Population model
dP
dt = rP − bP2.
Here, r is the growth rate, b is the competition rate per each individual. It means that the species compete for the same resource and causes the reduction of population. Thus, this is an environmen-tal constraint. The initial population is P0. In this model, the quanities are P , t, P0, r and b. t is the independent variable, P the depedent variable, P0, r and b are parameters. The dimensions of them are
[t] = T, [P ] = P, [P0] = P, [r] = 1
T, [b] = 1 T P.
We see that there are only two fundamental dimensions T and P . We can rescale this equation by a proper scaling as the follows. We write the equation by
dP
d(rt) = P − b
rP2 = P
1 − P
K
, K = r b. Thus, we can introduce the following dimensionless variables
t0 = rt, P0 = P
K, P00 = P0
K, then the equation is equivalent
dP0
dt0 = P0(1 − P0), P0(0) = P00.
This equivalent equation depends only on 3 dimensionless quantities P0, t0 and P00, instead of 5 quantities.
A falling object 1. Quantities
(a) independent variables: t.
(b) dependent variables: x (c) parameters: g, V , m, h 2. Relations Newton’s law:
m¨x = −mg, x(0) = h, ˙x(0) = V, 3. Dimensional Analysis
(a) [x] = [h] = L
1.2. DIMENSIONAL ANALYSIS 5 (b) [t] = T ,
(c) [g] = L/T2. (d) [V ] = L/T
We will find a relation between x, t, g, h, V . Notice that there are only two fundamental dimensions, namely, L and T .
4. Scaling: A natural way is to introduce h and V as our fundamental dimensions, which are the observed (or controlled) parameters. Then we rescale x0 = x/h, v0 = v/V , t0 = t/T = t/(h/V ). With these, the equation becomes
d2x0
dt02 = −gh−1T2 = −ghV−2= − 1 Fr2. The initial conditions are
x0(0) = 1,dx0
dt0(0) = 1.
Here, we introduce a dimensionless parameter Fr = V
√gh = inertial force gravitational force,
called Froude number, which also appears in water wave theory. Then the dimensionless variables are (x0, t0) and the Froude number Fr. Thus, the number of variables and parameters are reduced.
The solution of the above dimensionless equation is x0(t0) = 1 + t0−1
2 t02 Fr2. The critical time that the particle starts to fall is
dx0
dt0(t0) = 0.
That is,
t0 = Fr2.
Thus, the larger the Froude number is (i.e. inertia force >> gravitation force), the longer time the particle starts to fall. Notice that if we drop an object from height h. The speed it touches the ground is√
2gh.
Notice that h can be the length we are interested in, needs not be the initial height. Further, the dimensionless parameters are not uniquely chosen. For instance, another way is to choose v0 = V T /h = V /√
gh. Then the rescaled system becomes d2x0
dt02 = −1
6 CHAPTER 1. DIMENSIONAL ANALYSIS with the initial conditions
x0(0) = 1, dx0
dt0(0) = v0, a relation involve the three dimensionless quantities x0, t0, v0.
People usually use the first scaling because (1) the initial velocity V is something we can con-trol, it is natural to scale velocity in terms of V ; (2)the parameter Fr appears in the equation, we can study solution property (bifurcation, for instance) in terms of the parameter without solving it directly, whereas the solution property in terms of initial condition usually requires solving the equation first.
Body mass index (BMI) The BMI is defined to be BMI = w
h2,
where w is the weight in kilogram and h is the height in meter. What is the dimension of BMI?
Why it measures the amount of fat of your body?
Rain droplet Why the viscous force of an object is propotional with the its area? With a fixed density, why the body force of the object is propotional to its volume? Can you answer what size of the raindrop will fall?
Projectile 1. Quantities:
• independent variable t
• Dependent variables: z,
• Parameters: m, g, R (earth radius), V initial velocity 2. The relation:
md2z
dt2 = −G M m (z + R)2 This implies
d2z
dt2 = − R2g (R + z)2 where g = GMR2 .
z(0) = 0,dz
dt(0) = V.
