5: Flow, Transport and Exponential

Last updated
Save as PDF

Page ID: 17158

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

This chapter focuses on phenomena involving real fluids with viscosity and electric circuits with resistance. The effect of resistance in both kinds of flow means that energy will be reduced in the fluid system while thermal energy systems will increase. Frequently the flow is described as being dissipative. We call the model/approach we use to make sense of dissipative flow the Steady-State Energy Density Model, where the flow is constant over time. In the second part of the chapter we generalize the underlying ideas about flow to flow phenomena in which changes in energy are not of paramount importance. Rather, the focus is simply on the “fluid” and medium properties and the “driving force” that keeps the flow going. The “thing” that flows can be a real fluid, electric charge, energy, or other things that diffuse – in short, any phenomenon in which the flow of something becomes constant can be understood with this approach/model, which we call the Linear Transport Model. Toward the end of the chapter we look at examples where the flow is no longer constant, but displays exponential decay behavior.

5.0: Overview of Flow, Transport and Exponential
Here we present an overview of the main concepts covered in this chapter.
5.1: Steady-State Energy-Density Model
Our goal in this present chapter is to understand fluids and the flow of electric charge in electric circuits using the Steady-State Energy Density Model. Historically, different words and symbols have been used for the description of each of these phenomena, making their similarity even more difficult to see. We will generally use conventional notation and vocabulary, but will continuously draw parallels between these physically different flow systems.
5.2: Static Fluids
We start with defining energies that vary in a fluid system which is static, where there is no flow. Pressure and gravitation potential energy-density is introduced and analyzed here in static fluid systems.
5.3: Fluid Flow
In this section we modify the fluid system to allow for flow. The flow is steady-state which means it is constant over time for a given fluid system. To describe flow, kinetic energy-density, in addition to pressure and gravitational potential energy-density, is added to the equation which describes energy-density conservation. We also look dissipative effects of resistance to flow which convert mechanical energy to thermal energy, and a source which adds energy to the fluid flow system, a pump.
5.4: Electric Circuits
In this section we introduce steady-state electric charge flow and make multiple analogies with fluid flow. We start by introducing the idea of a circuit, where a fluid (or charge) returns to its original location or flows in a circular manner. Wire acts as pipes and a batteries like pumps. Resistors introduce dissipative effects.
5.5: Resistors in Parallel and Series
In this section we look at more complex circuits where resistors are wired in different configurations, in series and parallel.
5.6: Circuit Problem Solving
We present circuit problem-solving techniques and demonstrate them with multiple examples.
5.7: The Linear Transport Model
There are many phenomena that involve the motion or transport of some quantity that behave similarly to the way fluids and electric charge flow in those sections of a circuit without sources of energy density. In steady-state, we can model these phenomena the same way we model fluids and electric charge flow. These are such common and general phenomena that it is useful to collect them together under their own specially named model—the linear transport model.
5.8: Exponential Change Model
Things grow (or decay) exponentially when the time rate of change in the quantity is proportional to the amount of “stuff” already present. Thus, bacteria grow exponentially because the more you have, the more they reproduce (until they run out of food, or their waste starts to poison the environment.) The decay of radioactive elements is exponential because as the number of radioactive atoms decreases, there are fewer available to decay, so less decay occurs.
5.9: Exponential Fluid Flow
You will see multiple examples of exponential decay of fluid flow.
5.10: Exponential Charge Flow
A new electronic element, a capacitor, is introduced. When a capacitor is part of an electronic circuit, exponential decay of current and voltage is observed. Analogies are made between circuits with capacitor and fluid flow scenarios from the previous section.
5.11: Wrap up