text: The hydrodynamic entrance region refers to the area of a pipe where fluid entering a pipe develops a velocity profile due to viscous forces propagating from the interior wall of a pipe. This region is characterized by a non-uniform flow. The fluid enters a pipe at a uniform velocity, then fluid particles in the layer in contact with the surface of the pipe come to a complete stop due to the no-slip condition. Due to viscous forces within the fluid, the layer in contact with the pipe surface resists the motion of adjacent layers and slows adjacent layers of fluid down gradually, forming a velocity profile. For the conservation of mass to hold true, the velocity of layers of the fluid in the center of the pipe increases to compensate for the reduced velocities of the layers of fluid near the pipe surface. This develops a velocity gradient across the cross-section of the pipe.