Shear rate at the tube wall

The corresponding shear rate at the tube wall (yw) is given by... 

This can be solved for the shear rate at the tube wall (yw) by first differentiating Eq. (6-92) with respect to the parameter rw by application of Leibnitz rule to give... 

The Hagen-Poiseuille equation [Eq. (6-11)] describes the laminar flow of a Newtonian fluid in a tube. Since a Newtonian fluid is defined by the relation r = fiy, rearrange the Hagen-Poiseuille equation to show that the shear rate at the tube wall for a Newtonian fluid is given by yw = 4Q/nR3 = 8 V/D. 

The measured values of polymer flow taken by capillary rheometers are often presented as plots of shear stress versus shear rate at certain temperatures. These values are called apparent shear stress and apparent shear rate at the tube wall. Corrections must be applied to these values in order to obtain true values. The corrected value of shear stress is determined by the Bagley correction [20]... 

Attention is drawn to the fact that the values of m and n for use in turbulent region are deduced from the data in the laminar range at the values of (SVp/D) which is only the nominal shear rate at the tube wall for streamline flow, and thus this aspect of the procedure is completely empirical. Dziubinski [1995] stated that equation (4.26) reproduced the same experimental data as those referred to earlier with an average error of 15%, while equation (4.28) correlated the turbulent flow data with an error of 25%. Notwithstanding the marginal improvement over the method of Dziubinski and Chhabra [1989], it is reiterated here that both methods are of an entirely empirical nature and therefore the extrapolation beyond the range of experimental conditions must be treated with reserve. 

Only for a Newtonian fluid, n = 1, is F equal to the true shear rate at the tube wall 7, For an n of a reasonable value for a polymer melt or solution, the correction term is 1.5, so the apparent shear rate F is 50% lower than the true shear rate at the tube wall Rabinowitsch correction accounts for the fact... 

The power of this technique is two-fold. Firstly, the viscosity can be measured over a wide range of shear rates. At the tube center, symmetry considerations require that the velocity gradient be zero and hence the shear rate. The shear rate increases as r increases until a maximum is reached at the tube wall. On a theoretical basis alone, the viscosity variation with shear rate can be determined from very low shear rates, theoretically zero, to a maximum shear rate at the wall, yw. The corresponding variation in the viscosity was described above for the power-law model, where it was shown that over the tube radius, the viscosity can vary by several orders of magnitude. The wall shear rate can be found using the Weissen-berg-Rabinowitsch equation ... 

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 ... 

Actually, Metzner and Reed used an alternative formulation. Since the shear stress at the tube wall is a unique function of the apparent shear rate for laminar flow in cylindrical tubes, the power law may be written at the tube wall with the aid of (16.24) as... 

For the fully developed isothermal flow of an incompressible fluid in a pipe of radius R and length L where the pressure drop along the length of the tube is AP, the shear stress distribution is given by (AF/L)(r/2) hence, the shear stress at the tube wall is given by this equation with the radial position r = R. Also, the apparent shear rate, y, is given by 4Q/(ttR ), where Q is the volumetric flow rate, which can be calculated from the fluidity data from... 

The shear rate (velocity gradient) at the tube wall is obtained by differentiating equation 3.134 with respect to s, and then putting s = r. 

A velocity profile u(r) is obtained using MRI flow imaging and, with respect to radial position r, the values of shear rate y(r), ranging from zero at the tube center to a maximum at the tube wall, can be calculated from the velocity profile as local velocity gradients ... 

The second subscript N is a reminder that this is the wall shear rate for a Newtonian fluid. The quantity (8u/d,), or the equivalent form in equation 3.13, is known as the flow characteristic. It is a quantity that can be calculated for the flow of any fluid in a pipe or tube but it is only in the case of a Newtonian fluid in laminar flow that it is equal to the magnitude of the shear rate at the wall. 

A somewhat more general approach which has been shown to be at least partially useful for the latter two problems as well as for correlation of laminar-flow data is that of Metzner and Reed (Mil). Their work is based upon an equation developed by Rabinowitsch (Rl) and Mooney (M15) for the calculation of shear rates at the wall of a tube or pipe ... 

A study of the flow of a polyhedral foam in a regime of slip at the tube walls has been conducted [39]. It has been established that the rise in the dynamic viscosity of the foaming solution leads to diminishing the flow rate but to a much lesser extent at t0 = 1.25 Pa. Thus, a two fold increase in viscosity causes a 1.3 times decrease in the flow rate, while a 6 times increase in the dynamic viscosity only a 2.23 times decrease. This is probably related to the expanding of the effective thickness of the liquid layer 8 (ca. 3 times). The transition from plug flow (slip regime) to shear flow occurs at To = 9-10 Pa. This value of the shear stress is much smaller than the one obtained from Princen s formula for a two-dimensional foam (Eq. (8.18)) at a given expansion ratio and correlates well with To calculated from Eq. (8.24) and the experimental data of Thondavald and Lemlich [23],... 

The capillary tube viscometer is a flow-through-restriction type. When used carefully, it is capable of accuracies of better than 2 percent over its applicable shear-rate range (300 to 4000 s 1). For laminar flow of a nonnewtonian liquid in a capillary tube, it can be shown [4] that the wall shear stress xw and the shear rate at the wall % are given by... 

In such a viscometer, the motion is rectilinear and everywhere parallel thus, the shear rate is equal to the velocity gradient. In principle, this could be varied by applying various pressure differences over the tube. However, the most serious disadvantage for studying systems of any rheological complexity is that the shear rate varies from zero at the tube center to a maximum at the tube wall and is given by, for a Newtonian fluid,... 

For shear-thinning fluids, the apparent shear rate at the wall is less than the true shear rate, with the converse applying near the centre of the tube [Laim, 1983]. Thus at some radius, x R, the true shear rate of a fluid of apparent viscosity /r, equals that of a Newtonian fluid of the same viscosity. The stress at this radius, is independent of fluid properties and thus the true viscosity... 

The flow of viscoplastic fluids through beds of particles has not been studied as extensively as that of power-law fluids. However, since the expressions for the average shear stress and the nominal shear rate at the wall, equations (5.41) and (5.42), are independent of fluid model, they may be used in conjimction with any time-independent behaviour fluid model, as illuslrated here for the streamline flow of Bingham plastic fluids. The mean velocity for a Bingham plastic fluid in a circular tube is given by equation (3.13) ... 

As seen in Chapter 3, the quantity (8V/D) is the nominal shear rate at the wall (also see equation 5.35, for a circular tube, Dh = D, Kq = 2). Substituting for the nominal shear rate and wall shear stress from equations (5.41) and (5.42) in equation (5.48), slight re-arrangement gives ... 

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