Question

The velocity gradient at a distance $100mm$ from the boundary if the velocity profile is a parabola with the velocity at $200mm$ from the boundary between the boundary layer and free stream flow be $2m/s$ is $se{c}^{-1}$

• A.
  $1$
• B.
  $1.5$
• C.
  $2$
• D.
  $2.33$

Answer 1

The correct option is B

$1.5$
Let the equation of the parabolic velocity profile be
$u=A{y}^2+By+C$

Boundary Conditions

(i) $u=0aty=0$

(ii) $u=2m/saty=0.2m$

(iii) $\frac{du}{dy}=0$ between the boundary layer and force stream flow, at $y=0.2m$

From (i) $\Rightarrow c=0$
From (ii) $\Rightarrow 2=A(0.2{)}^2+B(0.2)....(iv)$
From (iii) $\Rightarrow 0=2A\left(0.2\right)+B......\left(v\right)$

Solving (iv) and (v)
$2=0.04A-(0.4\times 0.2)A$
$2=(0.04-0.08)$
$A=50$
From (iii)
$0=0.4A+B0=-20+BB=20$

Velocity profile, equation is

$u=-50y^2+20y$

Velocity gradient at 100 mm,

${\begin{array}{c}\frac{du}{dy}\end{array}|}_{100num}=-50(0.1{)}^2+20(0.1\left)\right)=-0.5+2=1.5se{c}^{-1}$