Three-dimensional magnetohydrodynamic (MHD) flow of Maxwell nanofluid containing gyrotactic micro-organisms with heat source/sink

Ali, Aamir; Sulaiman, M.; Islam, S.; Shah, Zahir; Bonyah, Ebenezer

doi:10.1063/1.5040540

This paper discusses the three-dimensional flow of Maxwell nanofluid containing gyrotactic micro-organisms over a stretching surface. The effects of magnetic field and heat source/sink are also considered. Theory of microorganisms is utilized to stabilize the suspended nanoparticles through bioconvection induced by the effects of buoyancy forces. HAM (homotopy analysis method) is used to acquire analytic solution for the governing nonlinear equations. The effects of Deborah number, Hartmann number, mixed convection parameter, buoyancy ratio parameter, bioconvection Rayeigh number, stretching ratio parameter, brownian diffusion and thermophoresis diffusion parameters, Prandtl number, Lewis number, micro-organisms concentration difference parameter, bioconvection Peclet number and the bioconvection Lewis number on velocity, temperature, density of motile microorganisms and nanoparticle concentration are discussed graphically. The local Nusselt, Sherwood and motile micro-organisms numbers are also analyzed graphically. The reduction of the boundary layer thickness and velocity due to magnetic field is noted. The heat source/sink parameter have opposite effects on the temperature profile. We found that In comparison to the case of heat sink the thermal boundary layer thickness and temperature increases in the case of heat source.

I. INTRODUCTION

In recent years the study of non-Newtonian fluids has importance due to many applications in engineering, industry and biological sciences e.g. in material processing and in the food and cosmetic industries.¹ These applications generally involve complex geometries and requires that the response of the fluid to given temperature conditions is controlled. The basic equations that govern the flow of non-Newtonian fluids are highly non-linear. The mathematical description of a non-Newtonian fluid requires a constitutive equation that determines the rheological properties of the fluid. Due to the complexity, the non-Newtonian fluids are generally divided into three types namely, the differential type, the rate type and the integral type. The simplest subclass of the rate type fluid is Maxwell fluid model which describes the properties of relaxation time. Maxwell fluid is one of those fluids which possess both viscous and elastic nature. There are several investigations in this direction for two-dimensional flow configurations have been done with different features of the non-Newtonian fluid models.^2–6 However, in practical applications sometimes, the flow is three-dimensional. In view of such motivation, researchers have studies the three-dimensional flow for different flow geometries.^7–11

The problem of heat and mass transfer analysis over a stretched boundary layer is of great interest for researchers due to its engineering and industrial applications like glass fiber, manufacture and drawing of plastics and rubber sheets, paper production, cooling of an infinite metallic sheet in a cooling bath, the continuous casting, cooling of continuous stripes, polymer extrusion process, heat treated materials travelling on conveyer belts, food processing and many others. Sakiadis¹² has done the pioneering study of the boundary layer flow on a moving surface, which was later extended by Crane¹³ by considering the linearly stretching surface and found the exact solution of the Navier-Stokes equations. After that several investigations has been done by considering different flow configurations such as MHD, suction/injection, Soret and Dufour effects etc.^14–18 The study of heat generation/absorption effects on heat transfer is of great significance because it crucially controls the heat transfer. Many researchers have studied the heat source/sink effects for different flow configurations e.g. Refs. 19–22.

With the advent of nanoscience; nanofluids have become a focus of attention in the study of fluid flow in the presence of nanoparticles. Nanofluids are the fluids that are arranged by scattering 109m sized substances such as nanoparticles, nanotubes, nanofibers, droplets etc in fluids. Actually, nanofluids are nanoscale shattered suspensions involving concise nanometer sized materials. These are two period systems the first one is solid phase and the second one is liquid phase. Nanofluids can be used to increase the thermal conductivity of the fluids and are more stable fluids having better writing, spreading and dispersion properties on solid surfaces. Choi and Eastman²³ initially presented the idea of suspension of nanoparticles in base fluids. Later, Buongiorno²⁴ extended the concept by considering Brownian motion and thermophoresis movement of nanoparticles in view of application in hybrid power engine, thermal management, heat exchanger, domestic refrigerators etc. They have enhanced heat conduction power as compared to base fluids and are useful in cancer therapy and medicine. With the presentation of a reliable model of nanofluid given by Buongiorno,²⁴ nanofluids have become a subject of great interest for researchers in the last few years.^25–35

Recently, Shah et al.^36–39 have studied nanofluid flow with impacts of thermal radiation and hall current with rotating system.

Bioconvection is a phenomenon that is used to describe the unstructured pattern and instability produces by the micro-organisms which are swimming to the upper part of a fluid and has a lesser density. Due to up swimming, these micro-organisms involved such as gyrotactic micro-organisms like algae tend to concentrate in the upper part of the fluid layer and causing unstable top heavy density stratification.⁴⁰ Nanofluid bioconvection phenomenon describes the unstructured pattern formation and density stratification due to synchronized interaction of the nanoparticles, thicker self propelled micro-organisms and buoyancy forces. Brownian motion and thermophoresis effect produces the motion of nanoparticles which is independent of the motion of motile micro-organisms. That is why the combined interface of bioconvection and nanofluids becomes significant for micro-fluidic appliances. Different types of micro-organisms, theoretical bio-convection models have been studied.^40–56 In this paper we will consider the three-dimensional bioconvection flow of Maxwell nanofluid over a bi-directional stretching surface. The combined effects of MHD and heat source/sink are also analyzed. The problem has been formulated in terms of governing equations of motion with appropriate boundary conditions, which will then be transformed into a set of ordinary differential equations by using similarity transformations. The analytical solution has been found using Homotopy Analysis Methods (HAM). The effects of involved physical parameters on the flow field have been analyzed with the help of graphs.

