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A thin plate is moving in still atmosphere air at a velocity of 5m/s.

A thin plate is moving in still atmosphere air at a velocity of 5m/s. The length of the plate is 0.6m and width 0.5m.

Calculate :

1. Thickness of Boundary Layer at the end of the plate

2. Drag force on one side of the plate.

Take density of air $1.24 kg/m^3$ and kinematic viscosity 0.15 stokes.

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Solution :

Given :

Velocity of plate, $\quad u=5 \mathrm{~m} / \mathrm{s}$

Length of plate, $\quad L=0.6 \mathrm{~m}$

Width of plate, $\quad b=0.5 \mathrm{~m}$

Density of air, $\quad p=1.24 \mathrm{~kg} / \mathrm{m}^{3}$

Kinematic viscosity, $\quad v=0.15$ stokes

$=0.15 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}$

Reynold number,

\begin{aligned} \quad R_{e}&=\frac{u L}{{v}}=\frac{5 \times 0.6}{0.15 \times 10^{-4}}\\ &=200000 \end{aligned}.

As $R_e$ is less than $5 \times 10^{5}$, hence boundary layer is laminar over the entire length of the plate.

(i) Thickness of boundary layer at the end of the plate by Blasius's solution is

\begin{aligned} \delta =\frac{4.91 x}{\sqrt{R_{e_{x}}}} &=\frac{4.91 L}{\sqrt{R_{e_{1}}}}\\ &=\frac{4.91 \times 0.6}{\sqrt{200000}}\\ &=0.00658 {~m}\\ &=6.58 {~mm} \end{aligned}

(ii) Drag force on one side of the plate is given by equation as

\begin{aligned} C_{D} &=\frac{F_{D}}{\frac{1}{2} \rho A u^{2}} \\ F_{D} &=\frac{1}{2} \rho A u^{2} \times C_{D} \end{aligned}

where $C_{D}$ from Blasius's solution,

\begin {aligned}C_{D}=\frac{1.328}{\sqrt{R_{e}}}&=\frac{1.328}{\sqrt{200000}}\\ &=0.002969\\ &=0.00297 \end{aligned}

\begin{aligned} F_{D} &=\frac{1}{2} \times 1.24 \times 0.6 \times 0.5 \times 5^{2} \times 0.00297 \\ &=0.01373 \end{aligned}

$\quad\{\because A=L \times b=0.6 \times 0.5\}$