Flow of nano-fluid over a sheet of variable thickness with non-uniform stretching (shrinking) and porous velocities

Nano-fluid flow is maintained over a non-uniform porous plate of variable thickness with non-uniform stretching (shrinking) velocity. In real engineering systems, the conduction resistance of sheets is necessarily important, whereas, in typical analysis very thin walls are undertaken. The surface thickness is ignored in the classical studies of flow, heat, and mass transfer problems. However, it the compulsory component in many physical problems, therefore, we thoroughly examined the perceptiveness of the wall thickness on the field variables and the transport of heat and nano-particle between solid surfaces and fluids. The phenomenon of variable wall thickness is extensively investigated with the combination of other boundary inputs. The variable stretching and shrinking velocities of the plate may have linear and non-linear forms and the sheet is uniformly heated whereas the nanoparticles are uniformly distributed over its surface. The diffusion of heat and nanoparticles in the fluid are governed using the boundary layer PDE's, which satisfy certain BC's. A set of unseen transformations is generated for solving the system of boundary value PDE's. In view of these new variables, we obtained a system of boundary value ODE's and it contains several dimensionless numbers (parameters). It is worthy noticeable that the problem describes and enhances the behavior of all field quantities in view of the governing parameters. All the field quantities, rates of heat and mass transfer are evaluated and effects all the parameters are seen on them and they are significantly changed with the variation of these dimensionless quantities. New results are presented in different graphs and tables and thoroughly examined. The Thermophoresis force enhances both the temperature and concentration profiles, however, the concentration distribution of nanoparticles is abruptly changed with a small variation in this force. The concentration profiles are bell-shaped on the right and behaves like a normal distribution. On the other hand, the addition of more nanoparticles into the base fluids increased (decreased) the temperature (concentration) profiles. Moreover, the two different attitudes of wall thickness are also examined on filed variables. The significant features and diversity of modeled equations are scrutinized and we recovered the previous problems of mass and heat transfer in Nano-fluid from a uniformly heated sheet of variable (uniform) thickness with variable (uniform) stretching/shrinking and injection/suction velocities. Moreover, two different numerical solutions of the modeled equations are found. These solutions are compared in a table and exactly matched with each other.