Pump A in Figure 1 forces water at constant pressure, P1, around a closed loop. As long as Valve B remains open or at least partially open, the water flowing in the pipe attains steady velocity, V1. The pump pressure, the viscosity of water, and the effective fluid-flow resistance of the pipes, valves, and elbows used in the circuit determine this velocity. The hydraulic circuit is analogous to a simple electrical circuit comprising a battery, a resistor, and a connection from the opposite end of the resistor back to the battery, represented by the tank. The electrostatic pressure the battery creates corresponds to water pressure at the head of the pump. The circulation of electrical current in the wires corresponds to the circulation of water in the plumbing.
The concept of resistance translates easily between the two circuits. If you define the fluid-flow resistance of a pipe as the pressure difference from end to end divided by its flow, then a long, skinny pipe makes a high resistance. A big, fat pipe, such as the one presently gushing oil into the Gulf of Mexico from the BP oil disaster, makes a low resistance. This fluid-flow equivalence works so naturally that most students, years later at the end of their careers, still think of a high resistance as a long, skinny pipe.
One major difference between the movement of water and that of electrons concerns linearity. The fluid-flow resistance of a piece of pipe varies almost quadratically with the flow, as opposed to the exquisitely linear relationship exhibited by an electrical resistance. Nobody brought that fact to my attention when we discussed the water-flow analogy in school, but, if you want to properly size your lawn-sprinkler system's pipes, you'd better check it out.
The lack of linearity in water-flow systems works to the advantage of a student's understanding. It focuses attention on two fundamental principles of circuit analysis. First, you must always maintain the correct relation of pressure drop to flow for every device throughout a circuit. Second, no water enters or exits a closed system. Those two rules are sufficient to develop a rich understanding of circuit behavior.
On the other hand, the simple linear nature of electronic components tempts electrical engineers to attack every problem with linear matrix analysis and Laplace transforms. These powerful tools confuse beginners. Rather than begin with linear analysis, every electrical-engineering student should first complete a thorough study of nonlinear circuits. Devices such as FETs and bipolar transistors are nonlinear and temperature-dependent anyway, so it makes sense to first introduce nonlinear behavior.
Water- and electron-based circuits, both ultimately dependent on the movement of myriad tiny particles, share many similarities beyond simple pressure and flow relations. For example, if you force too much water through a pipe, the required rise in pressure may burst the pipe, after which nothing flows through to the end and you have to mop up a huge mess. In an electrical circuit, an overtaxed resistor overheats and melts, after which no current flows and you must deal with the smell.
At the other end of the signal-amplitude scale, Brownian motion affects sensitive hydraulic systems, such as the diffusion of ions throughout a nerve ending. Thermal noise affects electrical circuits. It's the same effect either way. We are all just pushing particles around closed loops.