II. PROBLEM FORMULATION

We have considered the steady, incompressible three-dimensional flow of Maxwell nanofluid containing gyrotactic micro-organisms past a bidirectional stretching surface. A uniform magnetic field is applied normal to the flow. Due to the assumption of small Reynolds number, the induced magnetic field is ignored and applied magnetic field drags the magnetic field lines in the direction of the flow. The surface is maintained at constant temperature T_$w$, the density of micro-organisms n_$w$ and the nanoparticle volume fraction C_$w$, the ambient values of temperature, density of the micro-organisms and nanoparticle volume fraction are denoted by T_∞, n_∞ and C_∞ respectively while the reference values are by T₀, n₀ and C₀ respectively. The micro-organisms are taken into the nanofluid to stabilize the nanoparticles due to bioconvection. The effects of heat source/sink are also taken into account. The mechanical analog for the Maxwell fluid model and the geometry of the present analysis is shown in Fig. 1. The boundary layer form of the continuity, momentum, energy, nanoparticle concentration and density of microorganisms equations for three-dimensional Maxwell nanofluid are as follows:

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0,

(1)

\begin{matrix} u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} + λ_{1} (u^{2} \frac{\partial^{2} u}{\partial x^{2}} + v^{2} \frac{\partial^{2} u}{\partial y^{2}} + w^{2} \frac{\partial^{2} u}{\partial z^{2}} + 2 u v \frac{\partial^{2} u}{\partial x \partial y} + 2 v w \frac{\partial^{2} u}{\partial y \partial z} + 2 u w \frac{\partial^{2} u}{\partial x \partial z}) \\ = ν \frac{\partial^{2} u}{\partial z^{2}} - \frac{σ B_{0}^{2}}{ρ} (u + λ_{1} w \frac{\partial u}{\partial z}) + \frac{1}{ρ_{f}} ([ρ_{f} β (1 - C_{\infty}) (T - T_{\infty}) - (ρ_{p} - ρ_{f}) (C - C_{\infty}) \\ - γ (ρ_{m} - ρ_{f}) (n - n_{\infty}) (] g, \end{matrix}

(2)

\begin{matrix} u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} + λ_{1} (u^{2} \frac{\partial^{2} v}{\partial x^{2}} + v^{2} \frac{\partial^{2} v}{\partial y^{2}} + w u^{2} \frac{\partial^{2} v}{\partial x^{2}} + v^{2} \frac{\partial^{2} v}{\partial y^{2}} + w^{2} \frac{\partial^{2} v}{\partial z^{2}}) \\ (+ 2 u v \frac{\partial^{2} v}{\partial x \partial y} + 2 v w \frac{\partial^{2} v}{\partial y \partial z} + 2 u w \frac{\partial^{2} v}{\partial x \partial z}) = ν \frac{\partial^{2} u}{\partial z^{2}} - \frac{σ B_{0}^{2}}{ρ} (v + λ_{1} w \frac{\partial v}{\partial z}), \end{matrix}

(3)

u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} = α_{m} \frac{\partial^{2} T}{\partial z^{2}} + \frac{Q}{{(ρ c)}_{f}} (T - T_{\infty}) + τ [D_{B} (\frac{\partial C}{\partial z} \frac{\partial T}{\partial z}) + \frac{D_{T}}{T_{\infty}} {(\frac{\partial T}{\partial z})}^{2}],

(4)

u \frac{\partial C}{\partial x} + v \frac{\partial C}{\partial y} + w \frac{\partial C}{\partial z} = D_{B} \frac{\partial^{2} C}{\partial z^{2}} + \frac{D_{T}}{T_{\infty}} (\frac{\partial^{2} T}{\partial z^{2}}),

(5)

u \frac{\partial n}{\partial x} + v \frac{\partial n}{\partial y} + w \frac{\partial n}{\partial z} + \frac{b w_{0}}{Δ C} [\frac{\partial}{\partial z} (n \frac{\partial C}{\partial z})] = D_{m} \frac{\partial^{2} n}{\partial z^{2}},

(6)

and the boundary conditions are as below:

\begin{aligned} u & = a x = U_{w} (x), v = b y = V_{w} (y), w = 0, \\ T & = T_{w} = T_{0} + b_{1} x, C = C_{w} = C_{0} + d_{1} x, \\ n & = n_{w} = n_{0} + e_{1} x, when z = 0, \\ u & \to 0, v \to 0, C \to C_{\infty}, T \to T_{\infty}, n \to n_{\infty} when z \to \infty, \end{aligned}

(7)

in which the u, $v$ and $w$ are the components of velocity in the directions x, y and z respectively, λ₁ is the relaxation time, ν the kinematic viscosity, ρ_f the density of nanofluid, ρ_p the density of nanoparticles, ρ_m the density of micro-organisms particles, g the gravity, τ = (ρc)_p $/)$ (ρc)_f the heat capacity ratio, U_$w$ and V_$w$ are the stretching velocities, a, b > 0 are the stretching rate and b₁, d₁, e₁ are the dimensional constant. Let us introduce the following transformations:⁹

\begin{aligned} η & = \sqrt{\frac{a}{ν}} z, u = a x f^{'} (η), v = a y g^{'} (η), w = - \sqrt{a ν} [f (η) + g (η)], \\ θ (η) & = \frac{T - T_{\infty}}{T_{w} - T_{0}}, ϕ (η) = \frac{C - C_{\infty}}{C_{w} - C_{0}}, N (η) = \frac{n - n_{\infty}}{n_{w} - n_{0}} . \end{aligned}

