Effect of nano-particles on MHD flow of tangent hyperbolic fluid in a ciliated tube: an application to fallopian tube

K. Maqbool; S. Shaheen; A. M. Siddiqui; K. Maqbool; S. Shaheen; A. M. Siddiqui

doi:10.3934/mbe.2019144

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 2927-2941. doi: 10.3934/mbe.2019144

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Research article

Effect of nano-particles on MHD flow of tangent hyperbolic fluid in a ciliated tube: an application to fallopian tube

1.
Department of Mathematics & Statistics, International Islamic University, Islamabad 44000, Pakistan
2.
Department of Mathematics, York Campus, Pennsylvania State University, York, Pennsylvania 17403, U. S. A

Received: 28 December 2018 Accepted: 25 February 2019 Published: 10 April 2019

This study shows the effects of magnetic field and copper nanoparticles on the flow of tangent hyperbolic fluid (blood) through a ciliated tube (fallopian tube). The present study will be very helpful for those patients who are facing blood clotting in fallopian tube that may cause for infertility or cancer. The nanoparticles and magnetic field are very helpful to break the clots in blood flowing in fallopian tube. Since blood flows in fallopian tube due to ciliary movement, therefore medicines containing copper nanoparticles and magnetic field with radiation therapy help to improve the patient. Ciliary movement has a particular pattern of motion i.e., metachronal wavy motion which helps to fluid flow. For the forced convective MHD flow of tangent hyperbolic nano-fluid, momentum and energy equations are solved by the small Reynolds' number approximation and Adomian decomposition method by constructing the recursive relation of ADM and solved by software "MATHEMATICA". The effects of parameters such as nanoparticle volume fraction, Hartmann number, entropy generation and Bejan's number have been discussed through graphs plotted in software "MATHEMATICA". It is found that blood flow is accelerated and heat transfer enhancement is maximum in the presence of nano particles, also magnetic effects accelerates the blood flow and help to enhance the heat transfer whereas the presence of porous medium increases the fluid's velocity and reduce the transfer of heat through fluid flow.

Keywords:

Citation: K. Maqbool, S. Shaheen, A. M. Siddiqui. Effect of nano-particles on MHD flow of tangent hyperbolic fluid in a ciliated tube: an application to fallopian tube[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2927-2941. doi: 10.3934/mbe.2019144

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Abstract

1. Introduction

Heat transfer enhancement in biological fluids is an important area of modern biomechanics and biomedical engineering. The study of thermodynamics deals with properties of the substances associated with pressure, density, velocity, viscosity and temperature and their relationship with energy which can be studied from the literature ^[1,2]. Heat regulation is essential to all mammals and furthermore heat transfer has found many applications in modern biomedical engineering. For example heat flow in blood, heat transfer during eye treatment, respiratory thermal control and thermal bio convection ^[3,4,5]. Since some conventional fluids like water and blood have low thermal conductivities which decreases the enhancing performance. During last decade the field of nanoparticle was improved remarkably and a new kind of solid-liquid mixture is formed which is called nano fluid. By adding nanoparticles in the base fluids the thermal conductivity of base fluid will alter remarkably.

Nano-fluids are colloidal suspensions of nano-sized solid particles in a liquid, which was first studied by Choi ^[6]. More recently conducted experiments ^[7] have indicated that nano-fluids tend to have higher thermal conductivity than the base fluids. Among the many advantages of nano-fluids over conventional solid-liquid suspensions, the following are worth mentioning, higher specific surface area, higher stability of the colloidal suspension, reduced particle clogging compared to conventional colloids, and higher level of control of the thermodynamics and transport properties by varying the particle material, concentration, size and shape. Maiga et al. ^[8] studied the heat transfer enhancement in turbulent flow through tube using Al₂O₃ nanoparticle suspension. There are lot of nanoparticles in blood that are commonly one thousand times smaller than a human hair and the presence of nanoparticles produce many severe diseases such as neutropenia, blood cancer, eosinophilia, leukocytes etc. In many cases common method cannot be used to eliminate these particles. Presently nanotechnology is being used to separate these nanoparticles from plasma ^[9].

