15.4 Exercises [nanofluidics top]

1. Define a geometry-dependent effective electroosmotic mobility μ_EO,eff such that the flow rate per unit length Q′ of an electroosmotic flow between two infinite plates separated by a distance 2d is given by

   (15.56)

   Where d^* = d∕λ_D. This effective electroosmotic mobility thus gives the spatially-averaged flow rate in this system. μ_EO,eff is a function of d^*.

   1. Graph the velocity distribution for several values of d^* ranging geometrically from 0.2 to 50.
   2. Evaluate μ_EO,eff as a function of d^* and plot this relation from d^* = 0.2 to d^* = 50.

2. Consider the significance of viscous dissipation as compared to Joule heating in an electroosmotic system between two parallel plates separated by a distance 2d with a Debye length of λ_D. Define d^* = d∕λ_D. Assume that the Debye-Hückel approximation can be made.

   Assume a field of E is applied, and the fluid has conductivity and viscosity of σ and η, respectively. Assume the electroosmotic mobility of the system is μ_EO. Show that the ratio of viscous dissipation to Joule heating γ can be written as

   (15.57)

   where f(d^*) is a bounded function of d^* that peaks at values of d^* near one and has a peak magnitude on the order of unity.

   To do this, note that the local viscous dissipation for this 1D system is given by 2η².

   Evaluate this scaling relation for a system with η = 1×10^-3 Pa s, σ = 100 μS∕cm, L = 20 μm, and μ_EO = 4×10^-8 m²∕Vs.

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