The sloped open channel has steady laminar flow at a depth...

The sloped open channel has steady laminar flow at a depth $h$. Show that the Navier-Stokes equations reduce to $\partial^{2} u / \partial y^{2}=-(\rho g \sin \theta) / \mu$ and $\partial p / \partial y=-\rho g \cos \theta .$ Integrate these equations to show that the velocity profile is $u=[(\rho g \sin \theta) / 2 \mu]\left(2 h y-y^{2}\right) \quad$ and the shear-stress distribution is $\tau_{x y}=\rho g \sin \theta(h-y)$.

Key Concepts

Integration of Differential Equations with Boundary Conditions

After reducing the governing equations to second-order differential equations, solving them requires integration. The integration constants are determined using appropriate boundary conditions, such as the no-slip condition at solid boundaries and specified conditions at free surfaces. This step is vital for obtaining explicit analytical expressions for the velocity profile and shear stress distribution that satisfy the physical constraints of the problem.

Shear Stress Distribution

Shear stress in a fluid is a measure of the internal friction due to viscosity and is directly related to the velocity gradient across the fluid layers. In laminar flows, shear stress distributions can be derived by integrating the momentum equations. The resulting expressions often illustrate how shear stress varies with position in the flow, which is crucial for determining factors like frictional resistance and erosion potential in open channel flows.

Laminar Flow

Laminar flow refers to the regime where the fluid moves in orderly layers with minimal mixing between them, characterized by low Reynolds numbers. In laminar flow, the velocity profile is smooth and predictable, and the viscous forces play a dominant role. This property allows for the assumption that the flow is steady and simplifies the mathematical treatment such as reducing the Navier–Stokes equations for analysis.

Navier?Stokes Equations

These fundamental equations govern the motion of fluid substances by expressing the conservation of momentum. In fluid mechanics, they provide a balance between inertial forces, pressure gradients, viscous forces, and body forces such as gravity. In problems involving steady laminar flow, the inertial terms can often be neglected or simplified, leading to reduced forms that are easier to solve analytically.

Gravitational Force Decomposition

In many fluid flow problems involving inclined channels or surfaces, the gravitational force must be decomposed into components parallel and perpendicular to the surface. This decomposition is essential because the parallel component influences the momentum balance (driving the flow), while the perpendicular component contributes to the hydrostatic pressure gradient. Understanding this separation is key to setting up the proper momentum and pressure equations.

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