Problem 4 Consider a long and shallow microchannel of length L (-1cm) and height h (-100 m), as shown below. Due to a chemical coating the zeta potential on the channel walls is non-uniform and slowly varies according to (x) = sin(2x/L), where is negative constant (~-25 mV). The water (viscosity and permittivity) is set in motion by an applied voltage drop V. The channel ends are open to the atmosphere. Assume a thin double layer (compared to channel dimensions) and account for the electrokinetic effects by using the Helmholtz-Smoluchowski slip boundary conditions on the channel walls. V pa y Water, E ((x) pa h + X L I pa y Water, E ((x) pa h + L - a. Using order-of-magnitude analysis, find a value a characteristic pressure in the channel, p. b. Derive an expression for the pressure distribution in the channel. c. Derive an expression for the velocity component in x direction, u. d. Derive an expression for the velocity component in y direction, v, for an arbitrary distribution of (x). e. Derive an expression for the volume flux. f. Plot in Matlab the pressure distribution (normalized by p') as a function of normalized coordinate x/L. g. Plot in Matlab the flow streamlines in the channel. Use normalized coordinate x/L and y/h.