), u = v = 0. In the present context, we consider small, but nonzero, values of a/R, and recognize the Poiseuille flow solution as a first approximation in an asymptotic approximation scheme. In particular, if we assume that a solution exists for u in the form of a regular asymptotic expansion,... 

Viscosity can also be determined by measuring the total pressure drop (AO = AP + pgAz) and flow rate (Q) in steady laminar flow through a uniform circular tube of length L and diameter D (this is called Poiseuille flow). The shear stress at the tube wall (xj is determined from the measured pressure drop ... 

Statement of the problem. Let us consider laminar steady-state fluid flow in a circular tube of radius a with Poiseuille velocity profile (see Section 1.5). We introduce cylindrical coordinates 1Z, Z with the Z-axis in the direction of flow. We assume that for Z > 0 the temperature on the wall is equal to the constant T2. In the entry area Z < 0, the temperature on the wall is also constant but takes another value T. ... 

Let us briefly consider convective mass transfer accompanied by a surface reaction in a circular tube. Laminar steady-state fluid flow in a circular tube of radius a with Poiseuille velocity profile is outlined in Subsection 1.5-3. For... 

By substituting (6.4.8) into (6.4.4)-(6.4.7), one can find the basic characteristics of motion of a power-law fluid in a circular tube. The results of the corresponding calculations [452, 508] are presented in Table 6.5 and are shown in Figure 6.2. One can see that the velocity profiles become more and more filled as the rheological parameter n decreases. The limit case n -> 0 is characterized by a quasisolid motion of the fluid with the same velocity in the entire cross-section of the tube (it is only near the wall that the velocity rapidly decreases to zero). The parabolic Poiseuille profile corresponds to the Newtonian fluid (n = 1). The limit dilatant flow (n — oo) has a triangular profile, which is characterized by a linear law of velocity variation along the radius of the tube. 

PoiseuilWs Law Poiseuille flow is the steady flow of incompressible fluid parallel to the axis of a circular pipe or capillary. Poiseuille s law is an expression for the flow rate of a liquid in such tubes. It forms the basis for the measurement of viscosities by capillary viscometry. 

In the case of Poiseuille flow in a circular tube, Uj, = 2V (1 — j3 ). The vector V is equal in magnitude to the superficial velocity and points in the direction of net flow. 

Unless the contrary is explicitly stated, the following discussion of experimental and theoretical results is restricted to single, rigid, spherical particles freely suspended in a Poiseuille flow within a circular tube of effectively infinite length. Notation is as follows a = sphere radius 7 = tube radius (/ was used previously for this quantity) b = radial distance from tube axis to sphere center p = b/R = fractional distance from axis b = stable equilibrium distance of sphere from tube axis = b jR p = fluid density Pp = particle density p = viscosity v = pfp = kinematic viscosity. All velocities defined below are measured relative to the fixed cylinder walls V = mean velocity of flow vector (equal in magnitude to the superficial velocity and pointing parallel to tube axis in the direction of net flow) U = particle velocity vector—that is, the velocity of the sphere center (o = angular velocity of the sphere. The local velocity in the unperturbed Poiseuille flow is... 

In a direct attack upon the radial migration problem in essentially its full generality, Cox and Brenner (C18) succeeded in obtaining a first-order solution of the Navier-Stokes and continuity equations for the motion of a rigid spherical particle immersed in a Poiseuille flow within a circular tube of finite radius. No couple acts on the particle, so it is free to rotate. It is presumed in the analysis that the sphere center moves parallel to the tube axis. The lateral force required to maintain the sphere at a fixed distance from the axis is computed and converted into an equivalent radial migration velocity by application of Stokes law to this sidewise motion. 

We shall next consider the rigorous theory of Poiseuille flow, i.e., the steady laminar flow, caused by a pressure gradient, of an incompressible fluid through a tube of circular cross-section.< > We shall suppose that the tube is of infinite length so that end effects can be ignored. Let us choose a cylindrical polar coordinate system r(pz with z along the axis of the tube. In the steady state the only component of the velocity gradient is = dv/dr. It is natural to expect that the director is everywhere in the rz plane... 

As mentioned above, the first experimental work in which water through glass microtubes with an inner diameter down to 15 pm was tested was due to Poiseuille in 1840 [2]. The experimental data obtained by Poiseuille were used to state the famous Poiseuille law for laminar flows through circular tubes, according to which for Stokes flows the product of the frictirMi factor times the Reynolds number is a constant (equal to 16 for circular tubes) that depends on the cross-sectional geometry only. 

Because of the small diameters of microchannels, laminar flow can be assumed. The radial velocity profile in a single channel develops from the entrance to the position where a complete Poiseuille profile is established. The length ofthe hydro-dynamic entrance zone (Lg) in a circular tube depends on the Re (= pud /n) and can be estimated from the following empirical relation [51, 52] ... 

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... 

Hagen-Poiseuille equation (Poiseuille equation) n. The equation of steady, laminar, Newtonian flow through circular tubes ... 

Poiseuille flow n. Laminar flow in a pipe or tube of circular cross-section under a constant pressure gradient. If the flowing fluid is Newtonian, the flow rate will be given by the Hagen-Poiseuille equation. 

Capillary viscometers are the most extensively used instruments for the measurement of viscosity of liquids because of their advantages of simphcity of construction and operation. Both absolute and relative instruments were constracted. The theory of these viscometers is based on the Hagen-Poiseuille equation that expresses the viscosity of a fluid flowing through a circular tube of radius r and length L in dependence of the pressure drop AP and volumetric flow rate Q, corrected by terms for the so-called kinetic-energy and end corrections ... 

Clearly, Eq. (12) reduces to the analog of the Hagen-Poiseuille formula for the power-law fluid [i.e., Q = nnR /(3n+ lXAP7 /2mL) "] on noting that pressure flow in circular tubes corresponds to the limit of k and X both tending to zero. 

If a pressure variation AP = P+ -P-, is applied between the inlet and outlet of a circular cylindrical tube of length / and diameter D, the flow rate that passes through the tube is given by the solution of the Poiseuille flow for a fluid of dynamic viscosity ju ... 

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