Explanation:
The laminar boundary layer is a thin layer of fluid that forms on the surface of a solid body in contact with a fluid. The thickness of this boundary layer is an important parameter in the study of fluid mechanics.

The thickness of the laminar boundary layer over a flat plate can be determined using the concept of boundary layer theory. According to this theory, the thickness of the boundary layer varies with the distance from the leading edge of the plate.

The thickness of the laminar boundary layer at a distance x from the leading edge over a flat plate varies as x^1/2. This relationship can be derived using the following equation:

δ = (5x/Re)^1/2

where δ is the thickness of the laminar boundary layer, x is the distance from the leading edge of the plate, and Re is the Reynolds number.

The Reynolds number is a dimensionless parameter that describes the ratio of inertial forces to viscous forces in a fluid. It is defined as:

Re = ρUL/μ

where ρ is the density of the fluid, U is the velocity of the fluid, L is a characteristic length scale, and μ is the viscosity of the fluid.

The thickness of the laminar boundary layer is directly proportional to the square root of the distance from the leading edge of the plate. This means that as the distance from the leading edge increases, the thickness of the boundary layer increases at a slower rate.

The thickness of the laminar boundary layer is also directly proportional to the square root of the Reynolds number. This means that as the Reynolds number increases, the thickness of the boundary layer decreases.

Conclusion:
The thickness of the laminar boundary layer over a flat plate varies as x^1/2, where x is the distance from the leading edge of the plate. This relationship can be derived using the concept of boundary layer theory and the Reynolds number. As the distance from the leading edge increases, the thickness of the boundary layer increases at a slower rate. As the Reynolds number increases, the thickness of the boundary layer decreases.