A liquid is in steady state flow in an open trough of rectangular cross section inclined at an angle θ to the horizontal. On what variables would you expect the mass flow per unit time to depend? Obtain the dimensionless groups which are applicable to this problem.

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This problem is similar to Problems 1.11 and 1.12 although, here, the width of the trough and the depth of liquid are to be taken into account. In this case, the mass flow of liquid per unit time, G will depend on: fluid density, ρ; fluid viscosity, μ; depth of liquid, h; width of the trough, a; acceleration due to gravity, g and the angle to the horizontal, θ.
Thus: G=f(ρ,μ,h,a,g,Θ). The dimensions of each variable are: [latex]G= M / T , \rho= M / L ^3[/latex], [latex]\mu= M / L T , h= L , a= L , g= L / T ^2[/latex]and neglecting θ at this stage, with 6 variables with dimensions and 3 fundamental dimensions, there will be [latex](6-3)=3[/latex] dimensionless groups. Taking h, ρ and μ as the recurring set then:

[latex]h \equiv L[/latex], [latex]L =h[/latex]

[latex]\rho \equiv M / L ^3, \quad M =\rho L ^3=\rho h^3[/latex]

[latex]\mu \equiv M / L T , \quad T = M / L \mu=\rho h^3 /(h \mu)=\rho h^2 / \mu[/latex]

Thus: dimensionless group 1: [latex]G T / M =G \rho h^2 /\left(\mu \rho h^3\right)=G / \mu h[/latex]

dimensionless group 2: [latex]a / L =a / h[/latex]
dimensionless group 3: [latex]g T ^2 / L =g \rho^2 h^4 /\left(\mu^2 h\right)=g \rho^2 h^3 / \mu^2[/latex]

and: [latex]\underline{\underline{(G / \mu h)= f \left((a / h)\left(g \rho^2 h^3 / \mu^2\right)\right)}}[/latex]

Question 1.P.16: A liquid is in steady state flow in an open trough of rectan...

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