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.
In nonideal fluid dynamicsthe Hagen—Poiseuille equationalso known as the Hagen—Poiseuille lawPoiseuille law or Poiseuille equationis a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in lung alveolior the flow through a drinking straw or through a hypodermic needle. 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.
For velocities and pipe diameters above a threshold, actual fluid flow is not laminar but turbulentleading to larger pressure drops than calculated by the Hagen—Poiseuille equation.
Poiseuille’s Law Derivation
In standard fluid-kinetics notation: The yagen does not hold close to the pipe entrance. Low viscosity or a wide pipe may result in turbulent flow, making it necessary to use more complex models, such as Darcy—Weisbach equation. If the pipe is too deribation, the Hagen—Poiseuille equation may result in unphysically high flow rates; the flow is bounded by Bernoulli’s principleunder less restrictive conditions, by. 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.
However, the result for the pressure drop can be extended to turbulent flow by inferring an effective turbulent viscosity in the case of turbulent flow, even though the flow profile in turbulent flow is strictly deriivation not actually parabolic. In both cases, laminar or turbulent, the pressure drop is related to poiseuklle stress at the wall, which determines the so-called friction factor.
The wall stress can be determined phenomenologically by the Darcy—Weisbach equation in the field of hydraulicsgiven a relationship for the friction factor in terms of the Reynolds number. In the case of laminar flow, for a circular cross section:.
Hagen–Poiseuille flow from the Navier–Stokes equations
It proves more useful to define the Reynolds number in terms of the mean flow velocity because this quantity remains well defined even in the case of turbulent flow, whereas the maximal flow velocity may equatino be, or in any case, it may be difficult to infer. The theoretical derivation of a slightly hgen form of the law was made independently by Wiedman in and Neumann and E. Hagenbach in Hagenbach was the first who called this law the Poiseuille’s law.
The law is also very important in hemorheology and hemodynamicsboth fields of physiology. Poiseuille’s law was later in extended to turbulent flow by L.
Wilberforce, based on Hagenbach’s work. The Hagen—Poiseuille equation can be derived from the Navier—Stokes equations. Although more lengthy than directly using the Navier—Stokes equationsan alternative method of deriving the Hagen—Poiseuille equation is as follows.
Assume the liquid exhibits laminar flow. Laminar flow in a round pipe prescribes that there are a bunch of circular layers lamina of liquid, each having a velocity determined only by their radial distance from the center of the tube.
Also assume the center is moving fastest while the liquid touching the walls poiseuile the tube is stationary due to the no-slip condition. When two layers of liquid in contact with each other move at different speeds, there will be a shear force between them. The negative sign is in there deriivation we are concerned with the faster moving liquid top in figurewhich is being slowed by the slower liquid bottom in figure.
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. This equation assumes that the area of contact is so large that we can ignore any effects from the edges and that the fluids behave as Newtonian fluids. Assume that we are figuring out the force on the lamina with radius r. From the equation above, we need to know the area of contact and the velocity gradient.
The area of contact between the lamina and the faster one is simply the area of the inside of the cylinder: We don’t know the exact form for the velocity of the liquid within the tube yet, but we do know from our assumption above that it is dependent on the radius.
Therefore, the velocity gradient is the change of the velocity with respect to the change in the radius at the intersection of these two laminae. That intersection is at a dquation of r. So, considering that this force will be positive with respect to the movement of the liquid but the derivative of the velocity is negativethe final form of the equation becomes.
Next let’s find the force of drag from the slower lamina. We need to calculate the same values that we did for the force from the derivattion lamina.
Also, we need to remember that this force opposes the direction of movement of the liquid and will therefore be negative and that the derivative of the velocity is negative. To find the solution for the flow of a laminar layer through a tube, we need to make one last assumption. There is no acceleration of liquid in the pipe, and by Newton’s first lawthere is no serivation force. If there is no net force then we can add all of the forces together to get zero.
First, to get everything happening at the same point, use the first two terms of a Taylor series expansion of the velocity gradient:. The expression is valid for all laminae. Grouping like terms and dropping the vertical bar since all derivatives are assumed to be at radius r.
Finally, put this expression in the form of a differential equationdropping the term quadratic in dr. It can be seen that both sides of the equations are negative: Using the product rule derivwtion, the equation may be rearranged to:. It is subject to the following boundary conditions:.
Hagen–Poiseuille flow from the Navier–Stokes equations – Wikipedia
To find A and Bwe use the boundary conditions. Next the no-slip boundary condition is applied to the remaining equation:. Now we derivtion a formula for the velocity of liquid moving through the tube as a function of the distance from the center of the tube.
To get the total volume that flows through the tube, we need to add up the contributions from dfrivation lamina. To calculate the flow through each lamina, we multiply the velocity from above and the area of the lamina. Finally, we integrate over all lamina via the radius variable r. The Navier-Stokes equations reduce to.
The flow is essentially unidirectional because of infinite length. Joseph Boussinesq  derived the velocity profile and volume flow rate in for rectangular channel and tubes of equilateral triangular cross-section and for elliptical cross-section. Joseph Proudman  derived the same for isosceles triangles in The governing equation reduces to . For a compressible fluid in a tube the volumetric flow rate and the linear velocity are not constant along the tube.
The flow is usually expressed at outlet pressure. As fluid is compressed or expands, work is done and the fluid is heated or cooled. This means that the flow rate depends on the heat transfer to and from the fluid. For an equationn gas in the isothermal case, where poiseujlle temperature of the fluid is permitted to equilibrate with its surroundings, and when the pressure difference between ends of the pipe is small, the volumetric flow rate at the pipe outlet is given by.
Electricity was originally understood derjvation be a kind of equatiob. This hydraulic analogy is still conceptually useful for understanding circuits.
This analogy is also used to study the frequency response of fluid-mechanical networks using circuit tools, in which case the fluid network is termed a hydraulic circuit. This is the charge that flows through the cross section per unit time, i.
It follows that the resistance R is proportional to the length L of the resistor, which is true. However, it also follows that the resistance R is inversely proportional to the fourth power of the radius ri. 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. Electron gas is inviscid, so its velocity does not depend on the distance to the walls of the conductor.
The resistance is due to the interaction between the flowing electrons and the atoms of the conductor. Therefore, Poiseuille’s law and the hydraulic analogy are useful only within certain limits when applied to electricity. Both Ohm’s law and Poiseuille’s law illustrate transport phenomena. The Hagen—Poiseuille equation is useful in determining the flow rate of intravenous fluids that may be achieved using various sizes of peripheral and central cannulas.
The equation states that flow rate is proportional to the radius to the fourth power, meaning that a small increase in the internal diameter of the cannula yields a significant increase in flow rate of IV fluids. The radius of IV cannulas is typically measured in “gauge”, which is inversely proportional to the radius. As an example, the flow of a 14G cannula is typically twice that of a 16G, and ten times that of a 20G. It also states that flow is inversely proportional to length, meaning that longer lines have lower flow rates.
This is important to remember as in an emergency, many clinicians favor shorter, larger catheters compared to longer, narrower catheters. While of less clinical importance, the change in pressure can be used to speed up flow rate by pressurizing the bag of fluid, squeezing the bag, or hanging the bag higher from the level of the cannula.
It is also useful to understand that viscous fluids will flow slower e. From Wikipedia, the free encyclopedia. Surface tension Capillary action.
Hagen—Poiseuille flow from the Navier—Stokes equations. This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. September Learn how and when to remove this template message. Annual Review of Fluid Mechanics. Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices. Life in Moving Fluids: The Physical Biology of Flow. An introduction to fluid dynamics. Cambridge university press, Cambridge University Press,
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