# Effects of Thermo-Conductivity on Hydromagnetic Nanofluid Convective Flow through an Artificial Kidney past a Porous Stretching Cylinder

Authors: Christopher Venance; Mathew N. Kinyanjui; Kang’ethe Giterere
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The study of the effects of thermo-conductivity on hydromagnetic nanofluid convective flow through hollow fibers in an artificial kidney past a porous stretching cylinder has been considered. The flow is unsteady and magnetic field is perpendicular applied to the porous stretching cylinder in the artificial kidney. Metallic nanoparticles have been used to improve the heat conduction of the base fluids. The equations governing the hydromagnetic nanofluid convective flow past porous stretching cylinder in an artificial kidney are highly coupled and nonlinear; the equations are solved by finite difference method, a class of numerical techniques for solving differential equations by approximating derivatives with finite differences that applies Crank–Nicolson techniques, and the results obtained are presented graphically followed by discussion of the findings. The effects of each nondimensional parameter on flow variables such as velocity, temperature, and concentration profiles have been discussed. The nanoparticles dissolved in the base fluid improve thermo-conductivity of nanofluid; however, the axial velocity, radial velocity, and temperature as well as concentration of solutes increase as dialysis session of hemodialysis during the filtration of patient’s blood in an artificial kidney decreases from a minimum of six hours to two hours in a day.


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## Body

## 1. Introduction

The study of an artificial kidney in hemodialysis has attracted many researchers in the past decades. When the two natural kidneys present in humans are damaged, an artificial kidney, a dialysis machine, is used during hemodialysis to provide significant continuous blood purification and improve the quality of life of the kidney for patients. The main impact on a patient’s well-being using an artificial kidney in hemodialysis is that it improves adequacy of life and prolongs survival [1]. A dialysis machine is used in which the patient’s blood is exposed to one side of a membrane of a large surface area, on the other side of which is a fluid into which unwanted waste materials in the blood can pass through natural diffusion [2]. The artificial kidney is a man-made device. It is then made by hollow fiber and contains approximately 10000 hollow fibers, each with an inner diameter of about 200μm when wet, the membrane thickness of about 20−45μm, the length of about 160−125mm, pore area of 1.5m2, and mean pore size of 5−10nm [3]. An artificial kidney can be applied in biomedical engineering, such as the design of blood pumping machines, the design of peristaltic plasma pumping, and the design of modern artificial kidney machines. Neurologists and engineers who can operate biomedical machines and provide good services to patients with kidney problems in different communities were first considered. A medical study on how to improve renal kidney services in Sub-Saharan African countries, particularly in Tanzania [4], showed that the number of clinicians and experts in kidney problems increased to 14 in 2018 from one in 2006. Furthermore, Somji et al. [5] conducted a nonexperimental study on dialysis centers and embedded factors in patients with end-stage chronic kidney disease receiving hemodialysis. Different models were formulated to study more about how waste products and toxins can be filtered out of the blood to improve the life of patients, the difference between Newtonian and non-Newtonian fluid flow models of an artificial heart valve, [6]. There is an effective method of mass transfer through a dialyzer with different empirical approaches, [7].In the recent past, nanotechnology has played an important role in biomedical fields by dissolving nanoparticles in the base fluid to improve the thermal properties of the base fluid, [8]. Fluids with suspended nanoparticles, or nanometre-sized particles, are said to be nanofluids. Nanofluids are liquids or base fluids such as blood, water, or oils that contain a suspension of nanoparticles that give thermo-physical properties such as thermal conductivity, thermal diffusivity, viscosity, and convective heat transfer coefficients. Nanoparticles are made up of metals, oxides, nitrides, or carbides having very small diameters <100nm. In the targeted field of use, nanofluids have a variety of applications such as designing blood pumping machines, peristaltic plasma pumping, using gold nanoparticles in thermo-therapy, and getting rid of cancer cells. Various studies on the use of nanofluids have been carried out, such as [9] who developed a flow problem model for heat transfer and flow of nanofluids through a porous medium in the presence of a magnetic field. Recent advancements in the thermal stability, machining, solar energy, and biomedical fields used nanofluids [10]. A mathematical simulation model of a convective boundary layer of a nanofluid along an inclined angle plate considering the presence of a magnetic field was developed [11]. A mathematical model governed a chemically reactive porous flat vertical plate with a second-order slip condition of natural convective nanofluid flow [12]. However, a mathematical analysis of the thermal and mass of species of nanofluid flow along a thin membrane exponentially stretching sheet with partial slip was developed [13], a review of energy transfer through free convection using nanofluids in cavities by the influence of magnetic field [14]. Keller-box numerical investigation of micropolar nanofluid flows across an inclined surface [15]. A mathematical model for hydromagnetic nanofluid flow along a semiporous membrane with stretching hollow tubes was developed[16].Nanofluid flow problems past a porous stretching cylinder have received much attention in recent years because of their application in biomedical engineering. For example, blood dialysis in an artificial kidney, blood flow in the capillaries, the flow of blood in oxygenators, the design of filters in transpiration, cooling boundary layer control, and gaseous diffusion. A number of studies have been done with stretching cylinder, a numerical study on hydromagnetic Casson nanofluid flow with heat and mass properties over a porous stretching cylinder [17]. A mathematical model governing MHD nanofluid flow in non-Darcian Forchheimer past a porous stretching cylinder is analyzed [18]. We analyzed heat transfer characteristics on MHD nanofluid over-stretching cylinder [19]. A mathematical model governing the hydromagnetic mass transfer of Casson nanofluid over a thin permeable stretching cylinder considering the energy of isothermal radiation was developed [20]. The effects of heat dissipation on MHD nanofluid over a porous stretching cylinder through Darcy–Forchheimer medium were determined [21].For more than a century, scientists and engineers have made great efforts to enhance the poor thermal conductivity of conventional fluids. In 1873, Maxwell proposed the idea of using metallic particles to enhance the thermal conductivity of materials. Maxwell presented a theory for the effective conductivity of slurries by dispersing millimeter or micrometer-sized particles with a size between0.1μm and 100μm in liquids. Metallic nanoparticles proposed to improve the thermal conductivity of the bio-fluid must have the following factors: nanoparticle concentration, small nanoparticle size, cylindrical nanoparticle shape, and type of nanoparticle such as metallic particles, [22]. We analyzed studies with the effect of thermo-conductivity on nanofluid flow past stretching plates, such as the study on unsteady MHD slip fluid flow of non-Newtonian nanofluid using Steven’s power law model over a moving surface with temperature-dependent thermal conductivity [23]. This study explores the effects of combined thermal conductance and viscosity on Casson nanofluid flow with radiation and velocity infiltrate, [24]. Nanoparticles are dissolved in the base fluids to improve thermo-conductivity with various effective nondimensional parameters [25].Various studies have not been done on hydromagnetic nanofluid convective flow past a porous stretching cylinder in an artificial kidney with the effect of thermo-conductivity. The goal of the present investigation is to study the effects of nanoparticles in an electrically conducting nanofluid in order to improve the thermal conductivity past a porous stretching cylinder during hemodialysis in an artificial kidney.