Ciliary movement has been a subject of special interest in the society of experimental biology. Cilia presumably first discovered by famous Dutch microscopist Antoni van Leeuwenhoek in 1675. Cilia occurs in row, fields or tracks at the surface of living cells, where their usual function of propelling the fluid over the cell surface is achieved by unilateral lashing action. Cilia exhibits different beating patterns depending on the surrounding geometry as described in literature ^[10,11,12]. The cycle of beat of single cilium separated into effective and recovery stroke. During the effective phase the cilium is erect, maximizing its height, moving relatively quickly and bending to its base. Before a cilium can perform a second effective stroke the cilium must first progress through a recovery stroke. In the recovery phase the cilium maintains a low profile, moves more slowly and propagates a bend from base to tip drawing the cilium backwards in an unrolling action.

Different metachronal patterns can be recognized in protozoa and classified according to angle of power stroke in relation to the direction of metachronal transmission. In literature antiplectic wave pattern is widely used as compared to the symplectic metachrony because the prior is classified by the separation of collective cilia during the forward stroke, permitting them to push more fluid capacity and thus mounting the stroke effectiveness ^[13,14].

A thin layer of mucous is present at the inner surface of fallopian tube and ciliated cells of different lengths are present around the mucous membrane. In the fallopian tube ciliary motion helps to transport spermatozoa towards ampulla and also cilia induced flow transports ovum from ovaries at the time of ovulation. At ampulla, if fertilization occurs, then ciliary motion helps to transfer the embryo to the last region of fallopian tube i.e. intermural ^[15,16,17]. The ovary and uterus veins anastomose to form a network of blood vessels around the ovarian artery to facilitate the concentration of substances and communication within the female reproductive tract. Maqbool et al. ^[18,19,20] investigated the effect of ciliary motion on non-Newtonian fluid models.

Entropy generation in nano-fluid flow which is flowing through wavy channels are studied by many researchers ^[21,22]. Later on, many researchers ^[23,24] analyzed the heat transfer enhancement in MHD nano-fluids flowing in porous ducts or tubes. They found that entropy generation enhances due to the nanoparticles. A large number of attempts have been made to study the effect of nanoparticles with MHD but very less attention is paid to study the effect of nanoparticles in bio fluid.

It is evident that microorganisms when causes infection in human body, can die with the application of constant magnetic field strength. It is observed by the medical doctors that tuberculosis skin lesion is completely recovered by the antibiotic having constant negative strength of magnetic field. It is also observed by the physicians that for throat infection when pulmonary cilia are not working properly antibiotics containing magnetic field help to die out the virus that effects the pulmonary cilia movement. Recently, several researchers ^[25,26,27] introduced the study of ciliary movement where they used the magnetic field to observe the ciliary frequency. Later on, many researchers studied the effects of entropy generation of MHD flows in the presence of nanoparticles and porous medium ^{[28,29,30,31,32,33,34,35,36,37,38,39,40]}.

The activity of cilia and ciliated cells normally occurred in conditions where viscous forces greatly predominate over inertial forces i.e. Reynolds' number is less than one because of small size and low speed. Thus, the purpose of this work is to analyse the characteristics of MHD fluid flow of tangent hyperbolic nano-fluid with enhanced heat transfer due to ciliary motion in a porous medium. While the analysis could be applied to copper-blood nano-fluid due to the accessibility of its physical properties.

2. Mathematical model

Consider a tangent hyperbolic nano-fluid flow in a tube of mean radius a in a porous medium. Assume infinite number of continuously beating cilia are present at the inner walls of tube generating symplectic metachronal wave which moves towards positive z-axis with wave speed c. Uniform magnetic field of strength B₀ is applied in transverse direction i.e. along the perpendicular direction of fluid motion.

Figure 1. Geometry of ciliated tube.

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In existing literature many researchers ^[18,19] studied the motion of tips of cilia which are tracing the path of ellipse.

