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
with B.Cs as
where,
and
this gives,
On applying
and B.Cs reduces to,
Here,
Take
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
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
inverse Darcy number
thermal slip
permeability
similarity variable
radiation parameter
Prandtl number
constant
mass transpiration parameter
temperature
wall temperature
free stream temperature
wall mass transfer velocity
similarity variable
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.