## 2. Mathematical Model Formulation

A mathematical model governing the effects of thermo-conductivity on hydromagnetic nanofluid convective flow past porous stretching cylinder in an artificial kidney is established. A laminar flow of an incompressible electrically conducting fluid past a porous stretching cylinder subjected to a magnetic field applied in the direction perpendicular to the fluid flow is considered. An artificial kidney has a cylindrical shape with hollow tubes; each tube has two compartments, one for patient’s blood and the other one for a washing fluid called dialysis solution. A thin membrane with hollow fibers separates these two parts. From the flow problem geometry, Figure1, the cylinder has an axis of symmetry z-axis, any point in the cylinder is defined by three dimension coordinates. The projection of the perpendicular distance of the position vector from the z-axis on the rz-plane is r, the height of the point element from the rz-plane is z, and the projection angle from the θ -axis is θ. Therefore, in cylindrical polar coordinate, the coordinates are r,θ and z. The base fluid flows upward against the gravitational force due to a constant pressure gradient ∇P which is transmitted through the hollow fiber membrane of the artificial kidney by the stretching material with uniform velocity wz and the solutes flow through the second conduit as waste products.Figure 1
Flow problem geometry.


### 2.1. Flow Governing Equations

The conservation of mass, momentum, energy, and concentrations equations that govern the flow of an electrically conducting nanofluid past a porous stretching cylinder with the effect of thermo-conductivity are given in cylindrical polar coordinates as follows:(1)1r∂∂rwr+1r∂∂θwθ+∂∂zwz=0,∂wz∂t+wr∂wz∂r+wz∂wz∂z=ν1r∂wz∂r+∂2wz∂r2+ν∂2wz∂z2−νKpwz−σwzB02ρnf+βTgT−TR2+βCgC−CR2,and(2)∂wr∂t+wr∂wr∂r+wz∂wr∂z=ν1r∂wr∂r+∂2wr∂r2+ν∂2wr∂z2−νwrr2−νKpwr−σwrB02ρnf,ρCpnf∂T∂t+wr∂T∂r+wz∂T∂z=knf1r∂T∂r+∂2T∂r2+∂2T∂z2+2μnf∂wr∂r2+wrr2+∂wz∂z2+μnf∂wz∂r+∂wr∂z2+σB02wz2,∂C∂t+wr∂C∂r+wz∂C∂z=Dm∂2C∂r2+1r∂C∂r+∂2C∂z2+αnf∂2T∂r2+1r∂T∂r+∂2T∂z2.Initial and boundary conditionswr=0, wz=0.5ms−1, C=0, T=37K, at 0≤r≤R2, L=R2−R1 and R1≤r≤R2, when t=0.(3)wz=Nwzz,wr=w0,limr⟶R2wrR2,z,t=0,limr⟶R2CR2,z,t=CR2,limr⟶R2TR2,z,t=TR2,atr=R2whent>0,where wzz=w1ez/L is the stretching velocity, w0>0 is the suction velocity, w0<0 is the injection velocity, CR1 is the surface concentration, TR1 is the surface temperature, wr is the velocity component in radial direction, wz is the velocity component in the axial direction, and C is the concentration of solute such as urea and creatinine.N is the number of hollow fibers that every hollow fiber is assumed to have the same flow rate at the blood entrance in an artificial kidney, r denotes the inner radius of hollow fiber, L is the characteristics length, and w1>0 is the initial stretching rate of the porous stretching plate.αnf=knf/ρCpnf, αnf is the thermal diffusivity along the porous medium and knf is the effective thermal conductivity of nanofluid.ρnf is the density of nanofluid and Cpnf specific heat capacity at constant pressure of nanofluid. ν is the kinematic viscosity and it is given as ν=μnf/ρnf. wr, wθ, wz are the radial velocity, velocity in theta direction, and axial velocity. B0 is the magnetic field applied perpendicularly to the permeable and impermeable plates. Dm is the mass diffusion coefficient.


### 2.2. Nondimensionalization of the Flow Governing Equations

The flow governing equations of the problem are transformed to a nondimensional form; this makes sure that the flow governing equations are reduced to make it less complex so that can be solved much easily and the results are applicable to other geometrical similar configuration in the same set of flow boundary conditions of the problem.In order to transform the equation of motionmomentum, the equation of energy, and the equation of concentration into their respective nondimensional form, the following analysis is carried out defining first the nondimensional parameters:(4)w0⋆=w0wR2,wr⋆=wrwR2,wz⋆=wzwR2,r⋆=rL,R1⋆=R1L,R2⋆=R2L,t⋆=wR2tL,z⋆=zL,T⋆=T−TR2TR1−TR2,C⋆=C−CR2CR1−CR2,where wR2,L,C, and T are the characteristics velocity, length, concentration, and temperature.(5)∂wz⋆∂t⋆+wr⋆∂wz⋆∂r⋆+wz⋆∂wz⋆∂z⋆=1Re1r⋆∂wz⋆∂r⋆+∂2wz⋆∂r⋆2+∂2wz⋆∂z⋆2−λpwz⋆−Mwz⋆+GrTT⋆+GrCC⋆,and(6)∂wr⋆∂t⋆+wr⋆∂wr⋆∂r⋆+wz⋆∂wr⋆∂z⋆=1Re1r⋆∂wr⋆∂r⋆+∂2wr⋆∂r⋆2+∂2wr⋆∂z⋆2−wr⋆Rer⋆2−λpwr⋆−Mwr⋆,(7)∂T⋆∂t⋆+wr⋆∂T⋆∂r⋆+wz⋆∂T⋆∂z⋆=1PrRe1r⋆∂T⋆∂r⋆+∂2T⋆∂r⋆2+∂2T⋆∂z⋆2+2EcRe∂wr⋆∂r⋆2+wr⋆r⋆2+∂wz⋆∂z⋆2+EcRe∂wz⋆∂r⋆2+2∂wz⋆∂r⋆∂wr⋆∂z⋆+∂wr⋆∂z⋆2+HJRewz⋆2,(8)∂C⋆∂t⋆+wr⋆∂C⋆∂r⋆+wz⋆∂C⋆∂z⋆=ScRe1r⋆∂C⋆∂r⋆+∂2C⋆∂r⋆2+∂2C⋆∂z⋆2+SrRe1r⋆∂T⋆∂r⋆+∂2T⋆∂r⋆2+∂2T⋆∂z⋆2,wr⋆=0,wz⋆=0.5ms−1,C⋆=0,T⋆=37K,at0≤r⋆≤R2L,L=R2−R1andR1L≤r⋆≤R2L,wz⋆=Nw1wR2ez⋆,wr⋆=w0⋆wR2,for any value ofz,z⋆=∞,wr⋆=1,wz⋆=1,C⋆=1,T⋆=1,limr⋆⟶R2/Lwr⋆R2L,z,t=0,limr⋆⟶R2/LC⋆R2L,z,t=1,limr⋆⟶R2/LT⋆R2L,z,t=1,atr⋆=R2Lwhent⋆>0.