(8)

The continuity Eq. (1) is satisfied identically and Eqs. (2) to (6) takes the following form given below:

\begin{matrix} f^{'''} + (1 + M β) (f + g) f^{''} - f^{' 2} + β [2 (f + g) f^{'} f^{''} - {(f + g)}^{2} f^{'''}] - M f^{'} + λ (θ - N r ϕ - R b N) = 0, \end{matrix}

(9)

g^{'''} + (1 + M β) (f + g) g^{''} - g^{' 2} + β [2 (f + g) g^{'} g^{''} - {(f + g)}^{2} g^{'''}] - M g^{'} = 0,

(10)

θ^{''} + P r (f + g) θ^{'} - P r f^{'} θ + P r S θ + N_{b} ϕ^{'} θ^{'} + N_{t} {(θ^{'})}^{2} = 0,

(11)

ϕ^{''} + L e (f + g) ϕ^{'} - L e f^{'} ϕ + \frac{N_{t}}{N_{b}} θ^{''} = 0,

(12)

N^{''} + L b (f + g) N^{'} - L b f^{'} N - P e [ϕ^{'} N^{'} + (N + Ω) ϕ^{''}] = 0,

(13)

and the boundary conditions Eq. (7) reduced to:

\begin{aligned} f (0) & = 0 = g (0), f^{'} (0) = 1, g^{'} (0) = α, θ (0) = 1 = ϕ (0) = N (0), \\ f^{'} (\infty) & = 0, g^{'} (\infty) = 0, θ (\infty) \to 0, ϕ (\infty) \to 0, N (\infty) \to 0, \end{aligned}

(14)

where prime is the differentiation with respect to η. Furthermore, λ, β, M, Nr, Rb, Nb, Pr, Nt, S, Le, Lb, Pe, Ω and α are mixed convection parameter, the Hartmann number, the Deborah number, the buoyancy ratio parameter, the bioconvection Rayeigh number, the Brownian motion parameter, the Prandtl number, the thermophoresis parameter, the heat source/sink parameter, the Lewis number, the bioconvection Lewis number, the bioconvection Peclet number, the concentration difference parameter for micro-organisms and stretching ratio parameter, are the non-dimensional parameters and are defined as:

\begin{aligned} β & = λ_{1} a, M = \sqrt{\frac{σ β_{0}^{2}}{ρ a}}, N r = \frac{(ρ_{p} - ρ_{f}) (C_{w} - C_{0})}{ρ_{f} (1 - C_{\infty}) (T_{w} - T_{0}) β}, \\ R b & = \frac{γ (ρ_{m} - ρ_{f}) (n_{w} - n_{0}) g}{ρ_{f} (1 - C_{\infty}) (T_{w} - T_{0}) β}, P r = \frac{ν}{α_{m}}, N_{b} = τ D_{B} \frac{(C_{w} - C_{0})}{α_{m}}, \\ N_{t} & = τ D_{T} \frac{(T_{w} - T_{0})}{α_{m} T_{\infty}}, S = \frac{Q}{{(ρ c)}_{f}}, L e = \frac{ν}{D_{B}}, L b = \frac{ν}{D_{m_{0}}}, P e = \frac{b w_{m_{0}}}{D_{m_{0}}}, \\ Ω & = \frac{n_{\infty}}{n_{w} - n_{0}}, α = \frac{b}{a} . \end{aligned}

(15)

The expressions for local density number, local Sherwood number and local Nusselt number of the motile micro-organisms are defined as:

N n = \frac{x q_{n}}{D_{n} (n_{w} - n_{0})}, S h = \frac{x q_{m}}{D_{B} (C_{w} - C_{0})}, N u = \frac{x q_{w}}{k (T_{w} - T_{0})},

(16)

where

q_{n} = {(- D_{m} \frac{\partial n}{\partial z})}_{z = 0}, q_{m} = {(- D_{B} \frac{\partial C}{\partial z})}_{z = 0}, q_{w} = {(- k \frac{\partial T}{\partial z})}_{z = 0},

(17)

The following are the expressions of local sherwood number, local Nusselt number and local motile micro-organism in dimensionless form:

S h R e_{x}^{- 1 / 2} = - ϕ^{'} (0), N u R e_{x}^{- 1 / 2} = - θ^{'} (0), N n R e_{x}^{- 1 / 2} = - N^{'} (0),

(18)

where $R e_{x} = \frac{x U_{w}}{ν}$ is the Reynolds number. The non-dimensional ordinary differential Eqs. (9)–(13) with the boundary conditions (14) are solved analytically by using HAM (homotopy analysis method).

FIG. 1.

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Geometry of the Problem.

III. HOMOTOPY ANALYSIS METHOD (HAM)

The auxiliary linear operators and initial guesses for the dimensionless momentum, energy, concentration of motile micro-organisms and concentration of nanoparticles equations are denoted by $L_{f}, L_{g}, L_{θ}, L_{N}, L_{ϕ} f_{0}, g_{0}, θ_{0}, N_{0}, ϕ_{0}$ and and are defined as:

\begin{aligned} f_{0} (η) & = 1 - \exp (- η), \\ g_{0} (η) & = α [1 - \exp (- η)], \\ θ_{0} (η) & = \exp (- η), \\ ϕ_{0} (η) & = \exp (- η), \\ N_{0} (η) & = \exp (- η), \end{aligned}

