What a drag…

Irritated after a recent foray into the local High Street on a weekend, I began to reflect on the properties of a crowd. Most of the people I observed were meandering lethargically around the Pedestrianised Zone with no apparent sense of direction or purpose. This dearth of any determinable directing influence in the crowd as a whole puts me in mind of a Newtonian Fluid. Indeed the speed with which one can move through the High Street declines as the density of the crowd increases. I suggest therefore, that a person fighting through a crowd experiences a form of viscous drag, and it should hence be possible to determine a corresponding Drag Coefficient.

The Drag Equation for movement through a fluid is as follows:

Fd = ½pv²C_dA, where ‘F_d’is the drag force, ‘p’ is the fluid density, ‘v’ is the velocity of the object experiencing drag, ‘C_d’ is the Drag Coefficient, ‘A’ is the cross-sectional area of the object experiencing drag.

In the absence of any drag associated with a crowd, despite a constant application of force directed forward, one’s velocity does not increase without limit. Instead a constant velocity is achieved, due to an effective drag force created by air resistance, the impact of one’s footfalls on the ground, etc. This, from the Drag Equation, may be considered to be proportional to v². Let the constant of proportionality be ‘X’, then, for a forward motion driven by a force ‘F₀’, F₀ – Xv₁² = 0 (where v₁ is the velocity achievable in the absence of a crowd).

In the presence of a crowd, the additional drag force, ‘F_d’, must be factored in. For a forward velocity- fighting the crowd, of v₂:

F₀ – F_d – Xv₂² = 0

Therefore, F₀ - ½pv₂²C_dA – Xv₂² = 0

Rearranging gives: [2(F₀ – Xv₂²)]/[pv₂²A] = C_d

‘X’ may be determined by direct measurement of F₀ and v₁. The other variables must be modified to account for the fact that the crowd is essentially 2-dimensional for these purposes. ‘p’ then is defined in ‘people per m²’ and ‘A’ is given as the cross-sectional breadth of the person fighting the crowds (since their height is irrelevant for these purposes). ‘C_d’ then, ceases to be a dimensionless coefficient; dimensional analysis reveals it to have units: kg[people]^-1. (The constant ‘X’ has units: kgm^-1).

Biofuel:

A number of elaborate 'dual-fuel' systems are currently on the market, allowing the use of what is essentially vegetable oil in Diesel engines. Significant modifications are necessary because the viscosity of vegetable oil at room temperature is much higher than that of petro-Diesel. Furthermore, Diesel engines running on vegetable oil are prone to 'sooting'- the build-up of deposits of partially oxidised fuel within the cylinders. This is due to vegetable oil's comparatively low cetane number.

However, the cetane number of Diesel fuel is readily enhanced, eg by addition of amyl nitrate. Viscosity modifiers have also been long-since developed by the oil industry. Development of a suitable suite of additives to enhance cetane number and cold flow properties of vegetable oil would allow them to be used to fuel Diesel engines without any modification, and render the process of converting vegetable oil to biodiesel superfluous.

What are the Odds?

Murphy's Law is an oft quoted assertion to the effect that "anything that can go wrong, will go wrong". Clearly this is not strictly true, as an unlimited propensity of events to conspire against us would render life quite impossible. Seeking a more exact definition, I have attempted to define Murphy's Law mathematically, based on relative probabilities for given events, as a function of their 'undesirability':

The sum of the probabilities of all possible events from a given initial state is equal to ‘1’ (certainty), ie. it is certain that one of all the possible outcomes will occur. Let each possible event have a probability ‘P_n’, then the sum of all possible events: P₁ + P₂ + P₃ + P₄… = [SUM]P_n = 1.

Experience tells us that the probability of a given event occurring is to some degree directly proportional to the annoyance/inconvenience to the observer that is brought about by said occurrence.

However, experience also tells us that there are practical limits to the extent to which Murphy’s Law distorts the nominal probability of any given event:

For example, dropped toast may land butter-side down more frequently than it should- a moderate skewing of nominal probability, but the air in a room never conspires to spontaneously evacuate it- such an event would constitute a much greater distortion than Murphy’s Law brings about.

For a given event of probability ‘P’ then, we can conceive of a maximum coefficient by which the nominal probability can be altered. Let this probability be ‘M’ the Murphy’s Law Coefficient.

Now, for a simple system with two possible events resulting from a given initial state, P₁ and P₂, P₁ + P₂ = 1. Assume that the possible outcome ‘P₁’ is disfavoured by the observer, while outcome ‘P₂’ is favoured:

Incorporating Murphy’s Law, the probability ‘P₁’ then becomes P₁(1+N₁M), where ‘N’ is a number between -1 and +1, representing the annoyance/inconvenience inflicted upon the observer as a consequence of the event occurring. (In the event that an outcome is favoured by the observer, N is less than '0'). The sum of 'N's for all possible outcomes for a given event is necessarily '0': [SUM]N_n = 0.

In the ‘two possible outcome’ case then, P₁(1 + N₁M) + P₂(1 + N₂M) = 1.

Extending this reasoning to the general case then gives:

[SUM]P_n(1 + N_nM) = 1 ([SUM]N_n = 0).

What is the value of the Murphy’s Law Coefficient, ‘M’? By the conventions we employ here, and assuming Murphy’s Law always to have an adverse effect, ‘M’ must be greater than ‘0’. Our experience of the limited effect of Murphy’s Law suggests that ‘M’ must be significantly less than ‘1’.

However, any experiment intended to quantify the ‘Murphy’s Law’ coefficient would inevitably be thwarted by the influence of Murphy’s Law itself.