### 2.3. Numerical Methods

There are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences that applies Crank–Nicolson techniques. The numerical scheme will be implemented in MATLAB.The finite difference equation for the equation of motion (4) along the z-direction is given by(9)wzi,jm+1−wzi,jmΔt+wri,jmwzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm2Δr+wzi,jmwzi,jm+1−wzi,j−1m+1+wzi,jm−wzi,j−1m2Δz=1Re1rwzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm2Δr+1Rewzi+1,jm+1−2wzi,jm+1+wzi−1,jm+1+wzi+1,jm−2wzi,jm+wzi−1,jm2Δr2+1Rewzi,j+1m+1−2wzi,jm+1+wzi,j−1m+1+wzi,j+1m−2wzi,jm+wzi,j−1m2Δz2−λp2wzi,jm+1+wzi,jm−M2wzi,jm+1+wzi,jm+GrT2Ti,jm+1+Ti,jm+GrC2Ci,jm+1+Ci,jm,solving for wzi,jm+1 in equation (9) to get(10)wzi,jm+1=wzi,jm−wri,jmΔt/2Δr−wzi−1,jm+1+wzi,jm−wzi−1,jm−wzi,jmΔt/2Δz−wzi,j−1m+1+wzi,jm−wzi,j−1m+Δt/2rReΔr−wzi−1,jm+1+wzi,jm−wzi−1,jm+Δt/2ReΔr2wzi+1,jm+1+wzi−1,jm+1+wzi+1,jm−2wzi,jm+wzi−1,jm+Δt/2ReΔz2wzi,j+1m+1+wzi,j−1m+1+wzi,j+1m−2wzi,jm+wzi,j−1m−Δtλp/2wzi,jm−ΔtM/2wzi,jm+ΔtGrT/2Ti,jm+1+Ti,jm+ΔtGrC/2Ci,jm+1+Ci,jm1+wri,jmΔt/2Δr+wzi,jmΔt/2Δz−Δt/2rReΔr+Δt/2ReΔr2+Δt/2ReΔz2+λpΔt/2+MΔt/2.The finite difference equation for the equation of motion (10) along r-direction is given by(11)wri,jm+1−wri,jmΔt+wri,jmwri,jm+1−wri−1,jm+1+wri,jm−wri−1,jm2Δr+wzi,jmwri,jm+1−wri,j−1m+1+wri,jm−wri,j−1m2Δz=1Re1rwri,jm+1−wri−1,jm+1+wri,jm−wri−1,jm2Δr+1Rewri+1,jm+1−2wri,jm+1+wri−1,jm+1+wri+1,jm−2wri,jm+wri−1,jm2Δr2+1Rewri,j+1m+1−2wri,jm+1+wri,j−1m+1+wri,j+1m−2wri,jm+wri,j−1m2Δz2−12Rer2wri,jm+1+wri,jm−λp2wri,jm+1+wri,jm−M2wri,jm+1+wri,jm,solving for wri,jm+1 in equation (11) to get(12)wri,jm+1=wri,jm−wri,jmΔt/2Δr−wri−1,jm+1+wri,jm−wri−1,jm−wzi,jmΔt/2Δz−wri,j−1m+1+wri,jm−wri,j−1m+Δt/2rReΔr−wri−1,jm+1+wri,jm−wri−1,jm+Δt/2ReΔr2wri+1,jm+1+wri−1,jm+1+wri+1,jm−2wri,jm+wri−1,jm+Δt/2ReΔz2wri,j+1m+1+wri,j−1m+1+wri,j+1m−2wri,jm+wri,j−1m−Δt/2Rer2wri,jm−Δtλp/2wri,jm−ΔtM/2wri,jm1+wri,jmΔt/2Δr+wzi,jmΔt/2Δz−Δt/2ΔrrRe+Δt/2r2Re+Δt/ReΔr2+Δt/ReΔz2+λpΔt/2+MΔt/2.The finite difference equation for the equation of energy (6) is given by(13)Ti,jm+1−Ti,jmΔt+wri,jmTi,jm+1−Ti−1,jm+1+Ti,jm−Ti−1,jm2Δr+wzi,jmTi,jm+1−Ti,j−1m+1+Ti,jm−Ti,j−1m2Δz=1RePr1rTi,jm+1−Ti−1,jm+1+Ti,jm−Ti−1,jm2Δr+1RePrTi+1,jm+1−2Ti,jm+1+Ti−1,jm+1+Ti+1,jm−2Ti,jm+Ti−1,jm2Δr2+1RePrTi,j+1m+1−2Ti,jm+1+Ti,j−1m+1+Ti,j+1m−2Ti,jm+Ti,j−1m2Δz2+2EcRewri,jm+1−wri−1,jm+1+wri,jm−wri−1,jm2Δr2+EcRer2wri,jm+1+wri,jm2+2EcRewzi,jm+1+wzi,jm2Δr2+EcRewzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm2Δr2+2wzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm2Δrwri,jm+1−wri,j−1m+1+wri,jm−wri,j−1m2Δz+EcRewri,jm+1−wri,j−1m+1+wri,jm−wri,j−1m2Δz2+HJ2Rewzi,jm+1+wzi,jm2.solving for Ti,jm+1 in equation (13) to get(14)Ti,jm+1=Ti,jm−wri,jmΔt/2Δr−Ti−1,jm+1+Ti,jm−Ti−1,jm−wzi,jmΔt/2Δz−Ti,j−1m+1+Ti,jm−Ti,j−1m+Δt/2rRePrΔr−Ti−1,jm+1+Ti,jm−Ti−1,jm+Δt/2RePrΔr2Ti+1,jm+1+Ti−1,jm+1+Ti+1,jm−2Ti,jm+Ti−1,jm+Δt/2RePrΔz2Ti,j+1m+1+Ti,j−1m+1+Ti,j+1m−2Ti,jm+Ti,j−1m+EcΔt/2ReΔr2wri,jm+1−wri−1,jm+1+wri,jm−wri−1,jm2+EcΔt/Rer2wri,jm+1+wri,jm2+EcΔt/2ReΔz2wzi,jm+1−wzi,j−1m+1+wzi,jm−wzi,j−1m2+EcΔt/4ReΔr2wzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm2+EcΔt/2ReΔrΔzwzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm×wri,jm+1−wri,j−1m+1+wri,jm−wri,j−1m+EcΔt/4ReΔz2wri,jm+1−wri,j−1m+1+wri,jm−wri,j−1m2+ΔtHJ/2Rewzi,jm+1+wzi,jm21+wri,jmΔt/2Δr+wzi,jmΔt/2Δz−Δt/2rRePrΔr+Δt/2RePrΔr2+Δt/2RePrΔz2Similary, the finite difference equation for the equation of concentration (8) is given by(15)Ci,jm+1−Ci,jmΔt+wri,jmCi,jm+1−Ci−1,jm+1+Ci,jm−Ci−1,jm2Δr+wzi,jmCi,jm+1−Ci,j−1m+1+Ci,jm−Ci,j−1m2Δz=1ScRe1rCi,jm+1−Ci−1,jm+1+Ci,jm−Ci−1,jm2Δr+1ScReCi+1,jm+1−2Ci,jm+1+Ci−1,jm+1+Ci+1,jm−2Ci,jm+Ci−1,jm2Δr2+1ScReCi,j+1m+1−2Ci,jm+1+Ci,j−1m+1+Ci,j+1m−2Ci,jm+Ci,j−1m2Δz2+SrRe1rTi,jm+1−Ti−1,jm+1+Ti,jm−Ti−1,jm2Δr+SrReTi+1,jm+1−2Ti,jm+1+Ti−1,jm+1+Ti+1,jm−2Ti,jm+Ti−1,jm2Δr2+SrReTi,j+1m+1−2Ti,jm+1+Ti,j−1m+1+Ti,j+1m−2Ti,jm+Ti,j−1m2Δz2,solving for Ci,jm+1 in equation (15) to get(16)Ci,jm+1=Ci,jm−wri,jmΔt/2Δr−Ci−1,jm+1+Ci,jm−Ci−1,jm−wzi,jmΔt/2Δz−Ci,j−1m+1+Ci,jm−Ci,j−1m+Δt/2rΔrScRe−Ci−1,jm+1+Ci,jm−Ci−1,jm+Δt/2Δr2ScReCi+1,jm+1+Ci−1,jm+1+Ci+1,jm−2Ci,jm+Ci−1,jm+Δt/2Δz2ScReCi,j+1m+1+Ci,j−1m+1+Ci,j+1m−2Ci,jm+Ci,j−1m+SrΔt/2rΔrReTi,jm+1−Ti−1,jm+1+Ti,jm−Ti−1,jm+SrΔt/2Δr2ReTi+1,jm+1−2Ti,jm+1+Ti−1,jm+1+Ti+1,jm−2Ti,jm+Ti−1,jm+SrΔt/2Δz2ReTi,j+1m+1−2Ti,jm+1+Ti,j−1m+1+Ti,j+1m−2Ti,jm+Ti,j−1m1+wri,jmΔt/2Δr+wzi,jmΔt/2Δz−Δt/2rΔrScRe+Δt/Δr2ScRe+Δt/Δz2ScRe.