(19)

and

\begin{aligned} L_{f} & = \frac{d^{3} f}{d η^{3}} - \frac{d f}{d η}, \\ L_{g} & = \frac{d^{3} g}{d η^{3}} - \frac{d g}{d η}, \\ L_{θ} & = \frac{d^{2} θ}{d η^{2}} - θ, \\ L_{ϕ} & = \frac{d^{2} ϕ}{d η^{2}} - ϕ, \\ L_{N} & = \frac{d^{2} N}{d η^{2}} - N, \end{aligned}

(20)

with

\begin{aligned} L_{f} [A_{1} + A_{2} \exp η + A_{3} \exp - η] = 0, \\ L_{g} [A_{4} + A_{5} \exp η + A_{6} \exp - η] = 0, \\ L_{θ} [A_{7} \exp η + A_{8} \exp - η] = 0, \\ L_{ϕ} [A_{9} \exp η + A_{10} \exp - η] = 0, \\ L_{N} [A_{11} \exp η + A_{12} \exp - η] = 0, \end{aligned}

(21)

here A_i (i = 1 − 12) are the arbitrary constant. The deformation problems of order zero and m are:

A. Problem of zeroth-order

\begin{aligned} (1 - p) L_{f} [\hat{f} (η, p) - f_{0} (η)] = p h_{f} N_{f} [\hat{f} (η, p), ĝ (η, p), \hat{θ} (η, p), \hat{ϕ} (η, p), \hat{N} (η, p)], \\ (1 - p) L_{g} [ĝ (η, p) - g_{0} (η)] = p h_{g} N_{g} [ĝ (η, p), \hat{f} (η, p), \hat{θ} (η, p), \hat{ϕ} (η, p)], \\ (1 - p) L_{θ} [\hat{θ} (η, p) - θ_{0} (η)] = p h_{θ} N_{θ} [\hat{θ} (η, p), \hat{f} (η, p), ĝ (η, p), \hat{ϕ} (η, p)], \\ (1 - p) L_{ϕ} [\hat{ϕ} (η, p) - ϕ_{0} (η)] = p h_{ϕ} N_{ϕ} [\hat{ϕ} (η, p), \hat{f} (η, p), ĝ (η, p), \hat{θ} (η, p)], \\ (1 - p) L N [\hat{N} (η, p) - N_{0} (η)] = p h_{N} N_{N} [\hat{N} (η, p), \hat{f} (η, p), ĝ (η, p), \hat{ϕ} (η, p)], \end{aligned}

(22)

with

\begin{aligned} \hat{f} (0, p) & = 0, {\hat{f}}^{'} (0, p) = 1, {\hat{f}}^{'} (\infty, p) = 0, ĝ (0, p) = 0, ĝ^{'} (0, p) = α, ĝ^{'} (\infty, p) = 0, \\ \hat{θ} (0, p) & = 1, \hat{θ} (\infty, p) = 0, \hat{ϕ} (0, p) = 1, \hat{ϕ} (\infty, p) = 0, \hat{N} (0, p) = 1, \hat{N} (\infty, p) = 0, \end{aligned}

(23)

and

\begin{matrix} N_{f} [\hat{f} (η, p), ĝ (η, p), \hat{θ} (η, p), \hat{ϕ} (η, p), \hat{N} (η, p)] & = & \frac{\partial^{3} \hat{f} (η, p)}{\partial η^{3}} + (1 + M^{2} * β) (\hat{f} (η, p) + ĝ (η, p)) \frac{\partial^{2} \hat{f} (η, p)}{\partial η^{2}} \\ - {(\frac{\partial \hat{f} (η, p)}{\partial η})}^{2} + β [) 2 (\hat{f} (η, p) + ĝ (η, p)) \frac{\partial \hat{f} (η, p)}{\partial η} \frac{\partial^{2} \hat{f} (η, p)}{\partial η^{2}} \\ - {(\hat{f} (η, p) + ĝ (η, p))}^{2} \frac{\partial^{3} \hat{f} (η, p)}{\partial η^{3}}]) - M^{2} \frac{\partial \hat{f} (η, p)}{\partial η} \\ + λ [\hat{θ} (η, p) - N r \hat{ϕ} (η, p) - R b \hat{N} (η, p)], \end{matrix}

(24)

\begin{matrix} N_{g} [ĝ (η, p), \hat{f} (η, p), \hat{θ} (η, p), \hat{ϕ} (η, p)] & = & \frac{\partial^{3} ĝ (η, p)}{\partial η^{3}} + (1 + M^{2} * β) (\hat{f} (η, p) + ĝ (η, p)) \frac{\partial^{2} ĝ (η, p)}{\partial η^{2}} \\ - {(\frac{\partial ĝ (η, p)}{\partial η})}^{2} + β [) 2 (\hat{f} (η, p) + ĝ (η, p)) \frac{\partial ĝ (η, p)}{\partial η} \frac{\partial^{2} ĝ (η, p)}{\partial η^{2}} \\ - {(\hat{f} (η, p) + ĝ (η, p))}^{2} \frac{\partial^{3} ĝ (η, p)}{\partial η^{3}}]) - M^{2} \frac{\partial ĝ (η, p)}{\partial η}, \end{matrix}

(25)

\begin{matrix} N_{θ} [\hat{θ} (η, p), \hat{f} (η, p), ĝ (η, p), \hat{ϕ} (η, p)] & = & \frac{\partial^{2} \hat{θ} (η, p)}{\partial η^{2}} + P r (\hat{f} (η, p) + ĝ (η, p)) \frac{\partial \hat{θ} (η, p)}{\partial η} - P r \frac{\partial \hat{f} (η, p)}{\partial η} \hat{θ} (η, p) \\ + N b \frac{\partial \hat{ϕ} (η, p)}{\partial η} \frac{\partial \hat{θ} (η, p)}{\partial η} + N t {(\frac{\partial \hat{θ} (η, p)}{\partial η})}^{2} + P r * S \hat{θ} (η, p), \end{matrix}

