Nondimensionalizing the Rayleigh-Plesset Equation

In[11]:=

ndRPEq1 = RPEq/.{r[t] R0 R[τ], r '[t] R0 fn R '[τ], r''[t] fn^2 R0 R ' '[τ], F[x/cl - t] ->Fhat[τ], F '[x/cl - t] fn Fhat '[τ]}

General :: spell1 : Possible spelling error: new symbol name \"Fhat\" is similar to existing symbol \"Flat\". More…

Out[11]=

P0/ρ - (P0 Fhat[τ])/ρ - (P0 Pg0star (1/R[τ])^(3 k))/ρ + (2 σ)/(R ... (3 k)/2) R^′′[τ])/(cg0 cl ρ) + fn^2 R0^2 R[τ] R^′′[τ]

In[12]:=

ndRPEq2 = ndRPEq1 /. {σ->2 ρ (R0 fn)^2 R0/We, μ->ρ (R0 fn) R0/ReN, cg0->fn R0/Mg, P0->P0star ρ R0^2 fn^2/2}

Out[12]=

1/2 fn^2 P0star R0^2 - 1/2 fn^2 P0star R0^2 Fhat[τ] - 1/2 fn^2 P0star Pg0star R0^2 (1/R[& ... ])^(1/2 + (3 k)/2) R^′′[τ])/(2 cl) + fn^2 R0^2 R[τ] R^′′[τ]

Check Dimensional Consistency

In[13]:=

ndRPEq2/.{R0m, fn1/s, ρkg/m^3, clm/s, μkg/(m ... Fhat[τ] 1, Fhat '[τ] 1, Re1, P0star2, Pg0star1}

Out[13]=

(9 m^2)/(2 s^2) + (4 m^2)/(ReN s^2)

Created by Mathematica  (November 26, 2006)