Consider the flow of a Newtonian viscous fluid on an inclined flat surface as shown in
figure below. The momentum equation, for a fully developed steady laminar flow along
the z coordinate, is given by:
$-\mu \frac{d^2w}{dx^2} = \rho g \cos\beta$
where w is the z component of the velocity, \mu is the viscosity of the fluid, \rho is the density,
g is the acceleration due to gravity, and \beta is the angle between the inclined surface and the
vertical. The boundary conditions associated with the problem are that the shear stress is
zero at x = 0 and the velocity is zero at x = L.
$\left(\frac{dw}{dx}\right)_{x=0} = 0$, $w(x = L) = 0$
Velocity distribution w(x)
Direction of gravity
Use two quadratic elements in the domain (0, L)
(a) Find velocities at nodes and compare with the exact solution at x=0, L/2, and L.
$w_{exact}(x) = \frac{\rho gL^2 \cos\beta}{2\mu} \left[1 - \left(\frac{x}{L}\right)^2\right]$
(b) Evaluate the shear stress ($\tau_{xz} = -\mu dw/dx$) at the wall using the global equations, and
compare with the exact value.
(20 points)
