Continuum approach Motion descriptionBalance lawsTraffic flow behaviorGreenshields modelLinear traffic waves Method of characteristics
If we consider an artery with obstacles thesituation can be compared to a highwaypopulated by travelling vehicles where inthis case those cars are cells of the blood.
𝐁𝐁𝐁𝐁𝐁𝐁𝐁𝐁𝐁𝐁 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐁𝐁
µmµm
𝟏𝟏𝟏𝟏−𝟒𝟒 𝟏𝟏𝟏𝟏−𝟔𝟔𝟏𝟏𝟏𝟏−𝟑𝟑𝟏𝟏𝟏𝟏𝟏𝟏𝒎𝒎
Displacement
=0
Control Volume
Trajectory
Flow direction
1-D conservation equation
Traffic flow behavior
The flowMacroscopic)
Individual cars(Microscopic)
The observed behavior of cars involvesdriving speeds
driving lane
FactorsTraffic conditionsCars characteristics
Road conditionsExternal agents
𝒒𝒒
𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎
𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎
𝒒𝒒𝒎𝒎𝒎𝒎𝒎𝒎
𝒗𝒗
𝝆𝝆 𝝆𝝆
Constitutive law 𝒗𝒗 = 𝒇𝒇 𝝆𝝆
𝒗𝒗 𝝆𝝆 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 − 𝝆𝝆𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎
𝒒𝒒 𝝆𝝆 = 𝝆𝝆 𝒗𝒗 𝝆𝝆
Empirical dataGreenshields model
𝝏𝝏𝝆𝝆𝝏𝝏𝝏𝝏
+ 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 −𝟐𝟐𝝆𝝆𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎
𝝏𝝏𝝆𝝆𝝏𝝏𝒎𝒎
= 𝟏𝟏
𝝏𝝏𝝆𝝆𝝏𝝏𝝏𝝏
+ 𝒒𝒒𝒒(𝝆𝝆)𝝏𝝏𝝆𝝆𝝏𝝏𝒎𝒎
= 𝟏𝟏
𝒒𝒒 𝝆𝝆 = 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝝆𝝆 − 𝝆𝝆𝟐𝟐
𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎𝒗𝒗 𝝆𝝆 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 − 𝝆𝝆
𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎𝒒𝒒 𝝆𝝆 = 𝝆𝝆 𝒗𝒗 𝝆𝝆
𝒒𝒒′ 𝝆𝝆 = 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 −𝟐𝟐𝝆𝝆𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎
Quasi linearPDE
𝒒𝒒′ 𝝆𝝆𝟏𝟏 + 𝜹𝜹𝝆𝝆 = 𝒒𝒒′ 𝝆𝝆𝟏𝟏 + 𝒒𝒒′′ 𝝆𝝆𝟏𝟏 𝜹𝜹𝝆𝝆 + …
Linearized form
Solution form𝒒𝒒′ 𝝆𝝆𝟏𝟏 = 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 − 𝟐𝟐𝝆𝝆𝟏𝟏
𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎= 𝒄𝒄
𝒅𝒅𝒅𝒅𝒅𝒅𝝏𝝏𝒎𝒎𝒅𝒅𝒄𝒄𝒅𝒅 𝒎𝒎
𝟏𝟏𝒄𝒄
𝟏𝟏
𝒅𝒅𝒎𝒎𝒅𝒅𝝏𝝏 = 𝒄𝒄
𝝏𝝏𝒅𝒅𝒎𝒎𝒅𝒅𝝏𝝏
Thank you for your attention
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