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?

In the case of steady, laminar, two-dimensional flow over a flat plate, the thickness of the velocity boundary layer can be described by the Blasius equation: \[ \delta = \frac{5.0x}{\sqrt{\mathrm{Re}_x}} \] where \(\delta\) is the boundary layer thickness, \(x\) is the distance from the leading edge, and \(\mathrm{Re}_x\) is the Reynolds number based on distance: \[ \mathrm{Re}_x = \frac{U_\infty x}{\nu} \] Here, \(U_\infty\) is the free-stream velocity, and \(\nu\) is the kinematic viscosity. We will now find out how \(\delta\) depends on the given parameters.

As the distance from the leading edge, \(x\), increases, the Reynolds number based on distance also increases, assuming a constant free-stream velocity and kinematic viscosity. By calculating the derivative of \(\delta\) with respect to \(x\), we can analyze the effect of distance: \[ \frac{d\delta}{dx} = \frac{5.0}{\sqrt{\mathrm{Re}_x}} - \frac{5.0x}{2(\mathrm{Re}_x^{3/2})}\frac{d\mathrm{Re}_x}{dx} \] Since \(d\mathrm{Re}_x/dx = U_\infty/\nu\) is a positive constant, the first term in the derivative is positive, and the second term is negative. However, as \(x\) increases, the magnitude of the second term becomes smaller due to increasing Reynolds number. This indicates that the boundary layer thickness, \(\delta\), increases with increasing distance from the leading edge.

To analyze the effect of free-stream velocity, \(U_\infty\), on boundary layer thickness, we will calculate the derivative of \(\delta\) with respect to \(U_\infty\): \[ \frac{d\delta}{dU_\infty} = -\frac{5.0x}{2(\mathrm{Re}_x^{1/2})} \frac{d\mathrm{Re}_x}{dU_\infty} \] Since \(d\mathrm{Re}_x/dU_\infty = x/\nu\) is a positive constant, the negative sign in the formula means that as the free-stream velocity increases, the boundary layer thickness decreases.

Finally, we will analyze the effect of kinematic viscosity, \(\nu\), on the boundary layer thickness by calculating the derivative of \(\delta\) with respect to \(\nu\): \[ \frac{d\delta}{d\nu} = \frac{5.0x}{2(\mathrm{Re}_x^{1/2})} \frac{d\mathrm{Re}_x}{d\nu}. \] Since \(d\mathrm{Re}_x/d\nu = -U_\infty x/\nu^2\) is a negative constant, the formula shows that as kinematic viscosity increases, the boundary layer thickness also increases. In summary, the thickness of the velocity boundary layer increases with the distance from the leading edge and kinematic viscosity, and it decreases with the free-stream velocity.