From the velocity gradient equation above, and using the empirical velocity gradient limits, an integration can be made to get an expression for the velocity. Module 6: Navier-Stokes Equation. Lecture Tubular laminar flow and Hagen- Poiseuille equation. Steady-state, laminar flow through a horizontal circular pipe . In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the The theoretical derivation of a slightly different form of the law was made.
|Published (Last):||2 January 2013|
|PDF File Size:||2.42 Mb|
|ePub File Size:||14.30 Mb|
|Price:||Free* [*Free Regsitration Required]|
In the case of laminar flow:.
The third momentum equation reduces to:. Normally, Hagen-Poiseuille flow implies not just the relation for the pressure drop, above, but also the full solution for the laminar flow profile, which is parabolic. Hagen—Poiseuille flow from the Navier—Stokes equations.
The flow is usually expressed at outlet pressure. It follows that the resistance is proportional to the length of the resistor, which is true. To find A and Bwe use the boundary conditions. Hagen—Poiseuille equation Continuum mechanics. Shows the difference in the pressure between the two ends of the tube, determined by the fact any fluid will always flow from high pressure p1 to low pressure region p2 and the flow rate is determined by the pressure gradient P1 — P2.
This means that the flow rate depends on the heat transfer to and from the fluid. However, it also follows that the resistance R is inversely proportional to the fourth power of the radius ri.
To calculate the flow through each lamina, we multiply the velocity from above and the area of the lamina. Assume the liquid exhibits laminar flow. As fluid is compressed or expands, work is done and the fluid is heated or cooled.
In other projects Wikimedia Commons. Then the angular equation in the momentum equations and the continuity equation are identically satisfied. The assumptions of the 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.
By Newton’s third law of motionthe force on the slower liquid is equal and opposite no negative sign to the force on the faster liquid. If there is no net force then we can add all of the forces together to get zero. Although more lengthy than directly using the Navier—Stokes equationsan alternative method of deriving the Hagen—Poiseuille equation is as follows.
NPTEL :: Chemical Engineering – Fluid Mechanics
The liquid flow is inversely proportional to the length of the tube, therefore longer the tube, greater is the resistance to the flow. Unsourced material may be challenged and removed. It can be seen that both sides of the equations are negative: Low viscosity or a wide pipe may result in turbulent flow, making derivxtion necessary to use more complex models, such as Darcy—Weisbach equation. If the pipe is too short, the Hagen—Poiseuille equation may result in unphysically high flow rates; the flow is bounded by Bernoulli’s principleunder less restrictive conditions, by.
Hagen–Poiseuille flow from the Navier–Stokes equations
It is subject to the following boundary conditions:. Poiseuille’s law derivatio later in extended to turbulent flow by L. Two fluids moving past each other in the x direction.
The negative sign is in there because we are concerned with the faster moving liquid top in figurewhich is being slowed by the slower liquid bottom in figure.
We need to calculate the same values that we did for the force from the faster lamina. The area of contact between the lamina and the faster one is simply the area of the inside of the cylinder: Assume that we are figuring out the force on the lamina with radius. From the equation above, we need to know the area of contact and the velocity gradient.
The reason why Poiseuille’s law leads to a wrong formula for the resistance R is the difference between the fluid flow and the electric current. Rheology Viscoelasticity Rheometry Rheometer. Hagenbach in equagion, The flow is essentially unidirectional because of infinite length.
Take learning poissuille the go with our mobile app. Next let’s find the force of drag from the slower lamina. Normally, Hagen-Poiseuille flow implies not just the relation for the pressure drop, above, but also the full solution for the laminar flow profile, which is parabolic.
Cambridge university press, Congratulations You have selected the correct answer!!