First of all, I am not a mathematician and my dimensional analysis skills are almost non-existent; so I express my apologies in advance if this question is silly.

I am reading a fluid dynamics paper in which the dimensionless form of a matrix (mass matrix) contains a factor that I simply cannot understand.

The dimensional form of the matrix contains a terms of the form

$$M=\rho\int_{0}^{x}\frac{\intop_{0}^{x_{2}}g\left(x_{1}\right)\,\mathrm{d}x_{1}}{h\left(x_{2}\right)}\,\mathrm{d}x_{2} \tag{1}$$

being $x_{1}$ and $x_{1}$ dummy integration variables.

In the paper, the dimensionless form of $M$ is presented as

$$\bar{M}=\frac{4}{\rho L^{2}}M \tag{2}$$

the overbar indicates a non-dimensional quantity.

The non-dimensionalization is based on $$\overline{g}=\frac{g\left(x\right)}{h_{0}},\,\,\,\,\,\,\,\,\overline{h}=\frac{h\left(x\right)}{h_{0}},\,\,\,\,\,\,\,\,\overline{x}=\frac{x}{L} \tag{3}$$

being $[g\left(x\right)]=[h\left(x\right)]=[h_{0}]=[x]=L$.

If a inject the non-dimensional parameters (3) in Eq. (1) I get

$$M=\rho\int_{0}^{\hat{x}}\frac{\intop_{0}^{\bar{x}_{2}}\bar{g}\left(\bar{x}_{1}\right)h_{0}\,\mathrm{d}\left(L\bar{x}_{1}\right)}{\bar{h}\left(\bar{x}_{2}\right)h_{0}}\,\mathrm{d}\left(L\bar{x}_{2}\right)=\rho L^{2}\int_{0}^{\hat{x}}\frac{\intop_{0}^{\bar{x}_{2}}\bar{g}\left(\bar{x}_{1}\right)\,\mathrm{d}\bar{x}_{1}}{\bar{h}\left(\bar{x}_{2}\right)}\,\mathrm{d}\bar{x}_{2}=\rho L^{2}\bar{M} \tag{4}$$

which differs from (1) in the factor 4. My question is, where this factor came from?


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