---
title: "Shear stress, displacement thickness, drag on plate"
subject: "Fluid Mechanics"
chapter: 8
key: "semester_3/fluid_mechanics:flat-plate-shear-displacement-drag"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/flat-plate-shear-displacement-drag
site: BCT Engineering
---

# Shear stress, displacement thickness, drag on plate

Fluid Mechanics · Chapter 8: Boundary Layer theory

- Asked in: 2076 Asoj, 2+2+4 marks
- Video: [Introductory Fluid Mechanics L19 p7 - Displacement and Momentum Thickness](https://www.youtube.com/watch?v=h_vk0ZYMBVc) — Ron Hugo
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/flat-plate-shear-displacement-drag
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-8

## Question

Air flows over a flat plate 2 m long and 1.5 m wide at a velocity of 6.5 m/s. Determine the shear stress, and displacement thickness at distance of 1.8 m from the leading edge. Also determine the drag force on the face of the plate.

## Short answer

Re_x = 7.8×10⁵, so turbulent. δ = 0.37x/Re_x⁰·² = 0.0442 m, δ* = δ/8 = 5.5 mm. C_f = 0.0576/Re_x⁰·² = 0.00382, τ₀ = C_f·½ρU² = 0.0968 N/m². C_D = 0.074/Re_L⁰·² = 0.00480, drag force = C_D·½ρU²·(L·B) = 0.365 N on one face.

## Answer

Find the Reynolds number first to decide whether the boundary layer is laminar or turbulent, then use the matching formulas.

#### Assumptions

Air at about 20 °C: $\rho = 1.2\ \text{kg/m}^3$, $\nu = 1.5\times10^{-5}\ \text{m}^2/\text{s}$. $U = 6.5$ m/s, $L = 2$ m, $B = 1.5$ m, $x = 1.8$ m.

#### Reynolds numbers

$$
Re_x = \frac{Ux}{\nu} = \frac{6.5(1.8)}{1.5\times10^{-5}} = 7.8\times10^5,\qquad Re_L = \frac{6.5(2)}{1.5\times10^{-5}} = 8.67\times10^5
$$

Both exceed the critical value $5\times10^5$, so the boundary layer is taken as **turbulent** (1/7 power law, valid for $Re < 10^7$).

#### Boundary layer thickness and displacement thickness at x = 1.8 m

$$
\begin{aligned}
\delta &= \frac{0.37\,x}{Re_x^{0.2}} = \frac{0.37(1.8)}{(7.8\times10^5)^{0.2}} = 0.0442\ \text{m} \\
\delta^* &= \frac{\delta}{8} = 0.00552\ \text{m} = 5.52\ \text{mm}
\end{aligned}
$$

#### Shear stress at x = 1.8 m

$$
\begin{aligned}
C_f &= \frac{0.0576}{Re_x^{0.2}} = 0.003819 \\
\tau_0 &= C_f\cdot\tfrac12\rho U^2 = 0.003819(0.5)(1.2)(6.5)^2 = 0.0968\ \text{N/m}^2
\end{aligned}
$$

#### Drag force on one face

$$
\begin{aligned}
C_D &= \frac{0.074}{Re_L^{0.2}} = 0.004805 \\
F_D &= C_D\cdot\tfrac12\rho U^2\,(L B) = 0.004805(0.5)(1.2)(6.5)^2(2\times1.5) = 0.365\ \text{N}
\end{aligned}
$$

If the laminar length at the leading edge is allowed for, $C_D = 0.074/Re_L^{0.2} - 1700/Re_L = 0.002843$ and $F_D = 0.216$ N.

**Answer: $\tau_0 \approx 0.097$ N/m², $\delta^* \approx 5.5$ mm, drag on one face $\approx 0.37$ N (about 0.22 N if the initial laminar part is included).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How do we decide which formulas to use?
   A. Compare plate length with 1 m
   B. Always use Blasius formulas
   C. Compute Re_x and compare with 5 x 10^5; here 7.8 x 10^5 gives turbulent
   D. Compare velocity with 10 m/s
   Answer: C. Both Re_x and Re_L exceed 5 x 10^5, so turbulent (1/7 power law) formulas are used. (Key point: Deciding the regime)

2. How is delta* obtained in the turbulent case?
   A. delta* = delta/3
   B. delta* = delta/8 with delta = 0.37x/Re_x^0.2
   C. delta* = 1.72 x/sqrt(Re_x)
   D. delta* = delta
   Answer: B. For the 1/7 profile delta*/delta = 1/8, giving 5.52 mm. (Key point: Displacement thickness)

3. How is tau_0 found at x = 1.8 m?
   A. tau_0 = Cf x 1/2 rho U^2 with Cf = 0.0576/Re_x^0.2
   B. tau_0 = CD x 1/2 rho U^2 with CD = 0.074/Re_L^0.2
   C. tau_0 = mu U/x
   D. tau_0 = rho U^2
   Answer: A. Local shear uses the local coefficient Cf; the drag uses the average CD. (Key point: Wall shear stress)

4. Which formula gives the drag on one face of the plate?
   A. F_D = tau_0 x L B using the local shear at 1.8 m
   B. F_D = Cf x rho U^2
   C. F_D = 2 x tau_0
   D. F_D = CD x 1/2 rho U^2 x (L B) with CD = 0.074/Re_L^0.2
   Answer: D. Average CD over the full length L applies to the area L x B, giving about 0.37 N. (Key point: Drag force)

## Related in Fluid Mechanics

- [Jet on flat plate, force and efficiency](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Ajet-on-flat-plate-efficiency) — Fluid Mechanics, chapter 7
- [Counter weight for plate under air flow](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aflat-plate-balance-air-flow-mass) — Fluid Mechanics, chapter 9
- [Pressure change from displacement thickness in duct](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aduct-displacement-thickness-pressure) — Fluid Mechanics, chapter 8
- [BL zones sketch and displacement thickness](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Abl-characteristics-sketch-displacement) — Fluid Mechanics, chapter 8
- [Jet on hinged plate, force and angle](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Ajet-on-hinged-plate-force-angle) — Fluid Mechanics, chapter 7
- [BL, displacement thickness and ratio, 1/7 profile](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Abl-thickness-displacement-ratio-1-7) — Fluid Mechanics, chapter 8

## Related in other subjects

- [Laminar forced convection over a heated flat plate](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Alaminar-flow-over-heated-plate-air) — Heat Transfer, chapter 3
- [Force of jet on flat plate, stationary and moving](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Ajet-on-flat-plate-numerical) — Fluid Machines, chapter 1
- [Flat rolling draft, roll force and power](https://www.bctengineer.com/question.md?key=semester_4%2Fmanufacturing_and_production_processes%3Aflat-rolling-force-power-numerical) — Manufacturing and Production Processes, chapter 5
- [Diameter to thickness ratio for stress limit](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Apipe-diameter-thickness-ratio-hoop-stress) — Hydraulics, chapter 4