(26)

\begin{matrix} N_{ϕ} [\hat{ϕ} (η, p), \hat{f} (η, p), ĝ (η, p), \hat{θ} (η, p)] & = & \frac{\partial^{2} \hat{ϕ} (η, p)}{\partial η^{2}} + L e (\hat{f} (η, p) + ĝ (η, p)) \frac{\partial \hat{ϕ} (η, p)}{\partial η} \\ - L e \frac{\partial \hat{f} (η, p)}{\partial η} \hat{ϕ} (η, p) + \frac{N t}{N b} \frac{\partial^{2} \hat{θ} (η, p)}{\partial η^{2}}, \end{matrix}

(27)

\begin{matrix} N_{N} [\hat{N} (η, p), \hat{f} (η, p), ĝ (η, p), \hat{ϕ} (η, p)] & = & \frac{\partial^{2} \hat{N} (η, p)}{\partial η^{2}} + L b (\hat{f} (η, p) + ĝ (η, p)) \frac{\partial \hat{N} (η, p)}{\partial η} - L b \frac{\partial \hat{f} (η, p)}{\partial η} \hat{N} (η, p) \\ - P e . [\frac{\partial \hat{N} (η, p)}{\partial η} \frac{\partial \hat{θ} (η, p)}{\partial η} + \hat{N} (η, p) \frac{\partial^{2} \hat{θ} (η, p)}{\partial η^{2}}] + P e Ω \frac{\partial^{2} \hat{ϕ} (η, p)}{\partial η^{2}}, \end{matrix}

(28)

here p ∈ [0, 1] expresses the embedding parameter and h_g, h_f, h_θ, h_ϕ and h_N are the non-zero auxiliary parameters.

B. mth-order problem

\begin{aligned} L_{f} [f_{m} (η) - χ f_{m - 1} (η)] & = h_{f} R_{m}^{f} (η), \\ L_{g} [g_{m} (η) - χ g_{m - 1} (η)] & = h_{g} R_{m}^{g} (η), \\ L_{θ} [θ_{m} (η) - χ θ_{m - 1} (η)] & = h_{θ} R_{m}^{θ} (η), \\ L_{ϕ} [ϕ_{m} (η) - χ ϕ_{m - 1} (η)] & = h_{ϕ} R_{m}^{ϕ} (η), \\ L_{N} [N_{m} (η) - χ N_{m - 1} (η)] & = h_{N} R_{m}^{N} (η), \end{aligned}

(29)

with

\begin{aligned} f_{m} (0) = 0, f_{m}^{'} (0) = 0, f_{m}^{'} (\infty) = 0, g_{m} (0) = 0, g_{m}^{'} (0) = 0, g_{m}^{'} (\infty) = 0, \\ θ_{m} (0) = 0, θ_{m} (\infty) = 0, ϕ_{m} (0) = 0, ϕ_{m} (\infty) = 0, N_{m} (0) = 0, N_{m} (\infty) = 0, \end{aligned}

(30)

and

\begin{matrix} R_{m}^{f} (η) & = & f_{m - 1}^{'''} + \sum_{k = 0}^{m - 1} ((1 + M^{β}) f_{m - 1 - k} f_{k}^{''} + (1 + M^{β}) g_{m - 1 - k} f_{k}^{''} - f_{m - 1 - k}^{'} f_{k}^{'}) - M^{2} f_{m - 1}^{'} \\ + β \sum_{k = 0}^{m - 1} \sum_{l = 0}^{k} [) 2 (f_{m - 1 - k} f_{k - l}^{'} f_{l}^{''} + g_{m - 1 - k} f_{k - l}^{'} f_{l}^{''}) - () f_{m - 1 - k} f_{k - l} f_{l}^{'''} + g_{m - 1 - k} g_{k - l} f_{l}^{'''} \\ + 2 f_{m - 1 - k} g_{k - l} f_{l}^{'''}))]) + λ (θ_{m - 1} - N r ϕ_{m - 1} - R b N_{m - 1}), \end{matrix}

(31)

\begin{matrix} R_{m}^{g} (η) & = & g_{m - 1}^{'''} + \sum_{k = 0}^{m - 1} (1 + M^{β}) (f_{m - 1 - k} g_{k}^{''} + (1 + M^{β}) g_{m - 1 - k} g_{k}^{''} - g_{m - 1 - k}^{'} g_{k}^{'}) - M^{2} g_{m - 1}^{'} \\ + β \sum_{k = 0}^{m - 1} \sum_{l = 0}^{k} [) 2 (f_{m - 1 - k} g_{k - l}^{'} g_{l}^{''} + g_{m - 1 - k} g_{k - l}^{'} g_{l}^{''}) - () f_{m - 1 - k} f_{k - l} g_{l}^{'''} + g_{m - 1 - k} g_{k - l} g_{l}^{'''} \\ + 2 f_{m - 1 - k} g_{k - l} g_{l}^{'''}))]), \end{matrix}

(32)

\begin{matrix} R_{m}^{θ} (η) = θ_{m - 1}^{''} + P r \sum_{k = 0}^{m - 1} (f_{m - 1 - k} θ_{k}^{'} + g_{m - 1 - k} θ_{k}^{'} - f_{m - 1 - k}^{'} θ_{k} + N b θ_{m - 1 - k}^{'} ϕ_{k}^{'} + N t θ_{m - 1 - k}^{'} θ_{k}^{'}) + P r S θ_{m - 1}, \end{matrix}