Governing equations for an incompressible MHD tangent hyperbolic nano-fluid through porous medium are defined as ^[41]

$\nabla .\mathit{\boldsymbol{V}} = 0,$

(1)

where

$V = \left[ {u, 0, w} \right],$

(2)

and

${\rho _f}\frac{{d\mathit{\boldsymbol{V}}}}{{dt}} = div\mathit{\boldsymbol{S}} + \sigma \left( {\mathit{\boldsymbol{J}} \times \mathit{\boldsymbol{B}}} \right) + R,$

(3)

${\left( {\rho c} \right)_f}\frac{{dT}}{{dt}} = {k_{nf}}{\nabla ^2}T + trace\left( {\mathit{\boldsymbol{\tau }}.\mathit{\boldsymbol{L}}} \right)$

(4)

where V is the velocity components, w is the axial component of velocity, u is the radial component of velocity, S is the Cauchy stress tensor, $\frac{d}{{dt}} = \frac{\partial }{{\partial t}} + V.\nabla$ is the total derivative, ρ_f is the density of nano-fluid, σ is the electrical conductivity of fluid, J is the current density, B is the strength of magnetic field, R is the Darcy's resistance, T is the temperature profile and L is gradient of velocity.

Here the appropriate stress tensors for the tangent hyperbolic model are defined as ^[34,35]

$\mathit{\boldsymbol{S}} = - p\mathit{\boldsymbol{I}} + \mathit{\boldsymbol{\tau }},$

(5)

$\mathit{\boldsymbol{\tau }} = \left[ {\left( {{\eta _\infty } + \left( {{\eta _0} + {\eta _\infty }} \right)\tanh {{\left( {{\rm{\Gamma \mathsf{\dot γ} }}} \right)}^m}} \right)} \right],$

(6)

in which τ is the extra stress tensor, η₀ represents zero shear rate viscosity, η_∞ is the infinite shear rate viscosity, Γ is time constant and m shows the power law index.

${\rm{ \mathsf{\dot γ} }} = \sqrt {\frac{1}{2}\mathit{\boldsymbol{\pi }}} , \;\;\;\;\;where~\mathit{\boldsymbol{\pi }} = trace{\left( {grad\mathit{\boldsymbol{V}} + grad{\mathit{\boldsymbol{V}}^T}} \right)^2},$

(7)

where π is second order tensor we study the above equation for the case where η_∞ = 0 and $\Gamma \dot \gamma < 1$ . The elements of extra stress tensor can be written as

${{\dot \gamma }_i} = \mathit{\boldsymbol{L}} + {\mathit{\boldsymbol{L}}^T}$

(8)

$\bar \tau = {\eta _0}\left( {{{\left( {{\rm{\Gamma \mathsf{\dot γ} }}} \right)}^m}} \right){{\dot \gamma }_i} = {\eta _0}{\left( {1 + {\rm{\Gamma \mathsf{\dot γ} }} - 1} \right)^m}{{\dot \gamma }_i} = {\eta _0}\left( {1 + m\left( {{\rm{\Gamma \mathsf{\dot γ} }} - 1} \right)} \right){{\dot \gamma }_i}$

(9)

where thermal conductivity of Cu+blood nano-fluid is defined as follows

${\rho _{nf}} = \left( {1 - \phi } \right){\rho _f} + \phi {\rho _s}, \;\;\;\;\;\;{\eta _{nf}} = \frac{{{\eta _f}}}{{{{\left( {1 - \phi } \right)}^{2.5}}}},$

${\left( {\rho {c_p}} \right)_{nf}} = \left( {1 - \phi } \right){\left( {\rho {c_p}} \right)_f} + \phi {\left( {\rho {c_p}} \right)_s}, \;\;\;\;\;{\alpha _{nf}} = \frac{{{k_{nf}}}}{{{{\left( {\rho {c_p}} \right)}_{nf}}}}$

(10)

${k_{nf}} = {k_f}\left( {\frac{{{k_s} + 2{k_f} - 2\phi \left( {{k_f} - {k_s}} \right)}}{{{k_s} + 2{k_f} + 2\phi \left( {{k_f} - {k_s}} \right)}}} \right)$

where k_nf is the thermal conductivity of nano-fluid, k_f is the thermal conductivity of base fluid, k_s is the thermal conductivity of solid nano particles and ϕ is the solid volume fraction.

