Samuran Life in Moving Fluids: For electrical circuits, let be the concentration of free charged particles, ; let be the charge of each particle. Hagen—Poiseuille equation Continuum mechanics. To calculate the flow through each lamina, we multiply the velocity from above and the area poiseulile the lamina. Finally, put this expression in the form of a differential equationdropping the term quadratic in dr.
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The equation does not hold close to the pipe entrance. The theoretical derivation of a slightly different form of the law was made independently by Wiedman in and Neumann and E. Hagenbach in , The law is also very important specially in hemorheology and hemodynamics , both fields of physiology. Viscosity File:Poiseuille1. 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. 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.
Liquid flow through a pipe In a tube we make a basic assumption: the liquid in the center is moving fastest while the liquid touching the walls of the tube is stationary due to friction. File:Poiseuille abstraction. Those closest to the edge of the tube are moving slowly while those near the center are moving quickly. The pull from the faster lamina immediately closer to the center of the tube The drag from the slower lamina immediately closer to the walls of the tube.
The first of these forces comes from the definition of pressure. The other two forces require us to modify the equations above that we have for viscosity. In fact, we are not modifying the equations, instead merely plugging in values specific to our problem. From the equation above, we need to know the area of contact and the velocity gradient.
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. 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.
To calculate the flow through each lamina, we multiply the velocity from above and the area of the lamina. The flow is usually expressed at outlet pressure.
As fluid is compressed or expands, work is done and the fluid is heated and cooled. This means that the flow rate depends on the heat transfer to and from the fluid.
Electrical circuits analogy Electricity was originally understood to be a kind of fluid. 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 means that halving the radius of the tube increases the resistance to fluid movement by a factor of Hagen did his experiments in See also.