2.2. Governing Equation
Electric field equations should be combined with hydrothermal equations. Electric field equations are:
(1)(2)(3)Charge distribution can be modeled in two ways: mobility and conductivity models. The equation of electric current density is:
(4)Using the above equations leads to the following equation:
(5)Diffusion term is small, Equation (4) can be changed to:
(6)In existence of electric field, Coulomb forces should be added to momentum equations:
(7)
and
are defined as:
(8)(9)(10)(11)(12)
Table 1 illustrate the properties of the base fluid and nanoparticles. Effect of electric field on viscosity of nanofluid has been taken into account:
(13)
Table 2 shows the coefficient values of this equation. Non-dimensional parameters are presented as follow:
(14) where
![978-1-5225-7595-5.ch007.m17](https://igiprodst.blob.core.windows.net:443/source-content/9781522575955_208643/978-1-5225-7595-5.ch007.m17.png?sv=2015-12-11&sr=c&sig=Jl6MVkODUYOIQGPHS890Lmb3zuav%2BAuxRFzJimUO5Sw%3D&se=2019-12-27T19%3A23%3A08Z&sp=r)
and
![978-1-5225-7595-5.ch007.m18](https://igiprodst.blob.core.windows.net:443/source-content/9781522575955_208643/978-1-5225-7595-5.ch007.m18.png?sv=2015-12-11&sr=c&sig=Jl6MVkODUYOIQGPHS890Lmb3zuav%2BAuxRFzJimUO5Sw%3D&se=2019-12-27T19%3A23%3A08Z&sp=r)
are
![978-1-5225-7595-5.ch007.m19](https://igiprodst.blob.core.windows.net:443/source-content/9781522575955_208643/978-1-5225-7595-5.ch007.m19.png?sv=2015-12-11&sr=c&sig=Jl6MVkODUYOIQGPHS890Lmb3zuav%2BAuxRFzJimUO5Sw%3D&se=2019-12-27T19%3A23%3A08Z&sp=r)
and
![978-1-5225-7595-5.ch007.m20](https://igiprodst.blob.core.windows.net:443/source-content/9781522575955_208643/978-1-5225-7595-5.ch007.m20.png?sv=2015-12-11&sr=c&sig=Jl6MVkODUYOIQGPHS890Lmb3zuav%2BAuxRFzJimUO5Sw%3D&se=2019-12-27T19%3A23%3A08Z&sp=r)
, respectively. By eliminating the over bar, the equations are:
(15)Stream function and vorticity can be defined as:
(16)Stream function can satisfy the continuity equation. Vorticity equation can be derived by eliminating pressure sources.
and
along the lid wall can be obtained as:
(17)(18)