In Example 8.3 we derived the velocity profile for laminar...

In Example 8.3 we derived the velocity profile for laminar flow on a vertical wall by using a differential control volume. Instead, following the procedure we used in Example $5.9,$ derive the velocity profile by starting with the Navier-Stokes equations (Fqs. 5.27 ). Be sure to state all assumptions.

Key Concepts

Boundary Conditions

Boundary conditions are crucial for solving the simplified equations and obtaining the specific velocity profile. In the context of flow along a vertical wall, one usually assumes a no-slip condition at the wall, meaning the fluid velocity is zero at the surface. At the free surface or far into the fluid domain, appropriate conditions such as matching to ambient velocity or zero shear stress are applied. These conditions allow for integration of the governing equations and determination of the integration constants.

Simplifying Assumptions

A number of assumptions are necessary to simplify the Navier-Stokes equations for this problem. Common assumptions include steady-state conditions (no time dependency), incompressible flow (constant density), fully-developed flow (velocity profile does not change along the flow direction), and a Newtonian fluid behavior (linear relationship between shear stress and strain rate). Additionally, the effect of pressure gradients can be simplified based on the orientation of the wall and gravity, reducing the complexity of the momentum balance.

Laminar Flow

Laminar flow is characterized by smooth, orderly motion of fluid particles along well-defined streamlines, with minimal mixing between adjacent layers. This assumption simplifies the analysis by ensuring that the flow regime remains steady and predictable, allowing the use of simplified forms of the Navier-Stokes equations and enabling the derivation of an analytical velocity profile.

Navier-Stokes Equations

The Navier-Stokes equations are the fundamental equations of motion for viscous fluid flow, expressing momentum conservation in fluids. They combine the effects of inertial forces, pressure gradients, viscous stresses, and external forces (such as gravity) to describe how fluid velocity fields evolve in space and time. In the derivation of the velocity profile, these equations serve as the starting point for establishing the balance of forces in the fluid domain.

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