1. Tutorial-0
Q1: Give examples of phenomenon in Physics involving Steady Flow? How to formulate the model there? Solve?
Q2: Give examples of phenomenon where process approaches an end point asymptotically?

2. Steady Flow
Source
Sink

3. Examples: Steady Flow Electricity Heat Fluid Diffusion
𝒅𝑸 𝒅𝒕 =−𝑮 ( 𝑽 𝟐 − 𝑽 𝟏 )
Electricity
Ohm’s law
𝒅𝑯 𝒅𝒕 =− 𝑲( 𝑻 𝟐 − 𝑻 𝟏 )
Heat
Fourier’s law
𝒅𝑽 𝒅𝒕 =−𝓕( 𝒑 𝟐 − 𝒑 𝟏 )
Fluid
Hagen-Poiseuille law
𝒅𝒏 𝒅𝒕 =− 𝑪( 𝒏 𝟐 − 𝒏 𝟏 )
Diffusion
Brownian Diffusion
Does this point to an underlying generality of the phenomenon of flowing?
Additional Question: Dimensions of G, K, 𝓕, C ?

4. Intuition demystified!
Steady Flow
𝒅𝑸 𝒅𝒕 =−𝑮 ( 𝑽 𝟐 − 𝑽 𝟏 )
Something which flows
Cause of flow
𝒅𝑯 𝒅𝒕 =− 𝑲( 𝑻 𝟐 − 𝑻 𝟏 )
𝒅𝑽 𝒅𝒕 =−𝓕( 𝒑 𝟐 − 𝒑 𝟏 )
𝒅 ? 𝒅𝒕 =…
𝒅𝒏 𝒅𝒕 =− 𝑪( 𝒏 𝟐 − 𝒏 𝟏 )
Intuition demystified!

5. The process approaches an end point, asymptotically

6. Processes where rate becomes less and less over time
Emptying of a Water Tank
Discharging of a Capacitor
Radioactive Decay
Cooling of a warm object

7. Emptying of a Water Tank𝒉0 𝒉
𝒑 𝒂𝒕
𝒑 𝟐
𝒅𝑽 𝒅𝒕 =−𝓕( 𝒑 𝟐 − 𝒑 𝒂𝒕 )
𝐁𝐮𝐭, 𝒑 𝟐 = 𝒑 𝒂𝒕 +𝝆𝒈𝒉
As, 𝑽 = 𝑨𝒉,  d𝑽/dt = 𝑨 d𝒉/dt
∴ 𝒅𝑽 𝒅𝒕 =𝑨 𝒅𝒉 𝒅𝒕 =−𝓕𝝆𝒈𝒉
∴ 𝒅𝒉 𝒅𝒕 =− 𝓕𝝆𝒈𝒉 𝑨
 𝒅𝒉 𝒉 =− 𝓕𝝆𝒈 𝑨 𝒅𝒕
𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐧𝐠, 𝒍𝒏𝒉=− 𝓕𝝆𝒈 𝑨 𝒕+𝑪
∴𝒍𝒏 𝒉 𝒉 𝟎 =− 𝓕𝝆𝒈 𝑨 𝒕
At t = 0, 𝒉=𝒉0 , therefore, C = 𝒍𝒏𝒉𝟎
𝒉= 𝒉 𝟎 𝒆 − 𝓕𝝆𝒈 𝑨 𝒕
Finally,
𝓕 = π 𝒓 𝟒 𝟖η𝒍 is called the “flow” conductance

8. Emptying of a Water Tank𝒉0 𝒉
𝒑 𝒂𝒕
𝒑 𝟐 𝒉0
Water level, 𝒉
Time, 𝒕
𝒉= 𝒉 𝟎 𝒆 − 𝓕𝝆𝒈 𝑨 𝒕
Exponential Decay
 : relaxation time
As t → , 𝒉→𝟎
At t = A/𝓕𝝆g = , 𝒉= 𝟏 𝒆 𝒉0
Observations
At t = 𝒏, 𝒉= ( 𝟏 𝒆 ) 𝒏 𝒉0

9. Processes where rate becomes less and less over time
Emptying of a Water Tank
Discharging of a Capacitor
∴ 𝒅𝒉 𝒅𝒕 =− 𝓕𝝆𝒈𝒉 𝑨
𝒅𝑸 𝒅𝒕 =− 𝟏 𝑹𝑪 𝑸
𝓕 = π 𝒓 𝟒 𝟖η𝒍 is “flow” conductance
 = RC discharge time constant
Radioactive Decay
Cooling of a warm object
∴ 𝒅𝑵 𝒅𝒕 =−𝝀𝑵
𝒅𝑻 𝒅𝒕 =−𝜩(𝑻− 𝑻 𝒂 )
𝜩 depends on size, shape, specific heat, surface condition of an object, etc.
 is decay rate
t1/2 = 