Air flowing over a \(1-\mathrm{m}\)-long flat plate at a velocity of \(7 \mathrm{~m} / \mathrm{s}\) has a friction coefficient given as \(C_{f}=0.664(V x / \nu)^{-0.5}\), where \(x\) is the location along the plate. Determine the wall shear stress and the air velocity gradient on the plate surface at mid-length of the plate. Evaluate the air properties at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\).

Boundary Layer Theory is fundamental in understanding how fluids behave when they flow past solid objects. This theory describes the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.

In the context of the given exercise, the flat plate immersed in air flow creates a zone where air velocity changes from zero at the plate's surface to the free stream velocity at some distance away. This region is known as the boundary layer, and its thickness, denoted by \( \delta \), is a function of the distance from the leading edge of the plate, as well as the flow conditions.

The thickness of the boundary layer increases with distance along the surface, reflecting how the fluid's velocity gradient decreases as you move away from the plate. As for the exercise, calculating the boundary layer thickness at a certain point along the plate allows one to determine how sharply the air velocity changes with distance from the plate, which is essential for determining the air velocity gradient and the resulting shear stress on the plate.

The Reynolds number is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It is expressed as \( Re = \frac{{\rho V L}}{{\mu}} = \frac{{VL}}{{u}} \), where \( \rho \) is the fluid density, \( V \) is the flow velocity, \( L \) is a characteristic length, \( \mu \) is the dynamic viscosity and \( u \) is the kinematic viscosity of the fluid.

In the exercise, calculating the Reynolds number helps us understand the nature of the flow over the plate; whether it is laminar, turbulent, or in a transitional phase. For instance, lower values of Reynolds number typically indicate laminar flow conditions, while higher values suggest turbulent flow.

With a calculated \( Re_x = 2.3 \times 10^5 \) at the midpoint of the plate, this hints towards a transitional to turbulent boundary layer, affecting how the shear stress and velocity gradient would be computed. The selection of equations and coefficients pertaining to wall shear stress might differ based on the nature of the flow, influenced by the Reynolds number.

The velocity gradient, often denoted as \( \frac{{dV}}{{dy}} \), represents the rate at which the fluid velocity changes perpendicular to the flow direction. It is crucial in calculating the shear force experienced by the fluid as it flows along a surface.

The example provided calculates the boundary layer thickness to derive the air velocity gradient at a particular location on a plate. This result illustrates how fast the air velocity changes from the surface to the outer edge of the boundary layer. A high velocity gradient, like the one calculated in the exercise \( (426.83 \, \text{{m/s/m}}) \), often corresponds to a thin boundary layer, indicating a large change over a short distance.

This gradient is closely related to the shear stress on the wall, which is the force per unit area exerted by the fluid on the boundary wall. The shear stress is a product of the dynamic pressure and a friction coefficient, which is contingent on the behavior described by the boundary layer and the velocity gradient.