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Idea: Since the motion of fluids can be very complicated (due to turbulence, frictional forces between molecules etc.) we need to make the following simplifying assumptions:
The flow rate at each point along a (non-uniform) pipe or hose is defined as the volume of fluid passing through the pipe at that point per unit time:
FlowRate = = A = Av | (15) |
Idea: If the fluid is incompressible, the flow rate must be the same everywhere along the pipe (Conservation of fluid particles: what goes in must come out). This leads directly to the Continuity Equation:
Av = constant. | (16) |
Idea: Conservation of energy requires that the work done in
moving a volume of fluid through a portion of pipe must go into
kinetic energy
and/or potential energy of the fluid (assuming no frictional
forces). Consider a fluid moving through a pipe as in Fig.9.8.
The work done by the pressure P_{1} on the left in moving the fluid a distance x_{1} is:
W_{1} = F_{1}x_{1} = P_{1}A_{1}x_{1}.
On the right, the work done by the pressure is:W_{2} = - F_{2}x_{2} = - P_{2}A_{2}x_{2}
where the minus sign occurs because the force is in the opposite direction of the motion in this case. By the continuity equation, the amount of fluid entering the pipe on the left must equal the amount leaving on the right. Thus:A_{1}x_{1} = A_{2}x_{2}.
This volume of fluid, which we call V , can undergo both a change in kinetic energy and a change in potential energy as it is pushed through the pipe:KE | = | Vv_{2}^{2} - Vv_{1}^{2} | |
PE | = | Vgy_{2} - Vgy_{1}. |
P_{1} + v_{1}^{2} + gy_{1} = P_{2} + v_{2}^{2} + gy_{2} | (17) |
Note: If there is no height difference, Bernouilli's equation relates the pressure at two points along the flow to the fluid velocities:
P_{1} - P_{2} = (v_{2}^{2} - v_{1}^{2}). | (18) |
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