## 2.1. Flow Governing Equations

The conservation of mass, momentum, energy, and concentrations equations that govern the flow of an electrically conducting nanofluid past a porous stretching cylinder with the effect of thermo-conductivity are given in cylindrical polar coordinates as follows:(1)1r∂∂rwr+1r∂∂θwθ+∂∂zwz=0,∂wz∂t+wr∂wz∂r+wz∂wz∂z=ν1r∂wz∂r+∂2wz∂r2+ν∂2wz∂z2−νKpwz−σwzB02ρnf+βTgT−TR2+βCgC−CR2,and(2)∂wr∂t+wr∂wr∂r+wz∂wr∂z=ν1r∂wr∂r+∂2wr∂r2+ν∂2wr∂z2−νwrr2−νKpwr−σwrB02ρnf,ρCpnf∂T∂t+wr∂T∂r+wz∂T∂z=knf1r∂T∂r+∂2T∂r2+∂2T∂z2+2μnf∂wr∂r2+wrr2+∂wz∂z2+μnf∂wz∂r+∂wr∂z2+σB02wz2,∂C∂t+wr∂C∂r+wz∂C∂z=Dm∂2C∂r2+1r∂C∂r+∂2C∂z2+αnf∂2T∂r2+1r∂T∂r+∂2T∂z2.Initial and boundary conditionswr=0, wz=0.5ms−1, C=0, T=37K, at 0≤r≤R2, L=R2−R1 and R1≤r≤R2, when t=0.(3)wz=Nwzz,wr=w0,limr⟶R2wrR2,z,t=0,limr⟶R2CR2,z,t=CR2,limr⟶R2TR2,z,t=TR2,atr=R2whent>0,where wzz=w1ez/L is the stretching velocity, w0>0 is the suction velocity, w0<0 is the injection velocity, CR1 is the surface concentration, TR1 is the surface temperature, wr is the velocity component in radial direction, wz is the velocity component in the axial direction, and C is the concentration of solute such as urea and creatinine.N is the number of hollow fibers that every hollow fiber is assumed to have the same flow rate at the blood entrance in an artificial kidney, r denotes the inner radius of hollow fiber, L is the characteristics length, and w1>0 is the initial stretching rate of the porous stretching plate.αnf=knf/ρCpnf, αnf is the thermal diffusivity along the porous medium and knf is the effective thermal conductivity of nanofluid.ρnf is the density of nanofluid and Cpnf specific heat capacity at constant pressure of nanofluid. ν is the kinematic viscosity and it is given as ν=μnf/ρnf. wr, wθ, wz are the radial velocity, velocity in theta direction, and axial velocity. B0 is the magnetic field applied perpendicularly to the permeable and impermeable plates. Dm is the mass diffusion coefficient.


## 2.2. Nondimensionalization of the Flow Governing Equations

The flow governing equations of the problem are transformed to a nondimensional form; this makes sure that the flow governing equations are reduced to make it less complex so that can be solved much easily and the results are applicable to other geometrical similar configuration in the same set of flow boundary conditions of the problem.In order to transform the equation of motionmomentum, the equation of energy, and the equation of concentration into their respective nondimensional form, the following analysis is carried out defining first the nondimensional parameters:(4)w0⋆=w0wR2,wr⋆=wrwR2,wz⋆=wzwR2,r⋆=rL,R1⋆=R1L,R2⋆=R2L,t⋆=wR2tL,z⋆=zL,T⋆=T−TR2TR1−TR2,C⋆=C−CR2CR1−CR2,where wR2,L,C, and T are the characteristics velocity, length, concentration, and temperature.(5)∂wz⋆∂t⋆+wr⋆∂wz⋆∂r⋆+wz⋆∂wz⋆∂z⋆=1Re1r⋆∂wz⋆∂r⋆+∂2wz⋆∂r⋆2+∂2wz⋆∂z⋆2−λpwz⋆−Mwz⋆+GrTT⋆+GrCC⋆,and(6)∂wr⋆∂t⋆+wr⋆∂wr⋆∂r⋆+wz⋆∂wr⋆∂z⋆=1Re1r⋆∂wr⋆∂r⋆+∂2wr⋆∂r⋆2+∂2wr⋆∂z⋆2−wr⋆Rer⋆2−λpwr⋆−Mwr⋆,(7)∂T⋆∂t⋆+wr⋆∂T⋆∂r⋆+wz⋆∂T⋆∂z⋆=1PrRe1r⋆∂T⋆∂r⋆+∂2T⋆∂r⋆2+∂2T⋆∂z⋆2+2EcRe∂wr⋆∂r⋆2+wr⋆r⋆2+∂wz⋆∂z⋆2+EcRe∂wz⋆∂r⋆2+2∂wz⋆∂r⋆∂wr⋆∂z⋆+∂wr⋆∂z⋆2+HJRewz⋆2,(8)∂C⋆∂t⋆+wr⋆∂C⋆∂r⋆+wz⋆∂C⋆∂z⋆=ScRe1r⋆∂C⋆∂r⋆+∂2C⋆∂r⋆2+∂2C⋆∂z⋆2+SrRe1r⋆∂T⋆∂r⋆+∂2T⋆∂r⋆2+∂2T⋆∂z⋆2,wr⋆=0,wz⋆=0.5ms−1,C⋆=0,T⋆=37K,at0≤r⋆≤R2L,L=R2−R1andR1L≤r⋆≤R2L,wz⋆=Nw1wR2ez⋆,wr⋆=w0⋆wR2,for any value ofz,z⋆=∞,wr⋆=1,wz⋆=1,C⋆=1,T⋆=1,limr⋆⟶R2/Lwr⋆R2L,z,t=0,limr⋆⟶R2/LC⋆R2L,z,t=1,limr⋆⟶R2/LT⋆R2L,z,t=1,atr⋆=R2Lwhent⋆>0.


