Nanofluid flow in a converging and diverging channel of rectangular and heated walls

We have investigated the nanofluid flow in a converging and diverging channel of heated and inclined plane walls. The wellknown Buongiorno model has been undertaken for the transport of nanoparticles in a base fluid. The upper (lower) wall of the channel has the geometry and it is presented by an equation of the straight line y=mx+a0 (y=-mx-a0), in which m is the slope of the upper wall, a0 is the half width of the channel inlet (exit) for diverging (converging) flow, x and y are representing the Cartesian Coordinates. Note that the walls of the channel are uniformly and equally heated and the nanoparticle concentration at the walls is also constant. On the contrary, the flow problem is strictly described in Cartesian Coordinates. The problem is formulated for the upper half of the channel and later on the results are generalized for the lower half. The new observations are shown in different graphs. The four governing PD E’ s are simplified with the help of proper and appropriate similarity transformations and they constituted a system of ODE’s, whereas, it is equipped with several dimensionless parameters. Note that the convective terms in energy and concentration equations are disappeared in view of these similarity variables. The effects of all parameters are investigated on flow, heat and mass transfer characteristics. Moreover, effects of all parameters are seen on skin friction, rates of heat and mass transfer. In addition, classical models of nanofluid flow inside a converging and diverging channel in plane polar coordinates will be the special cases of the current modeled problem. Furthermore, the perturbation solution of the modeled equations is also determined for small values of the parameter. Note that slope m and Pr number are used as a perturbation parameters for the momentum equation and energy, species concentration equations and these solutions are valid for small values of all other parameters involved in the problem. In addition, exact solutions of the final ODE are also found for special ranges of parameters value. Moreover, the comparison is shown between the different solutions of the problem and classical solutions in tables and graphs. The temperature (concentration) profiles are increased (decreased) with the increasing of Nb, and Nt, however, both of them are increased with m. Moreover, the linear (non-linear) profiles of skin friction (rate of heat transfer) are decreased(increased) with the increasing of both Re (Pr,m) and m (Nb). In addition, the rate of mass transfer is increased linearly (non-linearly) against the increasing value of Nt (Pr) and Nb (Nb=Nt). Note that the temperature (concentration) profiles are effectively improved (declined) with the increase of thermophoretic forces, moreover, it is also rapidly increased (decreased) with the increase of concentration of nanoparticles (and their rapid fluctuations) in a converging and diverging channel.