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?

Steady Flow Source Sink

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 ?

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!

The process approaches an end point, asymptotically

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

Emptying of a Water Tank 𝒉0 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

Emptying of a Water Tank 𝒉0 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

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 = 