Question

Consider steady, laminar, two-dimensional flow over an isothermal plate. Does the wall shear stress increase, decrease, or remain constant with distance from the leading edge?

Short answer

Answer: The wall shear stress decreases with distance from the leading edge.

Step-by-step solution

Understand the problem and identify the governing equations

In this step, we need to understand the problem and identify the governing equations that will help us determine the behavior of the wall shear stress with distance from the leading edge. The boundary layer flows govern the problem, and we can use the boundary layer equations to find the wall shear stress.

Calculate the boundary layer thickness

For laminar flow over a flat plate, we can use the Blasius solution to calculate the boundary layer thickness. The boundary layer thickness can be determined using the formula: \( \delta(x) = \frac{5x}{\sqrt{Re_x}} \) where \( \delta(x) \) is the boundary layer thickness, \( x \) is the distance from the leading edge, and \( Re_x \) is the Reynolds number at the position x, which is given by: \( Re_x = \frac{Ux}{\nu} \) Here, U is the free stream velocity, x is the distance from the leading edge, and \( \nu \) is the kinematic viscosity of the fluid.

Calculate the wall shear stress

The wall shear stress is given by the formula: \( \tau_w = \mu \left(\frac{dU}{dy}\right)_{y=0} \) where \( \tau_w \) is the wall shear stress, \( \mu \) is the dynamic viscosity, and \( \frac{dU}{dy} \) is the velocity gradient at the wall (y=0). To calculate the wall shear stress, we need to find \( \frac{dU}{dy} \) at y=0. We can use the Blasius solution and the similarity variable \( \eta \) to determine the velocity gradient at the wall: \( \eta = \frac{y}{\sqrt{0.5\nu x/U}} \) Our goal is to find the value of \( \frac{dU}{dy} \) at y = 0 (at the wall). Using the Blasius solution, we have: \( \frac{dU}{dy} = U\frac{dF}{d\eta}\frac{d\eta}{dy} \) with \( F^{'}(\eta) = \frac{U(y)}{U} \) and \( \frac{d\eta}{dy} = \sqrt{\frac{0.5U}{\nu x}} \) We can now find the derivative of the similarity function F with respect to the similarity variable \( \eta \). From the Blasius solution, we know that: \( F^{'''}(\eta) + 0.5F(\eta)F^{''}(\eta) = 0 \) with the boundary conditions being: \( F(0) = 0, F^{'}(0) = 0, F^{'}(\infty) = 1 \) By integrating the Blasius equation numerically, or looking up values in tables, we find that: \( \frac{dF}{d\eta}(0) = F^{'}(0) = 0 \) and \( \frac{d^2F}{d\eta^2}(0) = F^{''}(0) = 0.332 \) Now, we can calculate the wall shear stress as: \( \tau_w = \mu U\frac{dF}{d\eta} \frac{d\eta}{dy} \) Substituting the known values, we get: \( \tau_w = \mu U \cdot F^{''}(0) \cdot \sqrt{\frac{0.5U}{\nu x}} \) \( \tau_w = 0.332\mu\sqrt{\frac{U^3}{\nu x}} \)

Determine the behavior of wall shear stress with distance from the leading edge

The wall shear stress is given by the formula: \( \tau_w = 0.332\mu\sqrt{\frac{U^3}{\nu x}} \) We can determine the behavior of wall shear stress with distance from the leading edge by analyzing the above equation. As x increases, the denominator in the wall shear stress equation becomes larger, causing the wall shear stress to decrease. Therefore, the wall shear stress decreases with distance from the leading edge.