In incompressible fluid flows over a flat plate with zero pressure gradient. The boundary layer thickness is 1 mm at a location where the Reynolds number is 1000. If the velocity of the fluid alone is increased by a factor of 4, then the boundary layer thickness at the same location, in mm, will be

Explanation:

Boundary Layer Thickness in Incompressible Flow

Definition: In fluid dynamics, the boundary layer is the thin region adjacent to a solid surface where the effects of viscosity are significant. For incompressible flow over a flat plate with zero pressure gradient, the boundary layer thickness depends on the Reynolds number and the velocity of the fluid. The Reynolds number (Re) is a dimensionless quantity that represents the ratio of inertial forces to viscous forces in the flow.

For laminar flow over a flat plate with zero pressure gradient, boundary layer thickness is inversely proportional to the square root of the Reynolds number, i.e.,

\( \delta \propto \frac{1}{\sqrt{Re}} \)

Calculation:

Given:

Initial Reynolds number, \( Re_1 = 1000 \)

Initial boundary layer thickness, \( \delta_1 = 1 \text{ mm} \)

Velocity increased by a factor of 4 ⇒ \( U_2 = 4U_1 \)

Reynolds number is directly proportional to velocity, so \( Re_2 = 4 \times Re_1 = 4000 \)

Now,

\( \frac{\delta_2}{\delta_1} = \sqrt{\frac{Re_1}{Re_2}} = \sqrt{\frac{1000}{4000}} = \frac{1}{2} \)

\( \delta_2 = 1 \times \frac{1}{2} = 0.5~\text{mm} \)