4.3 The Process of Nondimensionalisation
4.3.2 Terminal Velocity of a Falling Sphere in a Fluid
4.3 The Process of Nondimensionalisation 91
The 2nd general nondimensional scaling principle:
Select characteristic scales so that NO terms in the model diverge in the physical limit of interest.
(4.8)
The choice of scales is determined once the sub-set of Π’s to be normalised is made; hence other choices correspond to picking differentΠ’s. For problems withk characteristic scales, the scales will be determined from normalisingkΠ’s; this will be discussed further in connection with the Buckingham Pi theorem, see Sect.4.5.
To examine theY0→0 limit, consider settingΠ1=1 andΠ4=1 in (4.5). The length and time scales are then L=V20/gand T =V0/g. The revised form of the nondimensionalised problem is
d2y
dt2 = − 1
(1+ ˜Π2y)2, y(0)= ˜Π3, y(0)=1. (4.9a) Π˜2= L
RE = V20
gRE, Π˜3= Y0
L = Y0g
V20 . (4.9b)
This choice of scalings satisfies (4.8) since settingY0=0 setsΠ˜3=0 but does not cause any other parameters in the problem to diverge,
d2y
dt2 = − 1
(1+ ˜Π2y)2, y(0)=0, y(0)=1.
If we further specify thatRE→ ∞thenΠ˜2=0 and the ODE reduces toy= −1 yielding the solution
y(t)= −12t2+t =⇒ Y=V0T−12gT2.
ForV0>0 thetime of flightof the projectile is clearly seen to be 2V0/gwhile the maximum height reached isV20/(2g), both of which are directly proportional to the scales that we chose.
fluid has a different density,ρf and its viscosity isμkg/m s, which gives a measure of the frictional resistance to the ball moving through the fluid.
Noting that the forces acting on the ball are due to its weight, buoyancy and the drag of the fluid, we can use Newton’s second law to write down a force balance
MdV
dT =Mg−Fbuoy−Fdrag, V(0)=V0,
where we have expressed the acceleration as the derivative of the ball’s velocity. The magnitude of the buoyancy force is given by the weight of the fluid displaced by the ball,Fbuoy = 43πR3ρfg, while the frictional force acting on a sphere is given by Fdrag=6πμRV, a classic result derived by Stokes.
Consequently, the problem can be restated as 4
3πR3ρdV dT = 4
3πR3ρg− 4
3πR3ρfg−6πμRV. (4.10) Introducing the nondimensionalization,V(T)=Vv(t)withT=Tt, where V and T are characteristic scales for speed and time yields the nondimensional model
dv dt =
(ρ−ρf)gT ρV
Π1
− 9μT
2ρR2 Π2
v, v(0)= V0
V Π3
. (4.11)
SettingΠ1=1 andΠ3=1 yields the characteristic scales V=V0, T= ρV0
(ρ−ρf)g, (4.12)
and the nondimensionalized problem reads:
dv
dt =1−Stv, v(0)=1, (4.13) where theΠ2group has been renamed theStokes parameter
St= 9μV0
2(ρ−ρf)gR2. (4.14)
The “Stokes parameter” name attached to Π2 is a historical label used for this dimensionless parameter, but this classification has important value since it makes it easier to search for other results on related problems involving this parameter in books, journals, and other sources. Hence dimensionless parameters make it possi- ble to universally compare results from different studies (experiments, simulations, and theory) and communicate in a common terminology across many branches of
4.3 The Process of Nondimensionalisation 93
Table 4.1 A few commonly used dimensionless parameters
Name Formula Competing effects Area of study
Arrhenius E/(RT) Activation/potential energy Thermodynamics Damkohler kL/(UC0) Reaction/transport rates Thermodynamics
Lewis DT/DM Thermal/mass diffusivity Thermodynamics
Mach V/c Char. speed/sound speed Aerodynamics
Peclet UL/D Convection/diffusion Thermodynamics
Reynolds ρUL/μ Inertia/viscosity Fluid dynamics
Stokes (4.14) Drag/gravity Fluid dynamics
science and engineering. See Table4.1for a short list of some of the many other named parameters, also see [69].
