Question

1. Inviscid incompressible fluid pass a cylinder having a radius of R as shown in the figure below. The fluid velocity along the dividing streamline is found to be V = V, (1-R /x) a) Determine the pressure gradient (ap/ ôx) along the streamline b) Obtain the pressure p(x) for - sxs-R c) Plot the pressure distribution froma distance 15 m ahead of the cylinder to the stagnation point for flow speeds of Vo 0, 10 km/hr and 20 km/hr if p=0 and R = 1 m Stagnation point R

Answer 1

(a)The expression for the fluid velocity along the stream line is V=V₀(1-R²/x²)

here, V₀ is the upstream velocity and R is the radius of the cylinder.

Calculate the pressure gradient along the stream line using the following equation

-$\gamma$sin$\theta$ - $\partial$p/$\partial$s = $\rho$V $\partial$v/$\partial$s

$\partial$p/$\partial$s = $\gamma$sin$\theta$- $\rho$V $\partial$v/$\partial$s ..... (1)

here, $\gamma$ is the specific weight of the liquid.

since the stream line is horizontal substitute 0 to  $\theta$ in (1)

the acceleration term is given by the equation

V$\partial$v/$\partial$s = V $\partial$v/$\partial$x

substitute V₀(1-R²/x²) fpr V

V$\partial$v/$\partial$s = V₀(1-R²/x²) $\partial$ V₀(1-R²/x²)/ $\partial$x

= V₀(1-R²/x²) [ V₀(0-(-2R²/x³)]

= 2V₀²R²/ x³(1-R²/x²)

now substitute 2V₀²R²/ x³(1-R²/x²) for V$\partial$v/$\partial$s and 0 for $\theta$ in (1)

$\partial$p/$\partial$s = -$\gamma$sin 0 - $\rho$ [2V₀²R²/ x³(1-R²/x²)]

the pressure gradient is [-2$\rho$V₀²R²/ x³(1-R²/x²)]

(b) $\partial$p/$\partial$s = $\partial$p/$\partial$x

replace the s by x because the 2 coordinates are identical along stream line.

$\partial$p/$\partial$s = -2$\rho$V₀²R²/ x³(1-R²/x²)

Integrate the above equation

P1
$\partial$p = $\int$-2$\rho$V₀²R²/ x³(1-R²/x²) x$\partial$x

p(x) -p₁= -2$\rho$V₀²R²$\int$[1/x³ - R²/x⁵] $\partial$x

p(x) -p₁ = -2$\rho$V₀²R² [ -1/2x² + R²/4x⁴]

p(x) -p₁ = -2$\rho$V₀²R² - 1/2x² - (-2$\rho$V₀²R² * R²/4x⁴)

p(x) = p₁ + $\rho$V₀²R²/x² - $\rho$V₀²R⁴/2 x⁴

(c) Calculate the P (x = -a) = P₁ + $\rho$V₀²R²/-a² - $\rho$V₀²R⁴/2 -a⁴

= P₁ + $\rho$V₀² - $\rho$V₀²/2

= P₁ + $\rho$V₀²/2