## 2.3. Numerical Methods

There are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences that applies Crank–Nicolson techniques. The numerical scheme will be implemented in MATLAB.The finite difference equation for the equation of motion (4) along the z-direction is given by(9)wzi,jm+1−wzi,jmΔt+wri,jmwzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm2Δr+wzi,jmwzi,jm+1−wzi,j−1m+1+wzi,jm−wzi,j−1m2Δz=1Re1rwzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm2Δr+1Rewzi+1,jm+1−2wzi,jm+1+wzi−1,jm+1+wzi+1,jm−2wzi,jm+wzi−1,jm2Δr2+1Rewzi,j+1m+1−2wzi,jm+1+wzi,j−1m+1+wzi,j+1m−2wzi,jm+wzi,j−1m2Δz2−λp2wzi,jm+1+wzi,jm−M2wzi,jm+1+wzi,jm+GrT2Ti,jm+1+Ti,jm+GrC2Ci,jm+1+Ci,jm,solving for wzi,jm+1 in equation (9) to get(10)wzi,jm+1=wzi,jm−wri,jmΔt/2Δr−wzi−1,jm+1+wzi,jm−wzi−1,jm−wzi,jmΔt/2Δz−wzi,j−1m+1+wzi,jm−wzi,j−1m+Δt/2rReΔr−wzi−1,jm+1+wzi,jm−wzi−1,jm+Δt/2ReΔr2wzi+1,jm+1+wzi−1,jm+1+wzi+1,jm−2wzi,jm+wzi−1,jm+Δt/2ReΔz2wzi,j+1m+1+wzi,j−1m+1+wzi,j+1m−2wzi,jm+wzi,j−1m−Δtλp/2wzi,jm−ΔtM/2wzi,jm+ΔtGrT/2Ti,jm+1+Ti,jm+ΔtGrC/2Ci,jm+1+Ci,jm1+wri,jmΔt/2Δr+wzi,jmΔt/2Δz−Δt/2rReΔr+Δt/2ReΔr2+Δt/2ReΔz2+λpΔt/2+MΔt/2.The finite difference equation for the equation of motion (10) along r-direction is given by(11)wri,jm+1−wri,jmΔt+wri,jmwri,jm+1−wri−1,jm+1+wri,jm−wri−1,jm2Δr+wzi,jmwri,jm+1−wri,j−1m+1+wri,jm−wri,j−1m2Δz=1Re1rwri,jm+1−wri−1,jm+1+wri,jm−wri−1,jm2Δr+1Rewri+1,jm+1−2wri,jm+1+wri−1,jm+1+wri+1,jm−2wri,jm+wri−1,jm2Δr2+1Rewri,j+1m+1−2wri,jm+1+wri,j−1m+1+wri,j+1m−2wri,jm+wri,j−1m2Δz2−12Rer2wri,jm+1+wri,jm−λp2wri,jm+1+wri,jm−M2wri,jm+1+wri,jm,solving for wri,jm+1 in equation (11) to get(12)wri,jm+1=wri,jm−wri,jmΔt/2Δr−wri−1,jm+1+wri,jm−wri−1,jm−wzi,jmΔt/2Δz−wri,j−1m+1+wri,jm−wri,j−1m+Δt/2rReΔr−wri−1,jm+1+wri,jm−wri−1,jm+Δt/2ReΔr2wri+1,jm+1+wri−1,jm+1+wri+1,jm−2wri,jm+wri−1,jm+Δt/2ReΔz2wri,j+1m+1+wri,j−1m+1+wri,j+1m−2wri,jm+wri,j−1m−Δt/2Rer2wri,jm−Δtλp/2wri,jm−ΔtM/2wri,jm1+wri,jmΔt/2Δr+wzi,jmΔt/2Δz−Δt/2ΔrrRe+Δt/2r2Re+Δt/ReΔr2+Δt/ReΔz2+λpΔt/2+MΔt/2.The finite difference equation for the equation of energy (6) is given by(13)Ti,jm+1−Ti,jmΔt+wri,jmTi,jm+1−Ti−1,jm+1+Ti,jm−Ti−1,jm2Δr+wzi,jmTi,jm+1−Ti,j−1m+1+Ti,jm−Ti,j−1m2Δz=1RePr1rTi,jm+1−Ti−1,jm+1+Ti,jm−Ti−1,jm2Δr+1RePrTi+1,jm+1−2Ti,jm+1+Ti−1,jm+1+Ti+1,jm−2Ti,jm+Ti−1,jm2Δr2+1RePrTi,j+1m+1−2Ti,jm+1+Ti,j−1m+1+Ti,j+1m−2Ti,jm+Ti,j−1m2Δz2+2EcRewri,jm+1−wri−1,jm+1+wri,jm−wri−1,jm2Δr2+EcRer2wri,jm+1+wri,jm2+2EcRewzi,jm+1+wzi,jm2Δr2+EcRewzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm2Δr2+2wzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm2Δrwri,jm+1−wri,j−1m+1+wri,jm−wri,j−1m2Δz+EcRewri,jm+1−wri,j−1m+1+wri,jm−wri,j−1m2Δz2+HJ2Rewzi,jm+1+wzi,jm2.solving for Ti,jm+1 in equation (13) to get(14)Ti,jm+1=Ti,jm−wri,jmΔt/2Δr−Ti−1,jm+1+Ti,jm−Ti−1,jm−wzi,jmΔt/2Δz−Ti,j−1m+1+Ti,jm−Ti,j−1m+Δt/2rRePrΔr−Ti−1,jm+1+Ti,jm−Ti−1,jm+Δt/2RePrΔr2Ti+1,jm+1+Ti−1,jm+1+Ti+1,jm−2Ti,jm+Ti−1,jm+Δt/2RePrΔz2Ti,j+1m+1+Ti,j−1m+1+Ti,j+1m−2Ti,jm+Ti,j−1m+EcΔt/2ReΔr2wri,jm+1−wri−1,jm+1+wri,jm−wri−1,jm2+EcΔt/Rer2wri,jm+1+wri,jm2+EcΔt/2ReΔz2wzi,jm+1−wzi,j−1m+1+wzi,jm−wzi,j−1m2+EcΔt/4ReΔr2wzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm2+EcΔt/2ReΔrΔzwzi,jm+1−wzi−1,jm+1+wzi,jm−wzi−1,jm×wri,jm+1−wri,j−1m+1+wri,jm−wri,j−1m+EcΔt/4ReΔz2wri,jm+1−wri,j−1m+1+wri,jm−wri,j−1m2+ΔtHJ/2Rewzi,jm+1+wzi,jm21+wri,jmΔt/2Δr+wzi,jmΔt/2Δz−Δt/2rRePrΔr+Δt/2RePrΔr2+Δt/2RePrΔz2Similary, the finite difference equation for the equation of concentration (8) is given by(15)Ci,jm+1−Ci,jmΔt+wri,jmCi,jm+1−Ci−1,jm+1+Ci,jm−Ci−1,jm2Δr+wzi,jmCi,jm+1−Ci,j−1m+1+Ci,jm−Ci,j−1m2Δz=1ScRe1rCi,jm+1−Ci−1,jm+1+Ci,jm−Ci−1,jm2Δr+1ScReCi+1,jm+1−2Ci,jm+1+Ci−1,jm+1+Ci+1,jm−2Ci,jm+Ci−1,jm2Δr2+1ScReCi,j+1m+1−2Ci,jm+1+Ci,j−1m+1+Ci,j+1m−2Ci,jm+Ci,j−1m2Δz2+SrRe1rTi,jm+1−Ti−1,jm+1+Ti,jm−Ti−1,jm2Δr+SrReTi+1,jm+1−2Ti,jm+1+Ti−1,jm+1+Ti+1,jm−2Ti,jm+Ti−1,jm2Δr2+SrReTi,j+1m+1−2Ti,jm+1+Ti,j−1m+1+Ti,j+1m−2Ti,jm+Ti,j−1m2Δz2,solving for Ci,jm+1 in equation (15) to get(16)Ci,jm+1=Ci,jm−wri,jmΔt/2Δr−Ci−1,jm+1+Ci,jm−Ci−1,jm−wzi,jmΔt/2Δz−Ci,j−1m+1+Ci,jm−Ci,j−1m+Δt/2rΔrScRe−Ci−1,jm+1+Ci,jm−Ci−1,jm+Δt/2Δr2ScReCi+1,jm+1+Ci−1,jm+1+Ci+1,jm−2Ci,jm+Ci−1,jm+Δt/2Δz2ScReCi,j+1m+1+Ci,j−1m+1+Ci,j+1m−2Ci,jm+Ci,j−1m+SrΔt/2rΔrReTi,jm+1−Ti−1,jm+1+Ti,jm−Ti−1,jm+SrΔt/2Δr2ReTi+1,jm+1−2Ti,jm+1+Ti−1,jm+1+Ti+1,jm−2Ti,jm+Ti−1,jm+SrΔt/2Δz2ReTi,j+1m+1−2Ti,jm+1+Ti,j−1m+1+Ti,j+1m−2Ti,jm+Ti,j−1m1+wri,jmΔt/2Δr+wzi,jmΔt/2Δz−Δt/2rΔrScRe+Δt/Δr2ScRe+Δt/Δz2ScRe.


