Question

Two immiscible, incompressible, viscous fluids having same densities but different viscosities are contained between two infinite horizontal parallel plates, 2 m apart as shown below. The bottom plate is fixed and the upper plate moves to the right with a constant velocity of 3 m/s. With the assumptions of Newtonian fluid, steady and fully developed laminar flow with zero pressure gradient in all directions, the momentum equations simplify to

$\frac{d^{2}u}{d{y}^2}=0$

If the dynamic viscosity of the lower fluid ${\mu }_2$, is twice that of the upper fluid ${\mu }_1,$ then the velocity at the interface (round off to two decimal places) is m/s.

• A. 1

Answer 1

The correct option is A 1

Velocity profile is laminar in both fluids

$\frac{d^{2}u}{d{y}^2}=0$
$\frac{du}{dy}=c_1$
$u=c_{1}y+c_2$
i.e. we can assume linear velocity profile. If velocity profile is linear shear stress will be constant in gap everywhere i.e. in fluid (1) and fluid (2)
Also at the interface shear stress will be constant.
${\tau }_1={\tau }_2$
${\mu }_2\frac{V_i}{h_2}={\mu }_1\frac{(V-V_i)}{h_1}$
where $V_i$ is velocity at the interface
$2{\mu }_1\frac{V_i}{1}=\frac{{\mu }_1(3-V_i)}{1}$
$2V_i=3-V_i$
$3V_i=3$
$V_i=1m/s$