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
and
are
and
, 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)