## 3. Discussion of the Results

From the findings, the effects of various nondimensional parameters such as Reynolds number, magnetic field parameter, permeability parameter, Grashof number, Prandtl number, Eckert number, Joule heating parameter, Schmidt number, and Soret number on velocity, temperature, and concentration profiles are effectively discussed.From Figure2(a), it is observed that an increase in Reynolds number Re results in an axial velocity Wz of the layers to decrease exponentially from the center of the hollow fibers in the blood compartment to the walls of the dialysis solution compartment in an artificial kidney. This is due to the friction that results in the resistance of the nanofluid against the motion produced on the wall adjacent to it. Since Reynolds number is the ratio of inertial force to viscous force. From Figure 2(b), it is observed that an increase in Reynolds number Re results in radial velocity Wr of nanofluid to decreases exponentially along the hollow fiber walls of the artificial kidney. It is observed from Figure 2(c) that an increase in Reynolds number Re leads to an increase of the temperature T profile in the direction of the flow of the nanofluid at the center of the hollow fiber at the radial distance r=0.945, the inner radius of the hollow tube in the blood compartment. This is because a higher laminar kinetic energy in the nanofluid particles increases along the axis symmetry of the flow of the electrically conducting nanofluid inside the blood compartment in an artificial kidney but decreases as it moves periphery near the wall of the stretching cylinder of an artificial kidney of the dialysis solution compartment and from Figure 2(d) it is observed that an increase in Reynolds number Re leads to an increase in concentration C profile as it approaches the radial distance r=0.945, the inner radius of the hollow fibers in an artificial kidney but decreases as it moves periphery near the wall of the stretching cylinder in an artificial kidney of the dialysis solution compartment. The concentration of solutes in nanofluid is higher at the center of the flow in the blood compartment but decreases as it moves near the wall of the membrane in the dialysate compartment because not all waste products will diffuse through the pores of the hollow fibers some have bigger size molecules like protein and blood cells to diffuse through membrane by diffusion method.Figure 2
Effects of Reynolds numberRe on axial velocity Wz, radial velocity Wr, temperature T, and concentration C profiles.
(a)(b)(c)(d)It is noted from Figure3(a) that as the permeability parameter λp increases there is a decrease in axial velocity Wz; this is because permeability parameter λp is inversely proportional to the normal permeability number Kp leads the flow to decelerate and figure b shows that the radial velocity Wr is found to increase with an increase in permeability parameter λp; this is because the velocity is an increasing function of permeability parameter λp. An increase in the porosity of hollow fibers in an artificial kidney allows nanofluid and other solutes to pass through after blood filtration during hemodialysis in an artificial kidney, the higher the porosity, the higher the permeability that acts as the drag force of the permeability. It is observed from the Figure 3(c) that as the permeability parameter λp increases the temperature profile increases and decreases as it approaches the surface of the dialysis solution compartment when the radial distance is r=0.945, the temperature of nanofluid can be determined by the boundary layer velocity rather than thermal boundary layer when the temperature of nanofluid at the impermeable stretching plate in an artificial kidney and that of the dialysis solution compartment varies. From Figure 3(d) as permeability parameter λp increases concentration profile increases in the direction of electrically conducting nanofluid, at the center of the hollow fiber in the blood compartment and decreases as it approaches near the surface of the stretching wall of the dialysis solution compartment when radial distance is at r=0.945. Therefore, the concentration gradient between blood and dialysate helps the diffusion of solutes to move through the pores of hollow fibers in an artificial kidney to the dialysis solution compartment near the surface of the stretching cylinder.Figure 3
Effects of permeability parameterλp on axial velocity Wz, radial velocity Wr, temperature T, and concentration C profiles.
(a)(b)(c)(d)It is noted from Figure4(a) that as the magnetic field parameter increases the axial velocity Wz profile decreases, because of the force called the Lorentz force generated by the magnetic field applied in opposite direction to the flow direction of the electrically conducting nanofluid which leads to a decrease in velocity. Figure 4(b) shows that the radial velocity Wr profile increases as the magnetic field increases. This is because the magnetic field is applied in the direction of radial distance. It is observed from Figure 4(c) that an increase in the magnetic field parameter leads to an increase in the temperature profile in the blood compartment and decreases exponentially when it reaches a radial distance of 0.945 as it approaches the surface of the hollow fiber tubes stretching from the dialysis solution compartment in an artificial kidney. The metallic nanoparticles used in the study improve thermal conductivity that increases fluid temperature as a result of the interaction of the atomic ions of the metallic particles. Lorentz force generated by magnetic force has the tendency to slow the motion of the nanofluid in the boundary layer and increases its temperature profiles and Figure 4(d) shows that an increase in the magnetic field parameter leads to an increase in the concentration profile for the blood compartment and decreases exponentially when it reaches a radial distance of 0.945 as it approaches the dialysis solution compartment in an artificial kidney. Decreasing the velocity of the nanofluid tends to slow the movement of solutes and other waste products from the blood during hemodialysis, causing a decrease in concentration.Figure 4
Effects of magnetic parameterM on axial velocity Wz, radial velocity Wr, temperature T, and concentration C profiles.
(a)(b)(c)(d)From Figure5(a), it is observed that as the Grashof number due to the temperature gradient GrT increases, the axial velocity Wz profile also increases, as the Grashof number is the ratio between buoyant to viscous forces acting on the nanofluid in the velocity boundary layer. The role of the Grashof number is to describe the effects of natural convection currents due to temperature change of the nanofluid or cooling of the surface. Figure 5(b) shows that an increase in Grashof number due to temperature change GrT leads to a radial velocity Wr profile near the surface of the hollow fibers in an artificial kidney. Therefore, such an increase in Grashof number due to temperature change GrT does not have a significant decrease in the radial velocity profile Wr. It is observed that in Figure 5(c) as the Grashof number due to temperature increases, the temperature profile also decreases along the axis symmetry of the nanofluid flow and then increases exponentially. This is because the nanofluid temperature within the boundary layer is enhanced by fluid temperature and Figure 5(d) shows that as Grashof number due to temperature increases leads concentration profile to decrease along the axis symmetry of the nanofluid flow and then sharply increases exponentially. Natural convectional currents help bringing more mass transfer of solutes from the blood compartment to the dialysis solution compartment of the hollow tubes in an artificial kidney and hence concentration profiles increase.Figure 5
Effects of Grashof number due to temperature gradientGrT on axial velocity Wz, radial velocity Wr, temperature T, and concentration C profiles.
(a)(b)(c)(d)In Figure6(a), it is observed that an increase in Grashof number due to the change in concentration GrC leads to an increase in axial velocity Wz profile, which means that the increase of the Grashof number due to the concentration leads to an increase of the buoyancy force. An increase in Grashof number leads to a decreased viscous force which opposes the flow of nanofluid. Figure 6(b) shows that an increase in the Grashof number due to a change in concentration results in decrease in the radial velocity Wr profile; this shows that there is no significant increase in radial velocity Wr profile when Grashof number due to the change in concentration increases. In Figure 6(c) it is noted that an increase in Grashof number due to a change in concentration results in a slight decrease in the temperature profile in the blood compartment but as it approaches the surface of the stretching cylinder in the dialysis solution compartment increases exponentially. The fact is that an increase in Grashof number due to concentration gradient results in a decrease in the thickness of the thermal boundary layer leading to a decrease in the temperature profile. From Figure 6(d) it is noted that an increase in Grashof number due to change in concentration results in a slight decrease in the concentration profile in the blood compartment but as it approaches the surface of the stretching cylinder in the dialysis solution compartment increases exponentially. The concentration profile decreases then increases as it approaches the dialysis solution compartment of the stretching surface in an artificial kidney due to an increase in buoyancy force acting over the nanofluid.Figure 6
Effects of Grashof number due to concentration gradientGrC on axial velocity Wz, radial velocity Wr, temperature T, and concentration C profiles.
(a)(b)(c)(d)It is observed from Figure7(a) that an increase in the Prandtl number Pr results in an increase in axial velocity Wz. Physically, the Prandtl number is the ratio between momentum diffusivity and the thermal diffusivity of the nanofluid properties. This implies an increase in Prandtl number, heat diffuses relatively slowly compared to the velocity of the nanofluid. The decrease in thermal diffusivity leads to a velocity boundary layer thickness past the porous stretching cylinder decreasing when the Prandtl number increases. From Figure 7(b) it is noted that an increase in the Prandtl number Pr results in an increase in the radial velocity profile Wr. It is noted that as the Prandtl number increases, there is no significant change in the velocity since the changes in the Prandtl number are approximately too small to cause the nanofluid radial velocity to increase significantly. Figure 7(c) shows that increasing the Prandtl number causes an increase in the temperature profile. It is due to the fact that thermal diffusion takes place at a slower rate compared to the diffusivity of kinematic viscosity diffusivity. Nanoparticles dissolved in the base fluid improve the thermal conductivity of base fluid by increasing thermal boundary layer thickness. From Figure 7(d) shows that increasing the Prandtl number leads to an increase in the concentration profile. This is because as the surface of the hollow tubes in the dialysis solution compartment of an artificial kidney becomes cooled, the viscous force increases the kinetic energy of the nanofluid particles.Figure 7
Effects of Prandtl numberPr on axial velocity Wz, radial velocity Wr, temperature T, and concentration C profiles.
