Dimensional scaling implementation

[Hai Nguyen]

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:

Scaling:

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

"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.

 

[jvicens]

I understand that you are struggling with understanding how equation 16 was obtained from equation 15 in the paper you are reading. Dimensional scaling is a common concept in science and engineering, where we use dimensionless groups to simplify and analyze complex equations. I will try to explain the process and steps involved in obtaining equation 16 from equation 15, so you can understand and apply the same method when changing Q0 from a constant to a variable function in time.

Firstly, let's look at equation 15, where the dimensionless groups ξ, ζ, τ, γ, λ, Π, Ψ, and Ω are introduced. These groups are obtained by dividing each variable by a characteristic quantity, which is a quantity that has the same dimensions as the variable. This step is crucial as it allows us to remove the units and focus on the relationships between the variables. In this case, the characteristic quantities are l*, h*, t*, p*, q, and w, which are defined in equation 16.

Next, the paper states that six of the dimensionless groups are set to unity, meaning they have a value of 1. This step is called normalization and is commonly done to simplify equations. The remaining two groups, Ψ and Ω, are defined as numbers that control the problem. This means that they are parameters that can be changed to study different scenarios or conditions.

After this, the paper performs some algebraic manipulation to obtain equations for the characteristic quantities in terms of the physical parameters in the problem. For example, l* is expressed in terms of H, Δσ, E', μ, and Q0 in the equation given in the paper. This step involves rearranging equations and using mathematical operations to isolate the characteristic quantities.

Finally, by substituting these expressions for the characteristic quantities into equation 15, we get equation 16, which expresses the dimensionless groups in terms of the physical parameters of the problem. This allows us to analyze the problem and study the effects of changing these parameters on the system.

I hope this explanation helps you understand how equation 16 was obtained from equation 15. Remember to always pay attention to the units and use dimensional analysis to simplify and understand complex equations. Good luck with your research!