2.3.1 The approximation for Re < 0.5

Let's select two ordered pairs, (log Re, log CD), to construct a line to approximate the curve for Re < 0.5.


  logre1 = -1.6;
  logcd1 = 3.0;
  logre2 = 0.5;
  logcd2 = 1.0;
  Clear[cd, re]
  logeqn = cd-logcd1 == (logcd2-logcd1)/(logre2-logre1) *
                              (Log[10,re] - logre1)

Solving for logy,


  newlogeqn = Solve[logeqn, cd]

Now let's clean it up.


  logdragcoef = Simplify[ newlogeqn[[1,1,2]] ]

Now let's get rid of the logarithms by taking the exponential of both sides of the equation, base 10.


  dragcoef = 10^logdragcoef

Clean this up, too.


  dragcoefa = Expand[dragcoef]


  dragcoefb = Simplify[%]

Oops! Mathematica didn't have a rule available to simplify this. We will have to simplify it ourselves. We know that Log[re]/Log[10] == Log[10,re], so
29.9358/10^(0.952381 Log[re]/Log[10])
== 29.9358 10^(-0.952381 Log[re]/Log[10])
== 29.9358 (10^Log[10,re])^-0.952381
== 29.9358 re^-0.952381
Therefore,


  Clear[cd,re]
  coefofcd = 29.9358;
  expofcd = -0.952381;
  cd[re_] = coefofcd re^expofcd

We can check our approximation with a graph.


  Clear[logcd]
  linplot = Plot[Log[10,coefofcd] + expofcd logre, 
          {logre,-2.0,0.5},
          PlotStyle -> {{RGBColor[0,0,1],Thickness[0.01]}},
          DisplayFunction -> Identity];
  Show[logplot,linplot, DisplayFunction -> $DisplayFunction];

The approximation looks great!

Exercise 2

Graph Stokes' theoretical result, CD = 24/Re, on the same log-log
plot as the experimental data for CD and Re. How well does it
fit the experimental data?

Up to 2.3 Piecewise linear approximation of the log-log graph