Local wall shear stress

For turbulent flow, we shall use the Chilton-Colburn analogy [12] to derive an expression for mass transfer to the spherical surface. This analogy is based on an investigation of heat and mass transfer to a flat plate situated in a uniform flow stream. At high Schmidt numbers, the local mass transfer rate is related to the local wall shear stress by... 

For turbulent flow, the local wall shear stress, xr, is given by Eq. (25). Substituting Eqs. (48H50) into Eq. (47) and making use of Eq. (25), one arrives at an expression for the Sherwood number based upon the radius of the rotating hemisphere ... 

Here A a represents the difference between the interfacial tension at the end and at the beginning of the path. When the refreshing of the elements of liquid is complete, A a is equal to the difference between the interfacial tension at the equilibrium concentration at the interface and the interfacial tension between the liquid phases at their bulk concentrations. The problem of the boundary layer that develops when a solid planar surface moves continuously was treated by several authors. Tsou et al. [115] have derived the following expression for the local wall shear stress r ... 

TwF being the local wall shear stress in forced convection. Dividing the above two equations then gives ... 

Whereas the circumferential variations of the local wall shear stress (i.e., the momentum flux) in itself are not of interest in the study of the BSR, the analogous variations in mass flux or surface concentration are indeed. In Ref. 15 a graph is presented of the local heat flux relative to the circumferential average, for the constant-temperature boundary condition, as a function of a and s/dp. These data are based on a semianalytical solution of the governing PDE, following the procedure described by Ref. 8 (see Section II.B.2). At a relative pitch of 1.2 the local flux at a = 0 is ca. 64% lower than the circumferential average at a relative pitch of 1.5 the flux at a = 0 is still ca. 20% lower than the circumferential average. In the case of a constant surface temperature, the local heat fluxes are directly proportional to the local Nusselt (or Sherwood) numbers. 

The ratio between the local wall shear stress and the flow kinetic energy per unit volume is defined as the local Fanning friction factor ... 

Micro- and Nanoscale Anemometry Implication for Biomedical Applications, Fig. 8 (a) An array of MEMS sensors embedded in a 3D bifurcation model, (b) Computational fluid dynamics (CFD) solutions for skin friction coefficient (Cf) at a Reynolds number of 6.7. Cf represents local wall shear stress values normalized by the... 

As shown above, in principle only one adjustable parameter for the Hassid-Poreh model is needed. This parameter k depends on the chain-length and concentration of the DR polymer and local wall shear stresses. The dependence is of the form k = f(c, M, rj, where c is active concentration of DRA, M is local apparent molar mass of DRA and r is local wall shear stress. Parameters are fitted against measured data and simulation results according to the following procedure ... 

Micro- and Nanoscale Anemometry Implication for Biomedical AppiicaUens, Rgure 8 (a) An array of MEMS sensors embedded in a 3D bifurcation model, (b) Computational fluid dynamics (CFD) solutions for skin friction coefficient (Cf) at a R nolds number of 6.7. Cj represents local wall shear stress values normalized by the upstream dynamic pressure. Cj values are shown along the interior surface of bifurcation, (c) Comparison of the CFD On blue), experimental On green), and theoretical On red) skin friction for the 180° edge. x/D/cos(12.5) is the x distance normalized to the diameter of the inlet pipe and parallel to the centerline of the outlet pipes... 

The Chilton-Colburn analogy can be also used to estimate the local mass transfer rate in laminar flow where the wall shear stress is related to the azimuthal velocity gradient by... 

The average t0 of rox over the path length x0 should be considered as the local turbulent shear stress at the wall. Hence... 

Many experiments have established that, as mentioned before, there is a region near the wall where the local turbulent shear stress depends on the wall shear stress and the distance from the wall alone and is largely independent of the nature of the rest of the flow. In this region, the mixing length increases linearly with distance from the wall except that near the wall there is a damping of the turbulence due to viscosity. In the wall region, it is assumed therefore that ... 

Obtain expressions for the local and mean values of the wall shear stress and friction factor (or drag coefficient) for the laminar boundary layer flow of an incompressible power-law fluid over a flat plate Compare these results with the predictions presented in Table 7.1 for different values of the power-law index. 

Transition from laminar to turbulent flow is identified by a departure from the laminar flow velocity profile and the presence of time-varying velocity component in the flow, especially near the wall. The wall shear stress becomes higher following the departure from the laminar flow. The transition occurs over a range of Reynolds number and is dependent on the local wall and flow conditions. Microchannels are defined as channels with the minimum channel dimension in the range from 1 to 200 pm [1]. 

The shear stresses over the flow boundaries can be rigorously derived as an integral part of the solution of the flow field only in laminar flows. The need for closure laws arise already in single-phase, steady turbulent flows. The closure problem is resolved by resorting to semi-empirical models, which relate the characteristics of the turbulent flow field to the local mean velocity profile. These models are confronted with experiments, and the model parameters are determined from best fit procedure. For instance, the parameters of the well-known Blasius relations for the wall shear stresses in turbulent flows through conduits are obtained from correlating experimental data of pressure drop. Once established, these closure laws permit formal solution to the problem to be found without any additional information. 

Rovinsky et al. obtained expressions for the local wall and interfacial shear stresses in terms of Fourier integrals [12]. It has been shown that the shear stress over the phases interface is not constant when the viscosity gap between the phases is large, the region where the interface meets the pipe wall is characterized by large variations of the interfacial shear as well as the wall shear stresses in both phases domains. Large variation of the liquid-wall shear stresses in the circumferential direction also were obtained in a recent experimental study by Paras et al. [63]. 

Y. Roh, Development at local global SAW season far measutemeal of wall shear stress in turbulent (lows. Pb.D. thesis. The Pennsylvania State University. 1990. 

A few numerical studies of flow and transport in the carotid artery bifurcation have been reported recently as summarized in Figure 12.4. The carotid artery bifurcation is a major site for the localization of atherosclerosis, predominantly on the outer wall (away from the flow divider) in the flow separation zone, which is a region of low and oscillating wall shear stress [26]. Perktold et al. [27] simulated Oj transport in a realistic pulsatile flow through an anatomically realistic three-dimensional carotid bifurcation geometry using a constant wall concentration boundary condition. Ma et al. [28] simulated... 

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the flow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects. 

Atherosclerosis, a disease of the vascular wall, is the substrate for the arterial forms of CVD. Atherosclerotic plaques exhibit a focal distribution along the arterial tree as a consequence of local conditions that favor their initiation and progression. Low or reversed shear stress, for example, contributes to plaque development, a process in which the regulation of several genes may be involved (Resnick and Gimbrone 1995). 

---