Entrance region velocity boundary layer

The velocity profiles shown in Figure 3.2.22 do not represent the situation at the entrance of the tube, and we have to examine the entrance region more closely. Consider a fluid entering a circular tube with a uniform radial velocity profile. The fluid layer directly in contact with the surface of the tube will come to a complete stop. This layer causes the fluid in the adjacent layers to slow down gradually as a result of friction. This leads to an increased velocity in the midsection of the tube to keep the mass flow through the tube constant, and thus a velocity boundary layer develops along the tube. The thickness of this layer increases in flow direction until the layer reaches the tube center and fills the entire tube (Figure 3.2.23). 

The hydrodynamic entrance region of a microchannel is that region where the velocity boundary layer is developing from zero thickness at the entrance up to cover the whole cross-section far downstream. Usually, this region coincides with the part of the channel near the inlet section. The flow in this region is designated as hydrodynamically developing (Fig. 1). 

When the Reynolds number based on tube diameter is greater than 2100, the boundary layer becomes turbulent at some distance from the inlet. The transition usually occurs at a Reynolds number, based on distance from the entrance, Rcj, of between 10 and 10, depending on the roughness of the wall and the level of turbulence in (he mainstream. As shown in Fig, 4,11, the deposition rate tends to follow the development of the turbulent boundary layer. No deposition occurs until Re is about 10- the rate of deposition then approaches a constant value at Re = 2 x 10 in the region of fully developed turbulence. On dimensional ground.s. the deposition velocity at a given pipe Reynolds number can be assumed to be a function of the friction velocity, if, kinematic viscosity, v, and the particle relaxation time, m/f ... 

Transition length for laminar and turbulent flow. The length of the entrance region of the tube necessary for the boundary layer to reach the center of the tube and for fully developed flow to be established is called the transition length. Since the velocity varies not only with length of tube but with radial distance from the center of the tube, flow in the entrance region is two dimensional. 

Simultaneously developing flow is fluid flow in which both the velocity and the temperature profiles are developing. The hydrodynamic and thermal boundary layers are developing in the entrance region of the duct. Both the friction factor and Nusselt number vary in the flow direction. Detailed descriptions of fully developed, hydrodynamically developing, thermally developing, and simultaneously developing flows can be found in Shah and London [1] and Shah and Bhatti [2],... 

Figure 3.24 Development of the boundary layer and velocity profile for laminar flow in the entrance region of a pipe...

If the velocity profile at the entrance region of a tube is flat, a certain length of the tube is necessary for the velocity profile to be fully established. This length for the establishment of fully developed flow is called the transition length or entry length. This is shown in Fig. 2.10-6 for laminar flow. At the entrance the velocity profile is flat i.e., the velocity is the same at all positions. As the fluid progresses down the tube, the boundary-layer thickness increases until finally they meet at the center of the pipe and the parabolic velocity profile is fully established. 

For pipe flow, velocity and temperature profiles in a developing and fully developed region are illustrated in Figure 22.9. The entrance length where the merging of the momentum boundary layer access is given by... 

Qiu assumed uniform axial velocity and concentration profiles (with no secondary flow) at the curved tube entrance (0°), there is an entrance region ( 25°) where Sh essentially follows a L v que boundary layer development Eventually ( 50°), the secondary flow effects become manifest, and marked differences in transport rates between the inside wall (low transport Sh 2) and the outside wall (high transport Sh 55) develop. The regions of high and low transport in the curved vessel geometry cannot be associated with axial flow separation (as in the expansion, stenosis, and bifurcation) because flow separation does not occur at the modest curvature levels in the coronary artery simulation. 

At some distance away from the entrance, the boundary layers meet and flow is assumed as viscous over the entire cross section of the channel. The internal flow is categorized into two distinct regions (i) hydrodynamic entrance region where velocity profile varies with the axial length of the channel and (ii) hydro-dynamic fully developed region where velocity profile remains invariable with the longitudinal distance along the channel, or becomes fully developed. 

---