Fluid Mechanics

In a permanent regime :

\(\frac{{\partial \vec v}}{{\partial t}} = \vec 0\)

Convective acceleration is equal to , with the hypotheses of the statement : (\(\vec v=v(z)\;\vec e_x\))

\((\vec v.\overrightarrow {grad} )\vec v = \left( {{v}(z).\frac{\partial }{{\partial x}}} \right){v}(z) = 0\)

Therefore :

\(- \overrightarrow {grad} P + \rho \vec g + \eta \Delta \vec v = \vec 0\)

The previous equation gives, in projection ( and \(P\) does not depend on\( y\)) :

\(\frac{{\partial P}}{{\partial x}} = \rho g\sin \theta + \eta \frac{{{d^2}v}}{{d{z^2}}}\;\;\;\;\;\;\;;\;\;\;\;\;\;\;\frac{{\partial P}}{{\partial z}} = - \rho g\cos \theta\)

The second equation given by integration :

\(P(x,z) = - \rho g\cos \theta \;z + cste = \rho g\cos \theta (h - z) + {P_0}\)

The first becomes, noting that \(\frac{{\partial P}}{{\partial x}} = 0\) :

\(\rho g\sin \theta + \eta \frac{{{d^2}v}}{{d{z^2}}} = 0\;\;\;\;\;so\;\;\;\;\;v(z) = - \frac{{\rho g\sin \theta }}{{2\eta }}{z^2} + az + b\)

The velocity is zero at \(z = 0\) and the air does not exert tangential force on the liquid at \(z = h\), thus :

\(dv/dz=0\) at \(z=h\)

So :

\(v(z) = \frac{{\rho g\sin \theta }}{{2\eta }}(2h - z)z\)

A parabolic velocity profile with a maximum at \(z = h\) is obtained.

The volumetric flow rate is :

\({D_v} = L\int_0^h {\frac{{\rho g\sin \theta }}{{2\eta }}(2h - z)z} \;dz = \frac{{\rho gL{h^3}\sin \theta }}{{3\eta }}\)

And the average velocity of the flow is :

\({v_{moy}} = \frac{{{D_v}}}{{Lh}} = \frac{{\rho g{h^2}\sin \theta }}{{3\eta }}\)

We get :

\(v_{moy}=0,61\;mm.s^{-1}\)

And the Reynolds number is :

\(R_e = \frac {\rho h v_{moy}}{\eta}=6,5\;10^{-4}<<1\)

Viscosity forces are sufficient to impose a field of laminar speeds.