(33)

R_{m}^{ϕ} (η) = ϕ_{m - 1}^{''} + L e \sum_{k = 0}^{m - 1} (f_{m - 1 - k} ϕ_{k}^{'} + g_{m - 1 - k} ϕ_{k}^{'} - f_{m - 1 - k}^{'} ϕ_{k}) + \frac{N t}{N b} θ_{m - 1}^{''},

(34)

\begin{matrix} R_{m}^{N} (η) = N_{m - 1}^{''} + L b \sum_{k = 0}^{m - 1} (f_{m - 1 - k} N_{k}^{'} + g_{m - 1 - k} N_{k}^{'} - f_{m - 1 - k}^{'} N_{k}) - P e (N_{m - 1 - k}^{'} ϕ_{k}^{'} + N_{m - 1 - k} ϕ_{k}^{''}) + P e Ω ϕ_{m - 1}^{''}, \end{matrix}

(35)

if p = 0 and p = 1, then we can write

\begin{aligned} \hat{f} (η, 0) & = f_{0} (η), \hat{f} (η, 1) = f (η), \\ ĝ (η, 0) & = g_{0} (η), ĝ (η, 1) = g (η), \\ \hat{θ} (η, 0) & = θ_{0} (η), \hat{θ} (η, 1) = θ (η), \\ \hat{ϕ} (η, 0) & = ϕ_{0} (η), \hat{ϕ} (η, 1) = ϕ (η), \\ \hat{N} (η, 0) & = N_{0} (η), \hat{N} (η, 1) = N (η), \end{aligned}

(36)

when p varies from 0 to 1, then ĝ(η, p), $\hat{f} (η, p)$ ⁠, $\hat{ϕ} (η, p)$ ⁠, $\hat{θ} (η, p)$ and $\hat{N} (η, p)$ vary from the initial solution g₀(η), f₀(η), ϕ₀(η), θ₀(η), and N₀(η) to the final solutions g(η), f(η), ϕ(η), θ(η) and N(η) respectively. Using Taylor’s series we have

\begin{aligned} \hat{f} (η, p) & = f_{0} (η) + \sum_{m = 1}^{\infty} f_{m} (η) p^{m}, f_{m} (η) = \frac{1}{m!} \frac{\partial^{m} \hat{f} (η, p)}{\partial p^{m}} |_{p = 0}, \\ ĝ (η, p) & = g_{0} (η) + \sum_{m = 1}^{\infty} g_{m} (η) p^{m}, g_{m} (η) = \frac{1}{m!} \frac{\partial^{m} ĝ (η, p)}{\partial p^{m}} |_{p = 0}, \\ \hat{θ} (η, p) & = θ_{0} (η) + \sum_{m = 1}^{\infty} θ_{m} (η) p^{m}, θ_{m} (η) = \frac{1}{m!} \frac{\partial^{m} \hat{θ} (η, p)}{\partial p^{m}} |_{p = 0}, \\ \hat{ϕ} (η, p) & = ϕ_{0} (η) + \sum_{m = 1}^{\infty} ϕ_{m} (η) p^{m}, ϕ_{m} (η) = \frac{1}{m!} \frac{\partial^{m} \hat{ϕ} (η, p)}{\partial p^{m}} |_{p = 0}, \\ \hat{N} (η, p) & = N_{0} (η) + \sum_{m = 1}^{\infty} N_{m} (η) p^{m}, N_{m} (η) = \frac{1}{m!} \frac{\partial^{m} \hat{N} (η, p)}{\partial p^{m}} |_{p = 0} . \end{aligned}

(37)

The values of auxiliary parameter is chosen in such a way that the series (A19) converge at p = 1 i.e.

\begin{aligned} f (η) & = f_{0} (η) + \sum_{m = 1}^{\infty} f_{m} (η), \\ g (η) & = g_{0} (η) + \sum_{m = 1}^{\infty} g_{m} (η), \\ θ (η) & = θ_{0} (η) + \sum_{m = 1}^{\infty} θ_{m} (η), \\ ϕ (η) & = ϕ_{0} (η) + \sum_{m = 1}^{\infty} ϕ_{m} (η), \\ N (η) & = N_{0} (η) + \sum_{m = 1}^{\infty} N_{m} (η), . \end{aligned}

(38)

The general solutions (f_m,g_m,θ_m,ϕ_m,N_m) of Eqs. (9)–(13) in terms of special solutions (⁠ $f_{m}^{*}, g_{m}^{*}, θ_{m}^{*}, ϕ_{m}^{*}, N_{m}^{*}$ ⁠) are given by:

\begin{aligned} f_{m} (η) & = f_{m}^{*} (η) + A_{1} + A_{2} \exp (η) + A_{3} \exp (- η), \\ g_{m} (η) & = g_{m}^{*} (η) + A_{4} + A_{5} \exp (η) + A_{6} \exp (- η), \\ θ_{m} (η) & = θ_{m}^{*} (η) + A_{7} \exp (η) + A_{8} \exp (- η), \\ ϕ_{m} (η) & = ϕ_{m}^{*} (η) + A_{9} \exp (η) + A_{10} \exp (- η), \\ N_{m} (η) & = N_{m}^{*} (η) + A_{11} \exp (η) + A_{12} \exp (- η), \end{aligned}

