6.8 Exercises [electroosmosis top]

1. Consider a 1D case where the cross-sectional area of a polycarbonate microchannel with μ_EO = 2.6×10^-8 m²∕Vs changes instantaneously between two regions which we will refer to as region 1 and region 2. Assume that the conductivity σ is uniform throughout. Assume that the magnitude of the electric field in region 1 is denoted using E₁ and the magnitude of the electric field in region 2 is denoted E₂, while the cross-sectional area in region 1 is given by A₁ = 200 μm² and the cross-sectional area in region 2 is given by A₂ = 400 μm², and the length of region 1 is given by L₁ = 1 cm and the length region 2 is given by L₂ = 1.5 cm. Assume a potential difference of ΔV = 300 V is applied across the channel.

   1. using the techniques of Chapter 3, determine the resistance and potential drop in each region in terms of the input parameters. Determine E₁ and E₂.
   2. determine the magnitudes of the electric fields E₁ and E₂ in the two regions, and calculate the velocity magnitude in each region.
   3. Does this velocity solution satisfy conservation of mass? What is the relation between conservation of mass and conservation of current in this flow?

2. Consider flow of an aqueous solution (ε = 80ε₀, η = 1×10^-3 Pa s) through a L = 1 cm microchannel made of glass, as shown in Figure 6.10. Assume that the glass microchannel has uniform cross-section and is much longer than it is wide, and thus assume that the system is approximately one-dimensional. Assume that the glass is an insulator, and that the interface between the solution and the glass has a surface potential that is φ₀ = 70 mV lower than the potential of the bulk solution, as well as a surface charge density of q′′_wall = -0.05 C∕m². Assume that the voltage at the inlet is V₁ = 100 V and the voltage at the outlet is V₂ = 0 V.

   1. Formulate the governing equations and boundary conditions for a complete solution of this problem, including both electrical potential and fluid velocity both near to and far from the wall. Do not attempt to solve this system of equations.
   2. Execute a 1D integral analysis of the boundary layer near the wall, and determine the effective slip boundary condition that describes flow outside the electrical double layer.
   3. Given your effective slip boundary condition, reformulate the problem as follows:
      1. define the governing equation for the outer solution for fluid flow in this system.
      2. prescribe the boundary conditions for the outer solution for the fluid flow in terms of the electric field.
      3. prescribe the governing equations and boundary conditions required to solve for the electric field distribution in the system.
      4. solve for the outer solution for the velocity distribution for this system.

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   Figure 6.10: Schematic of a microchannel.

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3. Most exercises are excluded from this web posting. Follow the links to buy the text at Cambridge or Amazon or Powell's or Barnes and Noble.

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