Subject: Fluid Mechanics$-$I

Complex Engineering Problem$($CEP$)$

CLO$3$

PLO$3$

One of the consequences of surface tension is that surface tension causes pressure difference

across the curved interface. This pressure difference depends not only upon the coefficient

of surface tension but also on the geometry of the curved interface. In the texts, expressions

for this pressure difference are developed for two special and common cases, namely

cylindrical interface $($with example of cylindrical jet$)$ and spherical interface $($with examples

of spherical droplet and spherical bubble$).$ Following CEP is intended to get an exposure to

develop expression for a general curved interface, and to investigate the shape and height of

water surface along a plane vertical wall.

a$)$

$($i$)$ Explore how the curvature of surfaces is described, and what are the principal radii of

curvature, mean curvature, and Gaussian curvature of a surface. What are the values of

these quantities for common surfaces of plane, cylinder, and sphere?

$($ii$)$ Draw an elemental area of a general surface, and write down its area expression in terms

of principal radii of curvature.

$($iii$)$ Add thickness to this area element to make a $3-$D patch

$($iv$)$ Take an element of the curved interface $($between two fluids$)$ of the shape this patch $($as

developed in the previous step$).$ Draw the free body diagram of this element.

$($v$)$ Apply the equilibrium equations to this free body and show that for an interface of shape

of a general surface, the pressure difference across the interface is given by the expression

Delta p$=$sigma$\_($s$)((1)/($R$\_(1))+(1)/($R$\_(2)))$

Where R$\_(1)$ and R$\_(2)$ are the principal radii of curvature of the surface of interface. Discuss its

special cases for circular cylindrical surface and spherical surface and verify the equations

for these surfaces derived in texts.

b$)$ Now consider a plane wall immersed in water.

i$)$ State about the general shape of surface of water with arguments

ii$)$ Reduce the formula for pressure difference, as derived in $($a$),$ for the general shape of free

surface of water in contact with wall

iii$)$ By assuming the surface deflections to be sufficiently small, approximate the radius of

curvature formula

iv$)$ By using the approximated formula for radius of curvature into the expression developed

in $($ii$),$ show that the equation for the fee surface of water near wall is given by

y$=$he$^(-$sqrt$(($rho g$)/($sigma$\_($s$))$x$))$

Where $\text{'}$ h $\text{'}$ is the height of surface at the wall

v$).$ By applying equation of equilibrium to the free body of liquid between the curved surface

and horizontal base, show that height of surface at wall $($or equivalently rise at the wall$)$ is

given by

h$=($cot theta$)/($sqrt$($rho g$//$sigma$\_($s$)))$

Where $\text{'}$ theta $\text{'}$ is contact angle at the wall, as shown. Note $\text{'}$ I want you to solve each part of CEP properly with required figure and derivation.