Question

Consider steady, laminar, two-dimensional flow over an isothermal plate. Does the thickness of the velocity boundary layer increase or decrease with \((a)\) distance from the leading edge, ( \(b\) ) free-stream velocity, and (c) kinematic viscosity?

Short answer

Answer: The thickness of the velocity boundary layer increases with increasing distance from the leading edge and kinematic viscosity but decreases with increasing free-stream velocity.

Step-by-step solution

Effects of distance from the leading edge on boundary layer thickness

The distance from the leading edge, \(x\), is present in the equation for the boundary layer thickness, so we can analyze its effect directly. Using the Blasius solution, we find that \(\delta\) is directly proportional to the square root of \(x\): \(\delta \propto \sqrt{x}\) This means that the thickness of the velocity boundary layer increases as the distance from the leading edge increases.

Effects of free-stream velocity on boundary layer thickness

The free-stream velocity, \(U_0\), is also present in the equation for the boundary layer thickness. Using the Blasius solution, we see that \(\delta\) is inversely proportional to the square root of \(U_0\): \(\delta \propto \frac{1}{\sqrt{U_0}}\) This means that the thickness of the velocity boundary layer decreases as the free-stream velocity increases.

Effects of kinematic viscosity on boundary layer thickness

The kinematic viscosity, \(\nu\), is the final factor in the equation for the boundary layer thickness. Using the Blasius solution, we find that \(\delta\) is directly proportional to the square root of \(\nu\): \(\delta \propto \sqrt{\nu}\) This means that the thickness of the velocity boundary layer increases as the kinematic viscosity increases. In summary, the thickness of the velocity boundary layer increases with increasing distance from the leading edge and kinematic viscosity but decreases with increasing free-stream velocity.