Wall stresses

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

A thin-walled pressure vessel is one in which the wall thickness t is small when compared to the local radius of curvature r. At a point in the wall of the vessel where the radius of curvature varies with the direction, the wall stresses are... 

A common pressure vessel application for pipe is with internal pressure. In selecting the wall thickness of the tube, it is convenient to use the usual engineered thin-wall-tube hoop-stress equation (top view of Fig. 4-1). It is useful in determining an approximate wall thickness, even when condition (t < d/10) is not met. After the thin-wall stress equation is applied, the thick-wall stress equation given in Fig. 4-1 (bottom view) can be used to verify the design (Appendix A PLASTICS DESIGN TOOLBOX). 

Excess fluid administration may increase right-ventricular (RV) wall stress, RV ischemia, tricuspid regurgitation, and cause a septal shift that may impair left-ventricular (LV) compliance and filling. Administer with caution in patients with documented severe RV dysfunction or when measured pressures are high... 

The gas-phase wall stress can be written in terms of the usual friction factor, C/G, to give... 

Fanning (Darcy) friction factor f(f or fD) e, D 2 V2L fo = 4f TW yv2 e, = friction loss (energy/mass) rw = wall stress (Energy dissipated)/ (KE of flow x 4L/D) or (Wall stress)/ (momentum flux) Flow in pipes, channels, fittings, etc. 

Reynolds number flows /vRe N -°Vp /vRe — pV2 pV/D AQp izDp PV2 Tw/8 Pipe flow rw =wall stress (inertial momentum flux)/ (viscous momentum flux) Pipe/internal flows (Equivalent forms for external flows)... 

As can be determined from Eqs. (5-47) and (5-49), the pipe wall stress can also be made dimensionless by dividing by the kinetic energy per unit volume of fluid. The result is known as the pipe Fanning friction factor, f ... 

Because the friction loss and wall stress are related by Eq. (5-47), the loss coefficient for pipe flow is related to the pipe Fanning friction factor as follows ... 

If the wall stress (rw) in Eq. (6-11) is expressed in terms of the Fanning friction factor (i.e., tw = /pF2/2) and the result solved for/, the dimensionless form of the Hagen-Poiseuille equation results ... 

Because these relations apply only in the vicinity of the wall, Prandtl also assumed that the eddy (Reynolds) stress must be of the same order as the wall stress, i.e.,... 

This is the power law equivalent of the Hagen-Poiseuille equation. It can be written in dimensionless form by expressing the wall stress in terms of the friction factor using Eq. (6-5), solving for /, and equating the result to 16/ N t (i.e., the form of the Newtonian result). The result is an expression that is identical to the dimensionless Hagen-Poiseuille equation ... 

The wall stresses are related to corresponding friction factors by... 

The action of nitrates appears to be mediated indirectly through reduction of MVo2 secondary to venodilation and arterial-arteriolar dilation, leading to a reduction in wall stress from reduced ventricular volume and pressure. Direct actions on the coronary circulation include dilation of large and small intramural coronary arteries, collateral dilation, coronary artery stenosis dilation, abolition of normal tone in narrowed vessels, and relief of spasm. 

Figure 3.18 Apparent shear rate as a function of the wall stress (tJ. The first derivative of the function is used to perform the Weissenberg-Rabinowitsch correction. The data are for the HDPE resin at 190°C as shown in Fig. 3.17...

When the rubber is heated from 253 K to 298 K, the E-modulus becomes 298/253 = 1.18 times higher. The wall stress in the balloon will increase by 18% and the diameter decreases to 20/1.18 = 17 cm. Gradually the air in the balloon will be heated up, and the pressure will increase proportionally to the absolute temperature. Eventually the old diameter of 20 cm will be reached. 

From this figure, safe wall stresses at various temperatures and loading times can be determined by extrapolation. We extrapolate the curves found at higher T to lower Ts. Here a ten years life (87,660 hours) at 80 °C is required. Extrapolation of the curve for 80 °C results in an estimated stress to break of 28 MPa. To remain at the safe side, we stay somewhat below this value, so that 20 MPa looks a reasonably safe wall stress for this purpose. 

Arteriolar and venous tone (smooth muscle tension) both play a role in determining myocardial wall stress (Table 12-1). Arteriolar tone directly controls peripheral vascular resistance and thus arterial blood pressure. In systole, intraventricular pressure must exceed aortic pressure to eject blood arterial blood pressure thus determines the systolic wall stress in an important way. Venous tone determines the capacity of the venous circulation and controls the amount of blood sequestered in the venous system versus the amount returned to the heart. Venous tone thereby determines the diastolic wall stress. 

It is conventional to take the Re/ product as a positive number, even though the wall stress (and hence the friction factor) are understood to point in the negative z direction. As stated, Eq. 4.68 produces a positive parabolic velocity profile with Re/ assumed to be a positive number. 

Evaluating the stress at the plates, the net wall stress is given as... 

Figure 5.16 shows the product Rej/ / as a function of Rev. For Rev less than approximately 2, the wall-injection has very little effect. In this case the wall friction approaches that of the Hagen-Poiseuille flow (i.e., Reyf = 24). For Rev greater than approximately 2, the V velocity serves to skew the axial velocity profile and thus affect the wall stress. For Rev greater than approximately 20, Re / approaches a linear relationship as... 

In this expression there is no flux from the north or south neighbors. Since the velocities are exactly zero along the walls, there are no velocity gradients along the walls. On the north solid wall, similar reasoning provides the following expression for wall stress ... 

A30 This cell contains the nondimensional wall stress associated with the quarter control volume in the lower-left-hand comer. Its value is specified as... 

Mann DL. Basic mechanisms of left ventricular remodeling the contribution of wall stress. J Card Fail. 2004 10(suppl 6) S202-S206. 

P.A. Langston, U. Tuzun, D.M. Heyes, Distinct element simulation of granular flow in 2D and 3D hoppers dependence of discharge rate and wall stress on particle interactions, Chem. Eng. Sci. 50 (1995) 967-987. 

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