Question

While studying "flow through pipes" we say that "as area of cross section decreases then velocity increases in turn pressure gets 'decreased'.( that is pressure and velocity are inversely proportional).However while studying "Hagen poiseuilli equation", we say "pressure and velocity has linear relation and "directly proportional" to each other. Why two different statements,? please help me to understand it.(Fluid mechanics)

Answer 1

Inside a pipe or in general inside a flow tube pressure and velocity change according to the Bernoulli equation that consists in the application of energy conservation principle to fluid flow. for any incompressible, non-viscous fluid, the relation between pressure and velocity is given by Bernoulli’s equation,

Where,

P is the pressure of the incompressible, non-viscous fluid .? is the density of the incompressible, non-viscous fluid .v is the velocity of the incompressible fluid.g is the acceleration due to gravity.h is the verticle height of the pipe from the reference level .

Pressure and velocity are inversely proportional to each others so when, pressure increases, the velocity decreases to keep the algebraic sum of potential energy, kinetic energy, and pressure constant. Similarly if velocity increases the pressure decreases to so as to keep the sum of potential energy, kinetic energy, and pressure constant also

For any flow between two points the fluid energy is represented in terms of total pressure or stagnation pressure. For an adiabatic flow without friction the stagnation pressure is constant. And since the stagnation pressure is composed of static and dynamic components, then any increase in dynamic pressure (velocity) is accompanied by a decrease in static pressure. This would occur only when the flow cross sectional area changes. Therefore the relation in adiabatic invicid flow is always inversely proportional.

The assumptions of the Hagen–Poiseuille equation are that the fluid is incompressible and Newtonian; the flow is laminar through a pipe of constant circular cross-section that is substantially longer than its diameter; and there is no acceleration of fluid in the pipe.

If the channel is straight (cross sectional area is constant) then nothing will affect the pressure or velocity if no friction or heat transfer occur.friction and heat transfer tend to change both static and dynamic pressure in a way that they change in proportional manner.