fa.functional analysis – Dimensional scaling implementation

I’m not familiar with the concept of dimensional scaling at all. Anyone please help me with this problem.

I need to understand how they obtained eq(16) from eq(15) in the paper here so I can do the same when I change Q0 from a constant to a variable function in time. Summary as below:


$ξ=x/l,ζ=z/h_* ,τ=t/t_* ,γ=l/l_* ,λ=h/h_* ,Π=p/p_* ,Ψ=q/q_* ,bar Ψ=bar q ̅/q_* ,Ω=w/w_* ,bar Ω = bar w /w_* (15)$

“The six characteristic quantities that have been introduced are identified by setting to unity, six of the dimensionless groups that emerge from the governing equations when they are expressed in terms of the dimensionless variables. The remaining two groups are numbers that control the problem. After some algebraic manipulation, we obtain the following expressions for the characteristic quantities:”

$ l_*=frac{πH^{4}∆σ^{4}}{4E^{‘3} μQ_0 } $, $ h_*=H $, $ t_*=frac{π^{2} H^{6} ∆σ^{5}}{4μE^{‘4}Q_0^{2} } $ , $ p_*=∆σ $, $ q=frac{Q_0}{2H} $, $w=frac{πH∆σ}{2E^{‘}}=M_0∆σ (16)$

Thanks a lot.