(a)(b)(c)(d)It is observed from Figure8(a) that as the Eckert number Ec increases, so does the axial velocity Wz profile increases. Since Eckert number is the ratio between the kinetic energy and thermal energy in the nanofluid flow. When thermal heating increases, it leads to a rise in the velocity of the nanofluid particles. Figure 8(b) shows that as the Eckert number Ec increases, so does the radial velocity Wr profile decreases. There is no significant increase in Eckert number with the velocity profile of solutes in the radial direction, since there is no entry of temperature into the equation of motion in the radial direction. Figure 8(c) shows that temperature T profile increases as Eckert number Ec increases. Physically, Eckert number is the change in kinetic energy in the nanofluid flow into the internal energy by work done against the viscocity of the nanofluid. When thermal heating increases, this leads to an increase in the temperature of the nanofluid particles. Figure 8(d) shows that, as the concentration C profile increases the Eckert number Ec increases. There is a significant increase in the concentration profile since there is a mutual relationship between the Eckert number and concentration gradient as the equation of concentration has an entry effect of temperature. Changes in temperature and kinetic energy of the particles translate into mass transfer of solutes from nanofluid.Figure 8
Effects of the Eckert numberEc on axial velocity Wz, radial velocity Wr, temperature T, and concentration C profiles.
(a)(b)(c)(d)Physically, the Joule heating parameter is the ratio between the amount of heat released from an electrical resistor to its resistance and the charge passing through it to produce thermal energy. An increase in the temperature of the nanofluid leads to an increase in the kinetic energy of the nanofluid particles that results in an axial velocity profile increasing. Pressure gradients created at the bottom wall of the pipe in the direction of flow results in an increased velocity distribution of the nanofluid particles. Figure9(b) shows that as Joule heating increases, so does radial velocity Wr increases. An increase in Joule heating has a minor effect on radial velocity because the equation of motion in the radial direction has no temperature entry. Figure 9(c) shows that as Joule heating increases temperature profile increases. The generated heat increases the temperature distribution inside the hollow tubes of the blood compartment in the artificial kidney. Figure 9(d) shows that as Joule heating increases concentration profile increases. The pressure gradient modified for nanofluid particles creates the velocity distribution inside the hollow tubes of the blood compartment to keep the mass flow rate of solutes.Figure 9
Effects of Joule heating parameterHJ on axial velocity Wz, radial velocity Wr, temperature T, and concentration C profiles.
(a)(b)(c)(d)From Figure10(a), it is observed that with an increase in Schmidt number Sc, there is a slight decrease in the axial velocity Wz. Physically, the Schmidt number is the ratio of kinematic viscosity diffusivity to mass diffusivity of solutes in a nanofluid. Schmidt number increases the nanofluid viscosity with results in an increase in the velocity boundary layer thickness and so reduces shear stress and however decreases the flow of the nanofluid during hemodialysis in an artificial kidney. It is noted that in Figure 10(b) with an increase in Schmidt number Sc, there is a slightly insignificant decrease in the radial velocity Wr. Figure 10(c) shows that with an increase in Schmidt number, there is an increase in temperature in the pipe of the blood compartment but as it approaches the surface of the hollow tubes of the dialysis solution compartment, temperature decreases exponentially. Figure 10(d) shows that with an increase in Schmidt number, there is a decrease in concentration. Schmidt number can be used to characterize the nanofluid flow of mass transfer analysis, i.e., momentum and mass diffusion convection processes. When Schmidt number is large, kinematic viscosity across a nanofluid is much more effective than solute particles with small diffusivity and becomes less relevant at the surface of the porous stretching cylinder in an artificial kidney [26]. This is why the concentration boundary layer of an artificial kidney is relatively thin as the Schmidt number increases near the end of the stretching cylinder of an artificial kidney.Figure 10
Effects of Schmidt numberSc on axial velocity Wz, radial velocity Wr, temperature T, and concentration C profiles.
(a)(b)(c)(d)From Figure11(a), it is observed that as the Soret number Sr increases, the axial velocity Wz profile increases. Physically, the Soret number is the ratio of temperature difference to the concentration gradient. Nanoparticles dissolved in the base fluid results in the increase in thermal conductivity of an electrically conducting nanofluid, which helps to increase the kinetic energy of the particles and however the axial and radial velocity of nanofluid increases in the boundary layer thickness as the Soret number increases. Figure 11(b) shows that as the Soret number Sr increases radial velocity Wr profiles increase. Figure 11(c) shows that as the Soret number Sr increases temperature T profile decreases in the direction of nanofluid flow and as it is approaching the wall of the hollow tube in the dialysis solution compartment, the temperature profile increases. As the value of the Soret number increases, the value of temperature difference increases. Figure 11(d) shows that as the Soret number Sr increases, the concentration C profile increases also. An increase in Soret number affects the temperature gradient of nanofluid during purification of blood in an artificial kidney; this signifies the improvement of mass diffusion of solutes from blood to the dialysis solution through a porous stretching cylinder in the artificial kidney hence increasing the in concentration profile at the wall surface membrane in the artificial kidney.Figure 11
Effects of Soret numberSr on axial velocity Wz, radial velocity Wr, temperature T, and concentration C profiles.
(a)(b)(c)(d)It is observed from Figure12(a) that, as time t decreases, the axial velocity Wz profile increases. This is to the fact that when nanoparticles dissolved in the base fluid during hemodialysis result in an increase in the kinetic energy of the particles, however, leading to the increase in the nanofluid velocity past porous stretching cylinder as the time decreases. It is noted from Figure 12(b) that, as the time t decreases, the radial velocity Wr profile increases. This is to the fact that, solutes move faster near the wall surface of the stretching cylinder of the hollow fibers in an artificial kidney during dialysis as time decreases. It is noted from Figure 12(c) that, as time t decreases, the temperature T profile increases. This is to the fact that when nanoparticles dissolved in the base fluid during hemodialysis result in an increase in the thermo-conductivity of the particles; hence, thermal boundary layer thickness increases that rises the temperature of the nanofluid past porous stretching cylinder. From Figure 12(d) it is observed that, as time t decreases, concentration C profile increases. This is to the fact that the concentration gradient of the blood and dialysate enhances the faster movement of the solutes from blood to the dialysis solution compartment as time decreases.Figure 12
Effects of timeSr on axial velocity Wz, radial velocity Wr, temperature T, and concentration C profiles.
(a)(b)(c)(d)The wall skin friction coefficient can be calculated from axial and radial velocity profiles. The laminar flow of an incompressible nanofluid has low wall skin friction that improves the nanofluid movement through the wall surface of the hollow fibers in an artificial kidney. The wall skin friction coefficients in axial and radial direction are given as(17)Csz=−μnfwR2/L∂wz⋆/∂r⋆r⋆=01/2ρnfwR22=−2μnfLρnfwR2∂wz⋆∂r⋆r⋆=0,where Re=LρwR2/μnf.Then, equation (17) is reduced to(18)Csz=2Re∂wz⋆∂r⋆r⋆=0.Again,(19)Csr=−μnfwR2/L∂wr⋆/∂r⋆r⋆=01/2ρnfwR22=−2μnfLρnfwR2∂wr⋆∂r⋆r⋆=0,where Re=LρwR2/μnfThen, equation (19) is reduced to(20)Csr=−2Re∂wr⋆∂r⋆r⋆=0.The wall mass transfer rate is determined from concentration profiles. It is given by Sherwood number past a porous stretching cylinder of hollow fibers in an artificial kidney.(21)Shm=−Dm/LCR1−CR2∂C⋆/∂r⋆r⋆=0Dm/LCR1−CR2=−∂C⋆∂r⋆r⋆=0.The rate of heat transfer is determined from temperature profiles and it is given by the Nusselt number.(22)Nuw=−knf/LTR1−TR2∂T⋆/∂r⋆r⋆=0knf/LTR1−TR2=−∂T⋆∂r⋆r⋆=0.Effects of Reynolds numberRe, magnetic field M, permeability parameter λp, Joule heating parameter HJ, Eckert number Ec and time t on skin friction coefficient Cs, heat transfer Shm, and mass transfer Nuw are given as follows.From Table1,(i)
It is observed that asRe increases both Csz and Csr decreases in magnitude, due to the fact that, the laminar flow of an incompressible nanofluid has low skin friction between fluid and wall surface of hollow fibers in an artificial kidney. Shm and Nuw decrease in magnitude as Re increases; this shows that there is conduction and mass transfer at the boundary layers.(ii)
An increase inM results in an increase in the magnitude of Csz due to Lorentz force that is applied perpendicular to the direction flow of an electrically conducting nanofluid but decreases in Csr due to the fact that the magnetic field is applied in the direction flow of the radial distance. It is observed that as Mincreases, the magnitudes of Shm and Nuwdecrease, due to a decrease in thermal and concentration boundary layers that improve heat and mass transfer of solutes from blood to dialysis solution..(iii)
It is observed that asλp increases results to an increase in Csz but decreases in Csr. Sherwood and Nusselt numbers decrease in magnitude due to a decrease in thermal and concentration boundary layer thickness that allows the mass transfer of solutes through pores of a thin membrane of hollow fibers in an artificial kidney from blood to dialysis solution compartment.(iv)
It is observed that as theEc increase Csz decreases but Csr increases by increasing velocity boundary layer thickness. Sherwood and Nusselt number decreases in magnitude as Ec increases.(v)
It is noted that an increases inHJ results to decrease in Csz but an increase in magnitude of Csr. Sherwood number and Nusselt number decreases in magnitude as Joule heating parameter increases. Heat distribution in the nanofluid of the hollow fibers due to viscous dissipation improves the mass transfer of solutes from blood to the dialysis solution compartment.(vi)
It is observed that ast decreases both Csz and Csr decreases but Shm and Nuw increases due to the application of nanoparticles in the base fluid that improves nanofluid thermo-conductivity by improving the mass transfer of solutes from blood during filtration.Table 1
Variation of various nondimensional parameters on skin friction, heat, and mass transfer.
ReMλpHJEctCszCsrShmNuw0.0210.590200.322.45918e + 053.42263e + 01−1.26345e + 07−1.99508e + 060.0310.590200.321.94594e + 052.62293e + 01−1.503368 + 07−2.23001e + 060.0211.590200.322.46994e + 053.42155e + 01−1.27244e + 07−2.00502e + 060.0210.5100200.322.56489e + 053.41217e + 01−1.35202e + 07−2.09327e + 060.0210.590210.322.45044e + 053.42358e + 01−1.27385e + 07−2.01106e + 060.0210.590200.422.41264e + 053.43302e + 01−1.50590e + 07−2.42248e + 060.0210.590200.332.60816e + 054.08138e + 02−7.00017e + 06−1.28864e + 060.0210.590200.320.00000e + 000.00000e + 000.00000e + 000.00000e + 00