(39)

where the constants A_i(i = 1 − 12) through the boundary conditions (14) are:

\begin{aligned} A_{2} = A_{5} = A_{7} = A_{9} = A_{11} = 0, A_{3} = \frac{\partial f_{m}^{*}}{\partial η} |_{η = 0}, A_{1} = - A_{3} - f_{m}^{*} (0), \\ A_{6} = \frac{\partial g_{m}^{*}}{\partial η} |_{η = 0}, A_{4} = - A_{6} - g_{m}^{*} (0), A_{8} = - θ^{*} (0), A_{10} = - ϕ^{*} (0), A_{12} = - N^{*} (0) . \end{aligned}

(40)

IV. RESULTS AND DISCUSSION

This section explores the results of three-dimensional flow of Maxwell nanofluid on a stretching sheet containing gyrotactic micro-organisms in the presence of magnetic field and heat source/sink effects. Homotopy analysis method ensures the convergence of derived series solution. The auxiliary parameter h has a important role in adjusting and controlling the region of convergence for the series solutions. The h− curves are plotted which are shown in Fig. 2, and found that the reasonable values of auxiliary parameters h_g, h_f, h_θ, h_ϕ and h_N are − 1.25 ≤ h_g ≤ −0.1, − 1.45 ≤ h_f ≤ −0.05, − 1.45 ≤ h_θ ≤ −0.05, − 1.3 ≤ h_ϕ ≤ −0.15 and − 1.5 ≤ h_N ≤ −0.2 respectively. The analytic results for the problem are presented graphically to study the behavior of some physical parameters on the velocities f′, g′, temperature θ, concentration of nanoparticles ϕ and concentration of micro-organisms N. Fig. 3a and Fig. 3b exhibit the effects of Deborah number β on the velocity component f′ and g′ respectively for fixed values of other parameters. The velocity profile decreases for increasing values of β. For Newtonian fluids β = 0. The viscosity of the fluid may increase by increasing the values of β. Nonzero values of β correspond to elastic effects which retards the flow and hence the boundary layer will be thinner which is noted. Fig. 4a and Fig. 4b reveals the effects of Hartmann number M on the velocity component f′ and g′ respectively for fixed values of other parameters. The velocity profile has decreasing behavior for increasing Hartmann number. Because by increasing the Hartmann number M, drag force increases as a result of which the velocity of the fluid reduces. In Fig. 5 the impact of (a) bioconvection Rayeigh number Rb and (b) buoyancy ratio parameter Nr on the velocity component f′ is shown. The graphical results shows when the values of Rb and Nr are increasing then a rapid decrease in the velocity profile is observed. When the values of Rb increases, the buoyancy force due to bioconvection causes the decrease in the velocity. The effects of λ on the velocity component f′ are shown in Fig. 6a. As λ is the ratio of buoyancy force to viscous force. We increase the values of buoyancy force by increasing the values of λ which decreases the velocity of the fluid. The influence of stretching ratio parameter α on the velocity component g′ is shown in Fig. 6b. It is observed that when the values of α are increased the velocity at the wall η = 0 and the related boundary layer thickness increases. The combined effects of the thermophoresis parameter Nt and Brownian motion parameter Nb on the temperature profile are shown in Fig. 7a. The temperature profile increases when we increase the values of Nt and Nb. Since increasing the magnitude of Brownian motion on the particles and thermal diffusivity of the nanoparticles will accelerates the temperature of the nanofluid. To analyze the concentration ϕ of the nanoparticals, the values of Brownian diffusion parameter and thermophoresis diffusion parameter are varied for fixed values of other parameters as shown in Fig. 7b, for larger values of Nt and Nb the concentration profile ϕ increases. Increase in the values of Nt and Nb refers to increase in number of nanoparticles. Thus higher concentration of nanoparticles corresponds to higher Brownian and thermophoresis diffusion which can also be seen from Fig. 7b. The impact of Prandtl number Pr on temperature profile is shown in Fig. 8a. The increasing values of Prandtl number decreases the temperature of the system. Fig. 8a clearly depicts that the larger value of Prandtl number Pr corresponds to the lower heat and thermal boundary layer thickness. The Prandtl number indeed plays an essential role in heat transfer; it controls the relative thickness of the thermal boundary layer. In order to characterize the mixed convection flow, it is also important to investigate the effect of the Prandtl number on the temperature profile. We observe that the temperature decreases with increasing Prandtl number. This may be explained by the fact that increasing the Prandtl number is equivalent to reducing the thermal diffusivity, which characterizes the rate at which heat is conducted. The effects of Lewis number Le on the concentration profile are shown in Fig. 8b. The Lewis number also decreases the concentration of the nanoparticles. The effects of heat source S > 0 and heat sink S < 0 parameter on the temperature profile are shown in Fig. 9a and Fig. 9b respectively. We observe that thermal boundary layer thickness and the temperature enhances in the case of heat source contrary to the results with the case of heat sink. The influence of dimensionless bioconvection Lewis number Lb on concentration of micro-organisms N is shown in Fig. 10a. The graphs illustrates a rapid decline in the profile, because the bioconvection lewis number opposes the motion of the fluid. For higher values of Lb the diffusivity of micro-organisms decreases and thus the motile density decays. The relation of microorganisms concentration difference parameter Ω and motile density of the microorganisms N is discussed in Fig. 10b. If Ω is increased then an increase is noticed in N. Increase in Ω enhances the concentration of micro-organisms in the ambient fluid and decrease in the density profile. The effects of Peclet number Pe on N is shown in Fig. 11, which shows the buoyancy parameter is found to be more pronounced for a fluid with greater values of bioconvection peclet number Pe. Increment in Pe causes decrease in the diffusivity of micro-organisms and so motile density of fluid decreases. Table I shows the comparison of the present results with the results available in the literature in the limiting case. It is found that our solution has good agreement with the limiting numerical solution. In Fig. 12a the local Nusselt number is plotted for varying thermophoretic diffusion, Brownian diffusion and Prandtl number showing the behavior of heat flux at the wall. It is observed that heat flux at the surface is a decreasing function of nanoparticle properties. In other words, heat flux at the surface reduces in the presence of nanoparticles and can be controlled by varying the quantity and quality of nanoparticles. In Fig. 12b, the local Sherwood number at the surface for varying thermophoretic diffusion, Brownian diffusion and Schmidth number is plotted. It is observed Sherwood number is an decreasing function of both the thermophoretic diffusion and Brownian diffusion parameter for all values of Schmidth number. In Fig. 12c, the local Density of motile micro-organisms at the surface for varying bioconvection Lewis number, micro-organisms concentration difference bioconvection Peclet number is plotted. It is observed that the density of motile micro-organisms is an increasing function of both the bioconvection Lewis number and bioconvection Peclet number for all values of micro-organisms concentration difference.

