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I have the following configuration:

enter image description here

in which a viscous fluid with dynamical viscosity $\mu$ and density $\rho$ slides down the inclined plane due to gravity $g$.

After having solved the Navier-Stokes equation for this extremely symmetric and idealized situation I am asked to find the force per unit area the inclined surfaces feels using dimensional analysis, and then to compare the result with the analytical one.

What I did was to build a "table of units" as follows:

enter image description here

where the leftmost column L, M and T stand for units of length, mass and time and the uppermost row indicates each of the parameters of the problem. $F$ is the net force, not per unit area. And $\nu = \mu / \rho$ is the kinematic viscosity. Watching at this table it seems obvious to propose that $F \propto p_0$. Thus $$ F = p_0 d^\alpha g^\beta \nu^\gamma $$ from where I obtain the following equations for each unit

enter image description here

This tuns to be an undetermined system so one of the exponents in $ F = p_0 d^\alpha g^\beta \nu^\gamma $ is free.

How do I know which one is free? Is all this ok?

Thanks!

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closed as off-topic by John Rennie, stafusa, Kyle Kanos, Jon Custer, Chris Mar 9 '18 at 0:11

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  • $\begingroup$ If one of the variables is free, you just sum over all possible configurations of the variable with undetermined coefficients. physics.stackexchange.com/questions/389586/… might help you $\endgroup$ – Yuzuriha Inori Mar 8 '18 at 4:14
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    $\begingroup$ I didn't see $\rho$ in your table. $\endgroup$ – JEB Mar 8 '18 at 4:37
  • $\begingroup$ Are you trying to find the normal component of the force or the shear component? $\endgroup$ – Chet Miller Mar 8 '18 at 12:24
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Dimensional analysis usually requires you to think about which physics is important to the problem.

In this case you are asked to find the force per area on the ramp, and I assume that means the force in the x direction due to viscosity. That force is equal to the force exerted by the ramp on the fluid.

Now the fluid is in steady state so the sum of all external forces is zero. The external pressure $p_0$ produces a force in the y direction not the $x$, so it's not important here. (By the way you could know that $F_x \propto p_0$ is wrong because even if $p_0=0$ the force on the ramp definitely should not be zero). The only other external force is gravity which depends on the mass and $g$. $\mu$ should not appear in your result since that describes the internal forces.

Now you only have $\rho, d, g$ to play with and dimensional analysis leads to a unique answer up to a dimensionless constant.

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A force per unit area has dimension $L^{-1}MT^{-2}$, as does $\rho(\nu/d)^2$. Since $d^3g/\nu^2$ is dimensionless, $\rho d g$ has the right units too. As octonion has noted, we expect a result $\propto dg$.

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  • $\begingroup$ Sorry to keep bugging you J.G., but if there were no gravity, would there be a force on the ramp? $\endgroup$ – octonion Mar 8 '18 at 9:54

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