The variation of convective thermal resistance with radius over a cylindrical surface is hyperbolic. This means that as the radius increases, the convective thermal resistance decreases, but at a decreasing rate.

Explanation:

Convective thermal resistance is a measure of how easily heat can be transferred through a fluid medium. In the case of convection over a cylindrical surface, the convective thermal resistance is influenced by the fluid flow and boundary layer formation.

When fluid flows over a cylindrical surface, it forms a boundary layer, which is a thin layer of fluid adjacent to the surface. The thickness of this boundary layer increases as the flow moves along the surface. As a result, the heat transfer from the surface to the fluid is hindered by the thicker boundary layer, leading to a higher convective thermal resistance.

As the radius of the cylindrical surface increases, the curvature of the surface decreases. This leads to a decrease in the velocity of the fluid near the surface, which in turn reduces the thickness of the boundary layer. As a result, the heat transfer from the surface to the fluid becomes more efficient, leading to a decrease in convective thermal resistance.

However, the decrease in convective thermal resistance is not linear with radius. Initially, as the radius increases, the decrease in convective thermal resistance is significant. But as the radius continues to increase, the decrease in convective thermal resistance becomes less pronounced. This is because the reduction in boundary layer thickness becomes less significant as the radius increases further.

This variation can be described by a hyperbolic function, where the decrease in convective thermal resistance is rapid initially and then gradually levels off. This is why the correct answer is option 'B', hyperbolic.