Fully Developed Flow
This concept refers to a flow regime in which the velocity profile does not change along the direction of flow, implying that all derivatives with respect to the streamwise coordinate are zero. In fully developed flow, the shape of the velocity profile remains constant and only the magnitude may change with pressure gradient, which simplifies the analysis of momentum balance.
Laminar Flow
Laminar flow is characterized by fluid particles moving in smooth, parallel layers with little or no mixing between them. It is dominated by viscous forces rather than inertial forces, leading to predictable, orderly flow behavior. This is crucial for analytical solutions, as it allows the use of simplified linear relationships between shear stress and rate-of-strain.
Control Volume Analysis
Control volume analysis is a method where a fixed or moving volume in space is used to apply conservation laws (mass, momentum, and energy) to a fluid flow. By considering the inflow, outflow, and surface forces, one can derive integral or differential equations that govern the behavior of the fluid, which is essential in deriving velocity profiles.
Pressure Forces
Pressure forces are exerted on the surfaces of the control volume by differences in fluid pressure at the inlet and outlet. In the context of flow through a pipe, these forces drive the flow and are balanced by viscous forces. Recognizing and quantifying these forces is fundamental when applying the momentum conservation equation.
Viscous Shear Stress
Viscous shear stress arises due to the fluid’s viscosity and manifests as a frictional force along the walls of the conduit. In pipe flows, it opposes the motion of the fluid and is balanced by the pressure driving force in fully developed laminar flow. This concept is central to deriving the parabolic velocity profile seen in Poiseuille flow.
Momentum Conservation Equation
The momentum conservation (or momentum balance) equation, derived from Newton’s Second Law for a control volume, relates the net force acting on the flow to the change in momentum. In the case of fully developed pipe flow, the equation balances the pressure force difference with the viscous shear forces, leading to a differential equation that can be solved for the velocity field.
Velocity Distribution in Pipe Flows
By applying the momentum conservation equation within a cylindrical control volume and accounting for both the pressure forces at the ends and the viscous shear stress at the wall, one can derive an expression for the velocity distribution. This results in a parabolic profile typical of laminar, fully developed flow in circular conduits, often referred to as Poiseuille flow.