The Mathematical model for geometry of cilia tips in the wave frame is

$h = 1 + \varepsilon \;\cos \;2\pi z$

(11)

$w\left( h \right) = - 1 - 2\pi \varepsilon \alpha \beta \cos 2\pi z$

where ε is the cilia length parameter, α is the eccentricity of elliptic wave and β is the wave number.

The governing equations of motion of tangent hyperbolic fluid model in a tube are specified as follows.

$\frac{{\partial u}}{{\partial r}} + \frac{u}{r} + \frac{{\partial w}}{{\partial z}} = 0,$

(12)

${\rho _f}\left[ {u\frac{{\partial w}}{{\partial r}} + w\frac{{\partial w}}{{\partial z}}} \right] = - \frac{{\partial p}}{{\partial z}} + \frac{1}{r}\frac{\partial }{{\partial z}}\left( {r{\tau _{rz}}} \right) + \frac{{\partial {\tau _{zz}}}}{{\partial z}} - \left( {\sigma {B_0}^2 + \frac{{{\eta _f}}}{k}} \right)\left( {w + c} \right),$

(13)

${\rho _f}\left[ {u\frac{{\partial u}}{{\partial r}} + w\frac{{\partial u}}{{\partial z}}} \right] = - \frac{{\partial p}}{{\partial r}} + \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r{\tau _{rr}}} \right) + \frac{{\partial {\tau _{zr}}}}{{\partial z}},$

(14)

${\left( {\rho c} \right)_f}\left[ {u\frac{{\partial T}}{{\partial r}} + w\frac{{\partial T}}{{\partial z}}} \right] = {k_{nf}}\left( {\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial T}}{{\partial r}}} \right) + \frac{{{\partial ^2}T}}{{\partial {z^2}}}} \right) + {\tau _{zr}}\frac{{\partial w}}{{\partial r}}.$

(15)

where ρ_f is the fluid density, u and w are the radial and axial components of velocity, c is the wave speed, η_f is the apparent viscosity of fluid and k is the permeability parameter.

The following non-dimensional parameters can be introduced for further analysis.

$\begin{array}{*{20}{c}} {z^*} = \frac{z}{\lambda }, {r^*} = \frac{r}{a}, {u^*} = \frac{u}{{\beta c}}, {w^*} = \frac{w}{c}, {h^*} = \frac{h}{a}, {p^*} = \frac{{a\beta }}{{c\mu }}p, \\ \beta = \frac{a}{\lambda }, {S_{ij}}^* = \frac{a}{{\mu c}}{S_{ij}}, {\lambda _1} = \frac{{c{\lambda _1}}}{a}, Re = \frac{{\rho a\beta }}{\mu }, We = \frac{{{\rm{\Gamma }}c}}{a}, {\alpha _f} = \frac{k}{{{{\left( {\rho c} \right)}_f}}}, \\ \theta = \frac{{T - {T_0}}}{{{T_0}}}, M = \sqrt {\frac{\sigma }{\mu }} a{B_0}^2, {D_a} = \frac{k}{{{a^2}}}, Br = \frac{{{a^2}{\eta _f}}}{{{k_f}{T_0}}}, Pr = \frac{{\mu {c_p}}}{{{k_f}}}, \\ Ec = \frac{{{c^2}}}{{{c_p}{T_0}}}. \end{array}$

(16)

where λ, a, c symbolize the wavelength, width of velocity and wave speed respectively, Re is the Reynolds' numbers, β is the wave number while D_a is the Darcy's number. In terms of dimensionless parameters the momentum equations and shear stresses are

$\frac{{\partial u}}{{\partial r}} + \frac{u}{r} + \frac{{\partial w}}{{\partial z}} = 0,$

(17)

