- Present the equations of linear viscous flow down an inclined plane
Figure 1. Viscous flow down an incline
For a fluid flowing down an inclined slope, the change in potential energy per unit area of the contact surface along some length of the slope \delta x is
\begin{align}
\Delta E_{\mathrm{P}} &= \rho g h \delta z\\
&= \rho g h (\bar{u} S)
\end{align}
where \Delta E_{\mathrm{P}} is the change in potential energy, \rho is the fluid density, g is the acceleration due to gravity, h is the thickness of the flow, \delta z is the elevation change over the distance x, \bar{u} is the average flow velocity, and S is the slope of the incline.
Figure 2. Viscous flow down an incline
The downslope component of the gravity force on the flow is thus \rho g h S, which must be opposed by the drag force at the base of the flow \tau_{0}. Thus, we can say
\tau_{0} = \rho g h S
We can also calculate the shear stress for any position z above the base of the flow, which is a function of the thickness of the overlying fluid (h - z)
\begin{align}
\tau_{z} &= \rho g S (h - z) && \text{Since we know that }\tau_{0} = \rho g h S\\
\tau_{z} &= \tau_{0} \left( 1 - \frac{z}{h} \right)
\end{align}
Attention!
What does this suggest about the shear strength as a function of depth in the fluid?
For a laminar flow, we know \tau = \eta du/dz, so we can rewrite the resistance equation as
\begin{align}
\tau_{z} &= \eta \frac{du}{dz}\\
\tau_{z} &= \rho g S (h - z)\\
\eta \frac{du}{dz} &= \rho g S (h - z)\\
\frac{du}{dz} &= \frac{\rho g S (h - z)}{\eta}\\
\frac{du}{dz} &= \frac{\rho g S}{\eta} (h - z)
\end{align}
If we integrate this equation with respect to z, we find
\begin{align}
\int \frac{du}{dz} &= \frac{\rho g S}{\eta} \int (h - z)\\
u &= \frac{\rho g S}{\eta} \left(zh - \frac{z^{2}}{2} \right) + c_{1}
\end{align}
If we assume the flow velocity u = 0 at z = 0 (the base of the flow), the constant c_{1} = 0, so we are left with
u = \frac{\rho g S}{\eta} \left(zh - \frac{z^{2}}{2} \right)
Attention!
- What would the velocity profile look like in this flow?
- Where is the maximum velocity?
- What happens if the viscosity decreases? Slope increases? Thickness increases?
- The flow is a balance between the gravitational force on the fluid and resistance (drag) at the base
- The flow velocity increases following a parabolic geometry from u = 0 at the base to \frac{\rho g S}{\eta} \frac{z^{2}}{2}
- Steady-state
- 1-D
- Laminar flow!
- Constants are constant
- No temperature dependence