Entrance Region and Fully Developed Flow

When fluid enters a pipe, it first When fluid enters a pipe, it first passes through the entrance region before reaching a state of fully developed flow. This progression is essential for understanding pressure drop and velocity profiles in fluid systems:

Entrance Region

As fluid enters the pipe, the velocity profile is initially flat (uniform velocity across the pipe’s cross-section). A boundary layer begins to form along the pipe walls, where velocity is influenced by viscous force. This boundary layer grows thicker as the fluid moves downstream, causing the central region (inviscid core) to shrink.

Fully Developed Flow

Eventually, the boundary layers from opposite walls merge, and the velocity profile becomes stable and parabolic for laminar flow. At this stage, the velocity distribution does not change with further distance along the pipe (hence fully developed). For turbulent flow, the velocity profile is flatter in the center but still stabilized.

The length required for flow to become fully developed is called the Entrance Length $l_e$, given by

$$l_e=f(\text{Re})=\{\begin{array}{ll}0.06\cdot \text{Re}& \text{(laminar)}\\ 4.4\cdot (\text{Re}{)}^{1/6}& \text{(turbulent)}\end{array}$$

Boundary Layer Development

The boundary layer thickness grows as fluid moves downstream, and its thickness is approximated by:

$$\delta (x)\sim {\text{Re}}_x=\frac{\rho Vx}{\mu }$$

where ${\text{Re}}_x$ is the local Reynolds number, which depends on the distance traveled in the pipe. In this region, viscous effects are important for shaping the velocity profile.

Shear Stress & Pressure

The shear stress ($\tau$) in the fluid is represented as a function of velocity gradient and fluid viscosity:

$$\tau =\mu \frac{du}{dy}$$

• $\mu$ is the fluid velocity
• $y$ is the perpendicular distance from the pipe wall.
• In the fully developed region, shear stress remains constant along the pipe length.

For fully developed laminar flow:

• The velocity in the axial direction is uniform ($V=u\mathbf{i}$).
• The change in velocity with respect to the axial direction $(\frac{\partial u}{\partial x}=0$) is zero, signifying steady, streamlined motion.