$Re\beta \left[ {u\frac{{\partial w}}{{\partial r}} + w\frac{{\partial w}}{{\partial z}}} \right] = - \frac{{\partial p}}{{\partial z}} + \frac{1}{r}\frac{\partial }{{\partial z}}\left( {r{\tau _{rz}}} \right) + \beta \frac{{\partial {\tau _{zz}}}}{{\partial z}} - \left( {{M^2} + \frac{1}{{{D_a}}}} \right)\left( {w + 1} \right),$

(18)

$Re{\beta ^2}\left[ {u\frac{{\partial u}}{{\partial r}} + w\frac{{\partial u}}{{\partial z}}} \right] = - \frac{{\partial p}}{{\partial r}} + \frac{\beta }{r}\frac{\partial }{{\partial r}}\left( {r{\tau _{rr}}} \right) + {\beta ^2}\frac{{\partial {\tau _{zr}}}}{{\partial z}} - \beta \frac{{\partial {\tau _{\theta \theta }}}}{{\partial z}},$

(19)

$\beta {\left( {\rho c} \right)_f}\left[ {u\frac{{\partial \theta }}{{\partial r}} + w\frac{{\partial \theta }}{{\partial z}}} \right] = {k_{nf}}\left( {\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial \theta }}{{\partial r}}} \right) + {\beta ^2}\frac{{{\partial ^2}\theta }}{{\partial {z^2}}}} \right) + {\tau _{zr}}\frac{{\partial w}}{{\partial r}},$

(20)

${\tau _{rz}} = \left( {1 + m\left( {We\frac{{\partial w}}{{\partial r}} - 1} \right)} \right)\frac{{\partial w}}{{\partial r}},$

(21)

Using long wavelength approximation (β→0) the governing equations and boundary conditions are as follows

$\frac{{\partial p}}{{\partial z}} = \left( {1 - m} \right)\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial w}}{{\partial r}}} \right) + mWe\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r{{\left( {\frac{{\partial w}}{{\partial r}}} \right)}^2}} \right) - \left( {{M^2} + \frac{1}{{{D_a}}}} \right)\left( {w + 1} \right),$

(22)

$- \frac{{\partial p}}{{\partial r}} = 0,$

(23)

$\frac{{{k_{nf}}}}{{{k_f}}}\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial \theta }}{{\partial r}}} \right) = - Br\frac{{{\eta _{nf}}}}{{{\eta _f}}}\left( {1 + m\left( {We\frac{{\partial w}}{{\partial r}} - 1} \right)} \right){\left( {\frac{{\partial w}}{{\partial r}}} \right)^2},$

(24)

$\frac{{\partial w}}{{\partial r}} = 0, \;\;\frac{{\partial \theta }}{{\partial r}} = 0\;\;\;at\;\;r = 0, \\w = w\left( h \right), \theta = 0\;\;\;at\;r = h.$

(25)

Integration of Eq. (17) over the tube width is given as:

$\mathop \smallint _0^h \left[ {\frac{1}{r}\frac{{\partial \left( {ru} \right)}}{{\partial r}} + \frac{{\partial w}}{{\partial z}}} \right]rdr = 0,$

(26)

$hu\left( h \right) + \frac{1}{2}\frac{{\partial q}}{{\partial z}} - h\frac{{\partial h}}{{\partial z}}w\left( h \right) = 0,$

(27)

where $q = 2\mathop \smallint _0^h rwdr$ Eq. (27) can be written as

$\frac{{\partial q}}{{\partial z}} = 2h\left( {\frac{{\partial h}}{{\partial z}}w\left( h \right) - u\left( h \right)} \right)$

(28)

The relation between q and dimensionless volume flow rate Q is given by

$Q = 2\mathop \smallint _0^h RWdR = 2\mathop \smallint _0^h \left( {w + 1} \right)rdr = q + {h^2},$

(29)

The mean volume flow rate for the time period $T = \frac{\lambda }{c}$ is

$\bar Q = \frac{1}{T}\mathop \smallint \limits_0^T Qd{t^*} = q + 1 + \frac{{{\varepsilon ^2}}}{2},$

(30)

where λ is the wavelength of metachronal wave, c is the wave speed and t^* is the mean averaged time.

