Problem Solving$-$ Chapter $1$

The velocity distribution for the laminar flow of a Newtonian fluid between two wide

parallel plates is given by

$u=\frac{3u_{av}}{2}[1-(\frac{y}{(h)}{)}^2]$

where $u_{av}$ is the average velocity, and y is the distance from the centerline.

Provide an expression for the shear stress at the wall.

Laminar F low between $2$ parallel plates

wide plates

$2$ walls: at $y=+-h$

Shear rate $=\frac{du}{dy}=\frac{3u_{av}}{2}(0-\frac{y}{h^2})=-\frac{2u_{av}y}{h^2}$

${}_w=|\frac{-3u_{av}y}{h^2}{|}_y=+-h=\frac{3u_{av}|y|}{h^2}{|}_y|=+-h=\frac{3u_{av}}{h}$

Provide an expression for the coefficient of thermal expansion $$ in the case of an ideal

gas.

Prove the following equation for liquids and moderate temperature change

$={}_0[1-(T-T_0)]$

Referring to Fig. $1\mathrm{.}4,$ calculate the shear rate in $s^{-1}$ and shear stress in Pa using the

following data:

$V_0=20c\frac{m}{s}$;$y_0=1cm$;$=1cP$

Plot schematically the velocity profile in case the top plate in Fig. $1\mathrm{.}4$ is stationary and

the bottom plate is moving from the right to the left.