Hagen-Poiseuille equation: A non-invasive tool for detecting renal pelvic pressure
Viscous laminar fluid flow in a pipe is described by the well known Hagen- Poiseuille relation. According to this relation, the velocity of the fluid is directly. Goals. Describe key differences between a Newtonian and non-Newtonian fluid; Identify examples of Hagen-Poiseuille (laminar Flow) for Power Law Fluid ↑. In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Normally, Hagen-Poiseuille flow implies not just the relation for the pressure.
The pressure force pushing the liquid through the tube is the change in pressure multiplied by the area: This force is in the direction of the motion of the liquid. The negative sign comes from the conventional way we define. Viscosity effects will pull from the faster lamina immediately closer to the center of the tube. Viscosity effects will drag from the slower lamina immediately closer to the walls of the tube.
Poiseuille’s law: IV fluids
Viscosity Two fluids moving past each other in the x direction. The liquid on top is moving faster and will be pulled in the negative direction by the bottom liquid while the bottom liquid will be pulled in the positive direction by the top liquid.
When two layers of liquid in contact with each other move at different speeds, there will be a shear force between them. 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.Poiseuille's equation for flow of viscous fluid - in HINDI - EduPoint
Faster lamina 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 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.
Poiseuille’s law: IV fluids
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 radius of. 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 where the vertical bar and subscript r following the derivative indicates that it should be taken at a radius of.
Slower lamina 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 faster 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. Putting it all together 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 net force. If there is no net force then we can add all of the forces together to get zero or 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 rulethe equation may be re-arranged to: The right-hand side is the radial term of the Laplace operatorso this differential equation is a special case of the Poisson equation.
It is subject to the following boundary conditions: The differential equation can be integrated to: For Newtonian fluid exists only in ideal condition, no real fluid fits the definition perfectly in fact.
Water and saline can be assumed to be Newtonian for practical calculations under ordinary conditions. The linear regression equation was established Equation 4: We can get the RPP by the formula using simple subtraction Equation 3. There are some points worth noting in our study. Firstly, our experiment put pressure measurement device and renal pelvis in the same horizontal position, avoiding the influence of gravity on the pressure.
Secondly, as mentioned above, irrigation systems like motorized pumps producing irregular flow rates may not be suitable for the Hagen- Poiseuille Equation and the regression model we built. Thirdly, theoretically the result of the experiment is not only limited to PolyScope and also suitable for various ureteroscope with similar circular working channel. Our study has a number of limitations.
The results come for in vitro model, so it is likely to be more complicated in vivo. Further investigation in vivo should be performed for verification. But it does not seem to affect the clinical implications of this experiment.
This experiment is not the pursuit of accurate prediction of the RPP. For the inner diameter of T-Connecter is not the same as that of PolyScope, it may contribute to the failure of fitting with true pressure drop.
- Poiseuille Flow
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