3. Solution of the problem

To obtain the solution of governing equations we use Adomian decomposition method ^[42].

${{\cal L}_{rw}} - \frac{{\partial p}}{{\partial z}} + mWe\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r{{\left( {\frac{{\partial w}}{{\partial r}}} \right)}^2}} \right) - \left( {{M^2} + \frac{1}{{{D_a}}}} \right)\left( {w + 1} \right) = 0.$

(31)

The Linear and inverse operator are chosen as follows

${{\cal L}_{rw}} = \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\left( {1 - m} \right)\frac{{\partial w}}{{\partial r}}} \right),$

(32)

${\cal L}_{rw}^{ - 1} = \mathop \smallint \nolimits^ \left[ {\frac{1}{{r\left( {1 - m} \right)}}\mathop \smallint \nolimits^ r\left[ . \right]dr} \right]dr.$

(33)

Applying ${\cal L}_r^{ - 1}$ on Eq. (31) we get

$\begin{array}{l} w = {c_1}\ln r + {c_2} - {\cal L}_{rw}^{ - 1}\\ \left[ {\frac{{{M^2}}}{{1 - m}}w + \frac{m}{{1 - m}}mWe\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r{{\left( {\frac{{\partial w}}{{\partial r}}} \right)}^2}} \right) + \frac{m}{{1 - m}}{{\cal L}^{ - 1}}_{rw}\left( {{M^2} + \frac{1}{{{D_a}}}} \right)} \right]. \end{array}$

(34)

Adomian decomposition method yields the following infinite series

$w = \mathop \sum \limits_{n = 0}^\infty {w_n}$

(35)

Initial guess and recursive relation can be chosen in a following manner.

${w_0} = {c_1}\ln r + {c_2}\frac{1}{{1 - m}}\left( {{M^2} + \frac{1}{{{D_a}}}} \right)\frac{{{r^2}}}{4}$

(36)

${w_{n + 1}} = - {\cal L}_r^{ - 1}\left[ {\frac{{{M^2}}}{{1 - m}}{w_n} + \frac{m}{{1 - m}}We\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r{{\left( {\frac{{\partial {w_n}}}{{\partial r}}} \right)}^2}} \right)} \right].$

(37)

With the help of boundary conditions and initial guess solution takes the following form.

${w_0} = \frac{1}{{4\left( {1 - m} \right)}}\frac{{dp}}{{dz}}\left( {{r^2} - {h^2}} \right) + w\left( h \right)$

(38)

$\begin{array}{l} w = w\left( h \right) + {A_1}\left( {{r^2} - {h^2}} \right) + {A_2}\left( {{r^3} - {h^3}} \right) + {A_3}\left( {{r^4} - {h^4}} \right) + {A_4}\left( {{r^5} - {h^5}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + {A_5}\left( {{r^6} - {h^6}} \right) + \cdots \end{array}$

(39)

Using Eq. (39) in Eq. (24) we get

$\begin{array}{l} \theta = {A_6}\left( {{r^4} - {h^4}} \right) + {A_7}\left( {{r^5} - {h^5}} \right) + {A_8}\left( {{r^6} - {h^6}} \right) + {A_9}\left( {{r^7} - {h^7}} \right) + {A_{10}}\left( {{r^8} - {h^8}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + {A_{11}}\left( {{r^9} - {h^9}} \right) + {A_{12}}\left( {{r^{10}} - {h^{10}}} \right) + {A_{13}}\left( {{r^{11}} - {h^{11}}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + {A_{14}}\left( {{r^{12}} - {h^{12}}} \right) + {A_{15}}\left( {{r^{13}} - {h^{13}}} \right) + {A_{16}}\left( {{r^{14}} - {h^{14}}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + {A_{17}}\left( {{r^{15}} - {h^{15}}} \right) + {A_{18}}\left( {{r^{16}} - {h^{16}}} \right) + {A_{19}}\left( {{r^{17}} - {h^{17}}} \right) + \cdots \end{array}$

(40)

Integrating Eq. (39) using software "MATHEMATICA" calculating pressure gradient as