3. Dimensional Analysis: the dimensions of the quantities are
[t] = T, [V ] = LT−1, [z] = L, [g] = LT−2, [R] = L.
1.2. DIMENSIONAL ANALYSIS 7 There are 5 quantities, but only 2 fundamental dimensions involved. Thus, we expect that there are 3 dimensionless quantities. Let us suppose the dimensionless quantity π is a combi-nation of the 5 quantities in the following form:
[π] = [tα1zα2Rα3Vα4gα5]
= Tα1−α4−2α5Lα2+α3+α4+α5
In order to have π to be dimensionless, i.e. [π] = 1, the admissible exponents leading to the following equations:
α1− α4− 2α5 = 0 α2+ α3+ α4+ α5 = 0.
Let us use α3, α4 and α5as free parameters. We then obtain three sets of solutions:
• α3 = −1, α4= α5 = 0. This gives α2 = 1 and α1 = 0. We then get π1= z/R.
• α4 = 1, α3 = α5 = 0. This gives α1 = −α2 = α4= 1. We get π2= tz−1V .
• α5 = 1, α3 = α4 = 0. This gives α1 = 2, α2 = −1. We get π3= t2z−1g.
Alternatively, we can use
πx = z/R, πt= tV /R, π = gR V2. Then the differential equation can be written as
d2πx dπ2t = R
V2 d2z
dt2 = −gR V2
1
(1 + πx)2 = − π (1 + πx)2. with πx(0) = 0 and dπdπx
t(0) = 1. This is the dimensionless relation, which is a relation among πx, πtand π. So, numbers of variables are reduced from 5 to 3.
The general strategy to select dimensionless variables are
• rescale the spatial variable z by z/R
• rescale the temporal variable t by t/(R/V ).
• Select the rest dimensionless parameters from the equations so that its form is simple.
Diffusion process Consider an explosion process in the space. We are interested in the heat propagation of this exploration. The variables and parameters of this process are:
• independent variables: x and t;
• dependent variables: u (temperature), q heat flux;
• parameters: energy released at the exploration center, k (conductivity),c (heat capacity at constant volume).
8 CHAPTER 1. DIMENSIONAL ANALYSIS Relations: based on conservation of energy, we have
d dt
Z
Ω
cu dx = Z
∂Ω
q · (−n) dS This yields
Z
Ω
cutdx = − Z
Ω
∇ · q dx This is valid for any arbitrary Ω. Thus, we get
cut+ ∇ · q = 0.
The heat flux is related to the gradient of the temperature. This is the Fourier law:
q = −k∇u.
Plug this into trhe above equation, The heat equation becomes ut= ∇ · k
c∇u = D 4 u.
Here, the notation 4 = ∇2 is the Laplacian. The dimensions of these quantities are [u] = Θ, [c] = EΘ−1L−3, [q] = ET−1L−2, [k] = ET−1L−1Θ−1, [D] = L2T−1.
In the diffusion equation, the only quantities are x, t, u and D. There are three fundamental dimensions: L, T and Θ. Thus, there is only one dimensionless quantity left. Suppose it is π = tα1xα2uα3Dα4. In order to be dimensionless, we get
π1 = x
√ Dt Another one is
π2 = uc
e x3 = uc
e π31(Dt)3/2
Suppose we have a relation: π2 = g(π1). This leads to a relation which can express the unknown u in terms of the independent variable x, t and the parameter D:
u = e
c(Dt)−3/2g( x
√
Dt). (1.1)
Using dimensional analysis, we can find possible relation such as (1.1), which says that (1) the diffusion length scale x is√
Dt; (2) the maximum of the heat decays like (Dt)−3/2 in three space dimensions. We obtain this without solving the PDE at all, simply a guess from dimensional analy-sis.
There is another technique that used commonly. We can plug this ansatz into equation. Then we will get an ODE for g. Eventually we will get g(π1) = e−π12.
1.3. THE BUCKINGHAM PI THEOREM 9