## 4. Conclusion

From the study of hydromagnetic nanofluid convective flow past a porous stretching cylinder with the effect of thermo-conductivity, an efficient artificial kidney is developed by improving an optimal artificial kidney and hemodialysis conditions using simulation by reducing the number of hollow fibers from 10,000 hollow fibers, 10 permeability parameter values and flow rates at the entrance and outlets of the dialysis solution compartment of200−400mil/min and 400−800mil/min used in literature [2] to 100 hollow fibers having a permeability parameter value of 90 values since nanoparticles dissolved in the base fluid improves the thermo-conductivity of nanofluid. A sensor to detect all dissolved nanoparticles in the base fluid so that none of the particles will remain in the blood after blood purification has to be included in the design of an optimal artificial kidney. Flow rates at the entrance and outlets of the dialysis solution compartment have been increased to 400−800mil/min and 700−1200mil/min. The following results have been noticed as it saves substantive time and experimental cost of the hemodialysis session when varying different nondimensional parameters with velocity, temperature and concentration profiles.(i)
The results of thermo-conductivity, when Reynolds number, permeability parameter, magnetic parameter, Eckert number, Schmidt number increase, axial velocity decreases, but when Grashof number due to temperature, Grashof number due to concentration, Prandtl number, Joule heating, Soret number increase, leads to an increase in axial velocity of nanofluid at the center of blood compartment.(ii)
The results of thermo-conductivity with an increase in Reynolds number, Grashof number due to temperature, Grashof number due to concentration, Eckert number, Schmidt number, there is a decrease in the radial velocity while an increase in permeability parameter, magnetic parameter, Prandtl number, Joule heating, Soret number leads to an increase in the radial velocity of nanofluid at the center of the blood compartment.(iii)
It is observed that an increase in Reynolds number, Eckert number, Schmidt number, permeability parameter, magnetic parameter, Prandtl number, and Joule heating results in an increase in temperature while an increase in Grashof number due to temperature, Grashof number due to concentration, Soret number leads to decrease in the temperature of nanofluid at the center of blood compartment with the results in thermo-conductivity of nanofluid.(iv)
It is observed that an increase in Reynolds number, Eckert number, permeability parameter, magnetic parameter, Prandtl number, Joule heating, and Soret number results in an increase in concentration while an increase in Grashof number due to temperature, Grashof number due to concentration, Schmidt number leads to decrease in the concentration of nanofluid at the center of blood compartment.(v)
Using nanoparticles that are dissolved in the base fluid helps improve dialysis session of hemodialysis during the filtration of a patient’s blood in an artificial kidney from a minimum of six hours to two hours. However, it helps a good number of patients with kidney problems from different dialysis centers, especially in Sub-Saharan countries to receive the service of the filtration of blood.



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*Source: 5468057-2022-10-10.xml*