$\frac{{dp}}{{dz}} = - \frac{{{C_2}}}{{3{C_1}}} - \frac{{{2^{\frac{1}{2}}}\left( { - {C_2}^2 + 3{C_1}{C_3}} \right)}}{{3{C_1}{{\left( {{C_5} + \sqrt {4{{\left( {{C_6}} \right)}^3} - {{\left( {{C_7}} \right)}^2}} } \right)}^{\frac{1}{3}}}}} + \frac{{\left( {{C_8} + {{\left( {{C_5} + \sqrt {4{{\left( {{C_9}} \right)}^3} - {{\left( {{C_{10}}} \right)}^2}} } \right)}^{\frac{1}{3}}}} \right)}}{{3 \times {2^{\frac{1}{3}}}{C_1}}}$

(41)

4. Entropy generation

Entropy Generation can be written as

${{S'''}_{gen}} = \frac{{{k_f}}}{{T_\infty ^2}}\left( {{{\left( {\frac{{\partial T}}{{\partial r}}} \right)}^2} + {{\left( {\frac{{\partial T}}{{\partial r}}} \right)}^2}} \right) + \frac{1}{{{T_0}}}{\tau _{zr}}\frac{{\partial w}}{{\partial r}}.$

(42)

Dimensionless form of Entropy Generation can be written as

${N_S} = \frac{{S_{gen}^{'''}}}{{S_G^{'''}}} = {\left( {\frac{{\partial \theta }}{{\partial r}}} \right)^2} + Br{\rm{\Lambda }}\left( {1 + m\left( {We\frac{{\partial w}}{{\partial r}} - 1} \right)} \right){\left( {\frac{{\partial w}}{{\partial r}}} \right)^2}$

(43)

$S_G^{'''} = \frac{{{k_f}T_0^2}}{{\bar \theta _0^2{a^2}}}, \;\;\;\;\;{\rm{\Lambda }} = \frac{{{{\bar \theta }_0}}}{{{T_0}}}$

(44)

5. Graphical results

5. Discussion

In this section the graphical results have shown the effects of different parameters of interest. The cilia induced flow of a tangent hyperbolic nano-fluid in circular tube is investigated. The effect of emerging parameters for the entropy generation, temperature and velocity profile are observed.

Axial velocity for various values of, nanoparticle volume fraction of the fluid ϕ, Hartmann number M and Darcy's number D_a are observed in figure (2a)−(2c). Figure (2a) shows that velocity increases by increasing nanoparticle volume fraction of the fluid because when nanoparticles are high in fluid, they will move quickly that increases the velocity of the fluid. Figure (2b) shows that velocity of fluid decreases by increasing Hartmann number M because Lorentz force always opposes the fluid motion and figure (2c) shows that velocity increases by increasing Darcy's number, velocity of fluid increases because as Darcy's number increases which means more porous is the medium and fluid permeability increases and fluid through porous layer meets little resistance.

Figure 2. Variation of axial velocity w with r.

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Temperature profile for different values of nanoparticle volume fraction of the fluid φ, Hartmann number M, and Darcy's number D_a are observed in figure (3a)−(3c). It can be seen that temperature is maximum at the centre of tube and minimum at the ciliated walls where it is effected by ciliated walls. Figure (3a) illustrates that by increasing nanoparticle volume fraction of the fluid φ temperature profile decreases. It can be seen in graphs that by inserting nanoparticles in blood heat transfer increases due to which wall temperature falls. Figure (3b) shows that by increasing Hartmann number M temperature profile increases as some supplementary work has to be done by the fluid to drag it against Lorentz force which in result increases kinetic energy which is dissipated as heat. Figure (3c) shows that by increasing Darcy's number temperature profile decreases. This is due to decrease in thermal boundary layer thickness.

Figure 3. Variation of temperature θ with r.

