The impacts of radiation, mass transpiration, and volume fraction of carbon nanotubes on the flow of a Newtonian fluid past a porous stretching/shrinking sheet are investigated. For this purpose, three types of base liquids are considered, namely, water, ethylene glycol and engine oil. Moreover, single and multi-wall carbon nanotubes are examined in the analysis. The overall physical problem is modeled using a system of highly nonlinear partial differential equations, which are then converted into highly nonlinear third order ordinary differential equations via a suitable similarity transformation. These equations are solved analytically along with the corresponding boundary conditions. It is found that the carbon nanotubes can significantly improve the heat transfer process. Their potential application in cutting-edge areas is also discussed to a certain extent.

The fluid flow with carbon nanotubes (CNTs) through porous media considering the mass transpiration is an important concept in the field of industry and medicine, especially in nano-medicine. The porous media approach is useful in the treatment of cancer tumors as the tumor growth is represented by the mathematical model where the mass transfer represents the process of growth and death. The model of tumor growth represented by the porous media approach studied in detail by Shelton [

Fazle et al. [

Hamad [

The intend of the current examination is to analyze the influences of radiation and mass transpiration with porous media on the flow of CNTs due to stretching/shrinking plate with the existence of Navier’s slip and physical model is as shown in

The steady 2-D flow and heat transfer of electrically conducting fluid due to stretching/shrinking sheet in porous media with permeability _{w} and _{∞} is the ambient temperature. The fluids considered are base liquids like water, ethylene glycol and engine oil with single and multi-walled carbon nanotubes. The governing equations for described momentum and energy conservations are as follows:

with B.Cs as

where,

and

this gives,

On applying

and B.Cs reduces to,

Here,

Take _{1}_{2} and use the function transformation as,

here,

By taking the substitutions

The exact analytical solution of

This gives the velocity as,

Using this in

From this equation the exponent

The exact solution of

here

The solution of

Then the temperature distribution will becomes,

The scaled Nusselt number is given by,

The influence of mass transpiration, Navier’s slip and thermal slips on the flow of incompressible viscous MWCNT/SWCNT is investigated. The system of PDEs is changed to system of nonlinear ODEs with constant coefficients by using the suitable similarity transformations for velocity and temperature. Then the analytical solution for velocity profile is obtained in exponential form and that for the temperature field was obtained in terms of incomplete gamma function. The concerned effects are analyzed by the help of different graphs for MWCNT/SWCNT. The model is shown for the flow over porous medium and the advantages of using porous medium for its applications in industry and medicine field. To study the effects of different base fluids for MWCNT and SWCNT, the different graphs are shown here.

In

To study the variation of skin friction with mass transpiration _{0} is studied for different conditions and for different roots values of

Variation of Pr with

The influence of mass transpiration and Navier’s slip on the flow of incompressible viscous MWCNT/SWCNT for different base fluids is studied. The concerned effects are analyzed by the help of different graphs for MWCNT/SWCNT. The effects observed are,

Analytical solution is obtained for velocity and temperature field.

Prandtl number is lower for water and highest for engine oil.

Prandtl number for SWCNT is more than that of MWCNT.

constant

constant

first and second order slip parameters

Biot number

constant

stretching/shrinking sheet parameter

^{−1}

inverse Darcy number

_{1}

thermal slip

_{P}

permeability

similarity variable

radiation parameter

Prandtl number

_{0}

constant

_{0}

mass transpiration parameter

temperature

_{w}

wall temperature

_{∞}

free stream temperature

_{w}

wall mass transfer velocity

similarity variable

_{1},

_{2}

first and second order slips

similarity variable for temperature

solid volume fraction

parameter for carbon nanotubes

parameter of base fluid

multi wall carbon nanotubes

single wall carbon nanotubes

boundary conditions

ordinary differential equation

partial differential equation

The author T. Anusha is thankful to Council of Scientific and Industrial Research (CSIR), New Delhi, India, for financial support in the form of Junior Research Fellowship: File No. 09/1207(0003)/2020-EMR-I.