(a) For laminar draining of a wide, thin sheet of water on...

(a) For laminar draining of a wide, thin sheet of water on pavement sloped at angle $\theta$, as in Fig. $\mathrm{P} 4.36$, show that the flow rate is given by \[ Q=\frac{\rho g b h^{3} \sin \theta}{3 \mu} \] where $b$ is the sheet width and $h$ its depth. $(b)$ By (somewhat laborious) comparison with Eq. $(10.13),$ show that this expression is compatible with a friction factor $f=24 / \mathrm{Re}$ \[ \text { where } \operatorname{Re}=V_{\mathrm{av}} h / \nu \].

(a) For laminar draining of a wide, thin sheet of water on pavement sloped at angle $\theta$, as in Fig. $\mathrm{P} 4.36$, show that the flow rate is given by
\[
Q=\frac{\rho g b h^{3} \sin \theta}{3 \mu}
\]
where $b$ is the sheet width and $h$ its depth.
$(b)$ By (somewhat laborious) comparison with Eq. $(10.13),$ show that this expression is compatible with a friction factor $f=24 / \mathrm{Re}$
\[
\text { where } \operatorname{Re}=V_{\mathrm{av}} h / \nu
\].

Key Concepts

Laminar Flow

Laminar flow refers to fluid motion in which the fluid particles move in parallel layers without disruption between them. In this regime, viscous forces dominate, and the flow exhibits a smooth, steady behavior that can often be described by analytical solutions of the Navier–Stokes equations. This concept is central when analyzing flows like the gravity?driven draining of water over an inclined surface, where the flow is orderly and viscosity governs the momentum transfer.

Thin Film (Lubrication) Theory

Thin film or lubrication theory is an approximation method used when the thickness of a fluid layer is small compared to its other dimensions. It simplifies the governing equations by reducing the complexity associated with the velocity gradients across the film thickness. This is particularly relevant in the problem, since the thin sheet of water allows for simplifications that lead to an expression for the flow rate based on the film thickness and the gravitational driving force.

Gravitational Driving Force

The gravitational driving force is the component of gravity that acts along the slope of the surface. In inclined flows, this component, often expressed as ?g sin ?, is responsible for initiating and sustaining the flow. It is crucial in deriving the flow rate of the draining fluid by balancing it against the viscous resistance offered by the fluid's viscosity.

Friction Factor in Viscous Flows

The friction factor quantifies the resistance to flow due to viscous effects in a channel or over a surface. In laminar flow regimes, especially for simple geometries, the friction factor can be expressed in terms of the Reynolds number. The result f = 24/Re in the problem exemplifies how viscous dissipation scales with the flow’s inertial characteristics, and it provides a bridge between the derived flow rate law and the conventional frictional description used in fluid mechanics.

Reynolds Number

The Reynolds number is a dimensionless quantity that compares the relative importance of inertial forces to viscous forces in a flow. It is defined here using the average velocity and a characteristic length (film depth), and plays a key role in determining whether the flow is laminar or turbulent. In the context of the problem, it serves as the basis for expressing the friction factor, linking the microscopic properties of the flow (viscosity, velocity gradients) to the overall flow behavior.

Please give Ace some feedback