Final answer:
Bernoulli's equation describes the inverse relationship between pressure and velocity in an ideal fluid, giving rise to phenomena like lift on airplane wings. However, within a boundary layer, friction effects and viscous stresses prevent the upward motion that might result from velocity gradients based on Bernoulli's principle alone.
Step-by-step explanation:
The relationship between velocity and pressure in a fluid is explained by the Bernoulli's equation, which is a statement of the conservation of energy for a steady, incompressible fluid in the absence of friction. According to Bernoulli's equation, increased velocity leads to decreased pressure and vice versa, and this equation holds true for fluid at a constant depth. In a boundary layer, there is indeed a velocity gradient, ∂u/∂y, but because the boundary layer also experiences friction and is not an idealized frictionless fluid, Bernoulli's equation cannot be applied directly. Moreover, the pressure gradient across the boundary layer is generally consistent, and the vertical velocity gradient does not necessarily produce sufficient pressure differences for the fluid to curve upward significantly.
Relevant to the airplane wing, Bernoulli's principle demonstrates that the faster air moving over the top of a wing compared to the bottom creates a pressure difference, resulting in lift. Similarly, sails on a boat can generate propulsion due to the pressure differences on either side created by their wing-like shape. However, when examining flow within the boundary layer itself, other forces, such as viscous stresses and friction effects, dominate the behavior and can inhibit the upward motion suggested by a pure Bernoulli effect.