The nondimensionalized problem (4.13) has an explicit solution which conse- quently depends only ontand the single St parameter:
v(t,St)= 1 St
1−e−tSt
+e−tSt. (4.15)
The solution of the original dimensional problem can then be written in terms of (4.15) and (4.12) as
V(T)=V0v(t/T,St)
= V0
St +V0
1− 1
St
exp
−St(ρ−ρf)g ρV0
t
= 2(ρ−ρf)gR2 9μ
1−e−
9μ 2ρR2t
+V0e−
9μ 2ρR2t
. (4.16)
The last expression is what would have been obtained from solving (4.10) directly.
As should be expected, its functional form is equivalent to (4.15), but (4.16) makes it difficult to see that the solution is actually only dependent on a single parameter.
Having (4.13) in a less cumbersome form than (4.10) facilitates using it to identify the steady state asv=1/St; this represents the “terminal” free-fall velocity in this problem (V0/St=2(ρ−ρf)gR2/(9μ)in dimensional form).
Here, as in most problems, the dimensionless parameters more efficiently and compactly capture the dependence of key properties of the system on its design para- meters. In addition, if this model had been compared against experimental data for V =V(T,R,V0,ρf,ρ,μ,g)then while the data could involve many experiments varying the six parameters, Eq. (4.16) shows that all of that data could be captured by the two characteristic scales and the Stokes parameter. Dramatic reduction in re-organising data to show its underlying fundamental structure is often calledcol- lapsing of data, namely when many data points from different runs are all shown to fall onuniversal curves.
In (4.11), the ODE was normalised by the coefficient of the inertial term, making Π1the ratio of buoyancy to inertial effects andΠ2the ratio of drag to inertial effects for given V,T. Setting the V,T characteristic scales to normaliseΠ1, Π3=1 gives a formula for St in terms of given quantities (4.14) which embodies this ratio,
St= 9μV0
2(ρ−ρf)gR2 = 6πμRV0 4
3πR3(ρ−ρf)g = drag force
net gravity force, (4.17) We want to re-iterate the very important point thatalldimensionless parameters are ratios of competing effects in models. This is a consequence of allΠ’s being formed by dividing equations in the model by scaling coefficients for the influence of one term. In general,
Π =strength of effect 1
strength of effect 2. (4.18)
The limiting cases of such ratios have clear interpretations:
• Π→0: effect 1 is very weak (relative to effect 2).
• Π→ ∞: effect 1 is very strong (relative to effect 2).
Between these extremes other specialcritical valuesΠc can exist that separate different qualitative regimesfor the solution’s behaviour. The model (4.13) has a critical Stokes number of Stc=1:
• For St<1 solutions describe balls starting from lower speeds, that will accelerate up to the terminal velocity.
• For St>1 solutions represent balls starting from speed highers than the terminal velocity decelerating for all times.
• For St=1, the solution starts at and maintains the terminal velocity.
This is a simple example of a critical value of a parameter marking abifurcation. In other systems, critical values can signify a change in the stability of a solution, or act as boundaries of parameter regimes in which different numbers of solutions coexist.
Careful examination of related issues will be given in upcoming chapters (see Chaps.6–10) using asymptotic analysis and perturbation methods to solve models in small parameter limits. For the moment, we note that apart from any physical interpre- tations, the limiting cases,Π =0 andΠ = ∞, have a clear practical difference—an infinite coefficient precludes any calculations using the “usual methods”. In (4.13), having St=0 yields a well-defined solution,2v(t)=1+t, while (4.13) with St= ∞ is asingular limit that can not be sensibly evaluated directly. As already described in connection with (4.8), the remedy for this is to rescale the problem.
2This limiting solution could also be obtained using L’Hopital’s rule for the St→0 limit of (4.15).
4.3 The Process of Nondimensionalisation 95 We note that the limit St→0 in (4.14) corresponds to several different physical limiting situations:
• Low fluid viscosity,μ→0,
• Low initial ball velocity,V0→0,
• High ball density,ρ→ ∞,
• Large ball size,R→ ∞.
For any of these physical regimes, (4.13) gives a well-defined model capable of representing the limiting behaviour. Conversely, for the opposite extremes (i.e. high viscosity or high speed or low ball mass), (4.13) would have a divergent coefficient and should not be used in current form.
For example, consider the limit of high fluid viscosity that would be problematic for (4.13). SettingΠ2=1 andΠ3=1 in (4.11) selects the characteristic scales
V=V0, T= 2ρR2
9μ , (4.19)
and the new form of the scaled model:
dv
dt = ˜Π1−v, v(0)=1, (4.20)
withΠ˜1=1/St and yields the limiting solutionv(t)=e−t for St= ∞.