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In Figure (4a)−(4c) entropy generation for different values of, nanoparticle volume fraction of the fluid φ, Hartmann number M, and Darcy's number D_a are observed. It can be depicted that entropy generation is maximum at the ciliated walls and minimum at the centre of tube. It is noted in figure (4a) entropy generation increases by increasing nano-fluid volume fraction. As ϕ increases effective thermal conductivity of blood rises due to which rate of heat transfer increases and temperature decreases therefore entropy generation decreases. In figure (4b) it can be seen by increasing Magnetic parameter entropy generation rises. Due to increase in Hartmann number temperature of fluid rises thus entropy generation is increased. In figure (4c) it is noticed that entropy generation increases by increasing Darcy's number. As by increasing Darcy's number thermal boundary layer thickness reduces therefore heat transfer increases and temperature decreases which rises entropy generation.

Figure 4. Variation of Entropy generation Ns with r.

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In Figure (5a)−(5c) Bejan number for different values of nanoparticle volume fraction of the fluid φ, Hartmann number M, and Darcy's number D_a are observed. Since Bejan number is defined as the ratio of heat transfer irreversibility to the net irreversibility (i.e. heat transfer irreversibility and fluid friction irreversibility). As heat transfer across a finite temperature difference is small and frictional forces are also negligible at centre due to which Bejan number decreases at the middle of the tube and at the walls of tube Bejan number is maximum. It is noted in figure (5a) that Bejan number decreases with an increase in nanoparticles volume fraction ϕ which shows that total entropy generation in blood flow is greater than entropy generation with the help of heat transfer. From figures (5b) and (5c) it can be noted that by increasing Hartmann number M and Darcy's number D_a Bejan number increases which shows that entropy generation due to heat transfer is greater than net entropy generation.

Figure 5. Variation of Bejan number Be with r.

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6. Conclusions

In this study we have developed a mathematical model of forced convective flow of tangent hyperbolic fluid through a ciliated axisymmetric tube in a porous medium. Effects of copper nano particles, MHD and porous media are observed for the blood flow (tangent hyperbolic fluid) in a fallopian tube. The fluid motion is produced by the ciliated surface which is considered as a continuous envelope formed by the coordinated cilia. The boundary conditions are considered at the centre of the tube and on the tip of cilia which is anchored in the wall of the tube and formed a wavy surface. The simulation shows that energy and momentum equations involves physical parameter like velocity, temperature, pressure, thermal conductivity to see the effects of nano particles, MHD and porous medium for the enhancement of heat transfer. The present study can be validated by the work of ^[43] if M→0 i.e. Tanh parameter is zero. Following observations are highlighted in the present study.

Ⅰ.The large distribution of nanoparticles into the base fluid (blood) enhance the heat transfer.

Ⅱ.The blood flow along the tube has been accelerated by increasing the volume fraction of nanoparticles.

Ⅲ.The speed of the fluid flow has been decelerated by imposing the applied magnetic field in the transverse direction of the flow whereas heat transfer has increased by applying the magnetic field.

Ⅳ.The blood flow requires less amount of pressure due to the presence of nano particles.

Ⅴ.The presence of nanoparticles in base fluid results in weak disorder which helps to reduce viscous dissipation effects.

The present study is very helpful to observe the blood flow in fallopian tube for those patients who are facing blood clotting in fallopian tube that may cause for infertility or cancer. Theoretical analysis of present study have shown that the nano particles and field of moving charged particles (magnetic field) help to break the clots in blood. Since flow of blood in fallopian tube is due to ciliary movement, therefore medicines containing copper nanoparticles and magnetic field with radiation therapy help to improve the patient by reducing the viscosity of blood.

Acknowledgements

We are very grateful to the editor and reviewers for their valuable and constructive comments to improve the quality of our paper.

Conflict of interest

All authors declare no conflicts of interest in this paper.

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Effect of nano-particles on MHD flow of tangent hyperbolic fluid in a ciliated tube: an application to fallopian tube

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2. Mathematical model

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4. Entropy generation

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6. Conclusions

Acknowledgements

Conflict of interest

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Effect of nano-particles on MHD flow of tangent hyperbolic fluid in a ciliated tube: an application to fallopian tube

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2. Mathematical model

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4. Entropy generation

5. Graphical results

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Acknowledgements

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