91Ó°ÊÓ

By using the Navier-Stokes equation and considering the balance of forces in the direction of flow, you can write the equation as \( dp/dx = \mu * d^2u/dy^2 \), where \( p \) is the pressure, \( x \) is the distance along the flow, \( \mu \) is the dynamic viscosity and \( u \) is the velocity. By integrating twice and solving for constants using the boundary conditions that velocity \( u = 0 \) at \( y = 0 \) and \( y = h \), the velocity profile can be obtained. For the bottom layer with viscosity \( \mu_{1} \), \( u_{1}(y) = -dp/dx * (y^2/2\mu_{1} - h * y/\mu_{1} ) \). Similarly for the upper layer with viscosity \( \mu_{2} = 4\mu_{1} \), \( u_{2}(y) = -dp/dx * (y^2/2\mu_{2} - 2h * y/\mu_{2} + 3h^2/2\mu_{2} ) \). Substituting the given values of \( h = 0.005m \), \( \mu_{1} = 0.1Ns/m^2 \) and \( dp/dx = -50000Pa/m \) will give \( u_{1}(h) = u_{2}(h) = u_{interface} \).

To find the maximum velocity, the derivative of the velocity equation should be equated to zero and solved for y. By differentiating the velocity profile for each fluid layer and setting it to zero, \( du_{1}/dy = -dp/dx * (y - h) = 0 \) yields the maximum velocity \( u_{1max} = -dp/dx * h^2 / 2\mu_{1} \) occurs at \( y = h \). And \( du_{2}/dy = -dp/dx * (y - 2h) = 0 \) yields the maximum velocity \( u_{2max} = -dp/dx * h^2 / 2\mu_{2} \) occurs at \( y = 2h \).

You plot the velocity profiles \( u_{1}(y) \) and \( u_{2}(y) \) with y ranging from 0 to 2h. The velocity should be zero at the plates and maximum at the interface, representing a parabolic profile.