A plate of 500 mm × 200 mm has been placed longitudinally in a stream of an oil which flows with a velocity of 1.0 m/s. The kinematic viscosity of the oil was 0.5 stoke. The thickness of the boundary layer at the trailing edge is _______.
25 mm
Understanding Boundary Layer Thickness
The question asks us to determine the thickness of the boundary layer at the trailing edge of a flat plate placed in a stream of oil. The boundary layer is the thin layer of fluid near the surface of the plate where viscous effects are significant.
To solve this problem, we need to use the concept of the Reynolds number to determine the flow regime (laminar or turbulent) and then apply the appropriate formula for boundary layer thickness.
Given Parameters for Boundary Layer Calculation
- Plate Length (L): 500 mm = \(0.5 \, \text{m}\) (This is the length along the flow direction)
- Plate Width: 200 mm
- Free stream velocity (U): \(1.0 \, \text{m/s}\)
- Kinematic viscosity of oil (\(\nu\)): \(0.5 \, \text{stoke}\)
Converting Kinematic Viscosity Units
The kinematic viscosity is given in stokes. We need to convert it to standard SI units (m²/s).
We know that:
- \(1 \, \text{stoke} = 1 \, \text{cm}^2/\text{s}\)
- \(1 \, \text{cm} = 10^{-2} \, \text{m}\)
- \(1 \, \text{cm}^2 = (10^{-2} \, \text{m})^2 = 10^{-4} \, \text{m}^2\)
Therefore,
\(\nu = 0.5 \, \text{stoke} = 0.5 \times 10^{-4} \, \text{m}^2/\text{s}\)
Calculating Reynolds Number at Trailing Edge
The Reynolds number (\(Re\)) is a dimensionless quantity that helps predict flow patterns. For flow over a flat plate, it is calculated as:
\(Re_x = \frac{U \times x}{\nu}\)
where x is the distance from the leading edge.
We need the boundary layer thickness at the trailing edge, which is at \(x = L = 0.5 \, \text{m}\). So, we calculate the Reynolds number at the trailing edge:
\(Re_L = \frac{U \times L}{\nu}\)
\(Re_L = \frac{1.0 \, \text{m/s} \times 0.5 \, \text{m}}{0.5 \times 10^{-4} \, \text{m}^2/\text{s}}\)
\(Re_L = \frac{0.5}{0.5 \times 10^{-4}}\)
\(Re_L = 10^4\)
Determining Flow Regime (Laminar or Turbulent)
For flow over a flat plate, the transition from laminar to turbulent flow typically occurs around a critical Reynolds number of \(Re_{crit} \approx 5 \times 10^5\). Since our calculated \(Re_L = 10^4\), which is much less than \(5 \times 10^5\), the flow over the entire plate is laminar.
Calculating Boundary Layer Thickness for Laminar Flow
For a laminar boundary layer over a flat plate, the thickness (\(\delta\)) at a distance x from the leading edge is given by the Blasius solution (often approximated as):
\(\delta(x) = \frac{5x}{\sqrt{Re_x}}\)
We want the thickness at the trailing edge, so we use \(x = L\):
\(\delta_L = \frac{5L}{\sqrt{Re_L}}\)
Substituting the values:
\(\delta_L = \frac{5 \times 0.5 \, \text{m}}{\sqrt{10^4}}\)
\(\delta_L = \frac{2.5}{\sqrt{10000}}\)
\(\delta_L = \frac{2.5}{100}\)
\(\delta_L = 0.025 \, \text{m}\)
Converting Thickness to Millimeters
The options are given in millimeters, so we convert the thickness from meters to millimeters:
\(\delta_L = 0.025 \, \text{m} \times \frac{1000 \, \text{mm}}{1 \, \text{m}}\)
\(\delta_L = 25 \, \text{mm}\)
Summary of Calculation Steps
| Step | Description | Calculation | Result |
|---|---|---|---|
| 1 | Convert kinematic viscosity | \(0.5 \, \text{stoke} = 0.5 \times 10^{-4} \, \text{m}^2/\text{s}\) | \(0.5 \times 10^{-4} \, \text{m}^2/\text{s}\) |
| 2 | Calculate Reynolds number at trailing edge | \(Re_L = \frac{1.0 \times 0.5}{0.5 \times 10^{-4}}\) | \(10^4\) |
| 3 | Confirm flow regime | \(Re_L < 5 \times 10^5\) | Laminar |
| 4 | Calculate boundary layer thickness (m) | \(\delta_L = \frac{5 \times 0.5}{\sqrt{10^4}}\) | \(0.025 \, \text{m}\) |
| 5 | Convert thickness to mm | \(0.025 \, \text{m} \times 1000 \, \text{mm/m}\) | \(25 \, \text{mm}\) |
Conclusion on Boundary Layer Thickness
Based on the calculations, the thickness of the boundary layer at the trailing edge of the plate is 25 mm.
Important Questions from Boundary Layer Theory
Friction drag is generally larger than the pressure drag in _______.
When a fluid flows along a solid boundary, more and more fluid gets retarded in the vicinity of the boundary. This deceleration is accompanied by:
As defined by Prandtl, in convective heat transfer, a thin layer next to the wall where shear stress is confined is called _____.
Thickness of laminar boundary layer on flat plate:
Separation of fluid flow is caused by
