(Solved):   Problem 1. Give answers to Problem 1 on separate sheets (Total Points: 50) A lubricant of ...

 

Problem 1. Give answers to Problem 1 on separate sheets (Total Points: 50) A lubricant of viscosity \( \mu \) flows along a pressure-fed bearing of length \( l \), clearance \( h \) and supply pressure \( p_{s} \), as shown in the figure below. Note that the upper surface represents a piston moving as indicated by the arrow, at velocity \( u_{h} \) : The governing differential equation and boundary conditions for the velocity profile \( u(x) \) are \[ \begin{array}{c} \frac{d}{d x}\left(\mu \frac{d u}{d x}\right)+\frac{p_{s}}{l}=0, \quad 0 \leq x \leq h \\ u(0)=0 ; \quad u(h)=u_{h} \end{array} \] Determine the finite element solution to Equation (1) using three unequally-spaced linear elements as follows as follows: a) First, obtain the \( 4 x 4 \) system of equations for the nodal values. Note that the element contributions can be determined by direct analogy with the 1-D model problem. (10 points) b) Incorporate the boundary conditions and solve for the unknown nodal value(s). (10 points) c) From the original global system, solve for the shear load reaction value(s) at the upper and lower surfaces. (5 points) d) Using your results for the nodal solution values, determine the velocity at the point \( x=3 h / 4 \). (5 points) Page 1 of 5 NAME e) Again, using your results for the nodal solution values, determine the shear stress, \( \tau=\mu \frac{d u}{d x} \), at the same point \( x=3 h / 4 \). (5 points) f) Based on your result for part e), explain what the calculated shear stress value at \( x=4 h / 5 \) would be and why? (5 points) g) Sketch a plot of the finite element solution for the velocity \( u(x) \) over the entire domain. Is the finite element solution exact? Clearly explain why or why not. (5 points) h) Write the differential equation that represents the time-dependent version of Equation (1). (5 points)