FIG. 2.

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h−curves for (a) f, (b) g, (c) θ, (d) ϕ and (e) N.

FIG. 3.

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Influence of β on (a) f′ and (b) g′.

FIG. 4.

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Influence of M on (a) f′ and (b) g′.

FIG. 5.

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Influence of (a) Rb (b) Nr on f′.

FIG. 6.

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Influence of (a) λ on f′ (b) α on g′.

FIG. 7.

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Influence of Nb and Nt on (a) θ (b) ϕ.

FIG. 8.

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Influence of (a) Pr and θ and (b) Le on ϕ.

FIG. 9.

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Influence of (a) S < 0 and (b) S > 0 on θ.

FIG. 10.

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Influence of (a) Lb (b) Ω on N.

FIG. 11.

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Influence of Pe on N.

TABLE I.

Comparison of − f^′′(0) for different values of β when α = M = λ = Nr = Rb = 0.

β	S. Mukhopadhyay¹⁹	Hayat et. al.¹¹	Present result
0.0	0.999963	0.999962	0.999960
0.2	1.051949	1.051948	1.051890
0.4	1.101851	1.101850	1.101859
0.6	1.150162	1.150160	1.150136
0.8	1.196693	1.196694	1.196692

β	S. Mukhopadhyay¹⁹	Hayat et. al.¹¹	Present result
0.0	0.999963	0.999962	0.999960
0.2	1.051949	1.051948	1.051890
0.4	1.101851	1.101850	1.101859
0.6	1.150162	1.150160	1.150136
0.8	1.196693	1.196694	1.196692

FIG. 12.

View large Download slide

(a) local Nusselt number (b) local Sherwood number and (c) local Density of motile micro-organisms for varying Brownina diffusion, thermophoretic diffusion parameters, Prandtl, Peclet, Lewis numbers and micro-organisms concentration difference parameters.

V. CONCLUDING REMARKS

In this manuscript the three-dimensional bioconvection flow of Maxwell nanofluid over a stretching surface with gyrotactic micro-organisms is examined. The effects of MHD and heat source/sink are also considered. Analytic technique is applied to investigate the problem and series solution is computed by using homotopy Analysis method. The velocity, temperature, concentration and micro-organisms profiles are plotted to analyze the effects of various physical parameters. We have observed that the velocity has decreasing behavior for increasing values of β, M, Rb, Nr and λ. The velocity component g′ and related boundary layer thickness increases due to stretching ratio parameter α. The combined effects of thermophoretic diffusion parameter Nt and Brownian diffusion parameter Nb increases the concentration and temperature profiles. The Prandtl number Pr and Le the Lewis number decreases the concentration and temperature profiles. In case of heat source the thermal boundary layer thickness and the temperature enhances compared to results of the heat sink case. The concentration of micro-organism N have decreasing behavior for increasing Lb and Pe while it has increasing behavior for increasing Omega. The local Nusselt number is decreasing while the local sherwood number is increasing function of nanoparticle properties. The density of motile micro-organisms is decreasing function of bioconvection Lewis number and bioconvection Peclet number.

ACKNOWLEDGMENTS

The authors declare that they have no conflict of interest.

NOMENCLATURE

x, y, z: dimensional coordinates (m)
u, $v$ ⁠, $w$: dimensional velocity components (ms⁻¹)
T: nanofluid temperature (K)
C: nanoparticles volume fraction (−)
n: number of motile micro-organisms (−)
T_∞: ambient fluid temperature (−)
C_∞: ambient nanoparticles volume fraction (−)
n_∞: ambient number of motile micro-organisms (−)
n_∞: ambient number of motile micro-organisms (−)
D_m: diffusivity of micro-organisms (m²s⁻¹)
U_$w$, V_$w$: stretching velocities (ms⁻¹)
C_$w$: surface nanoparticles volume fraction (−)
a, b: stretching rate (ms⁻¹)
C₀: reference nanoparticles volume fraction (−)
b₁, d₁, e₁: constants (−)
θ: dimensionless fluid temperature (−)
N: dimensionless number of motile micro-organisms (−)
λ: mixed convection parameter (−)
Rb: bioconvection Rayleigh number (−)
Nb: brownian motion parameter (−)
Sc: Schmidth number (−)
Pe: bioconvection Peclet number (−)
α: stretching ratio parameter (−)
λ: relaxation time (s)
ν: kinematic viscosity (m²s⁻¹)
ρ_f: density of nanofluid (kgm⁻³)
ρ_p: density of nanoparticle (kgm⁻³)
ρ_m: density of micro-organisms particle (kgm⁻³)
g: gravitational acceleration (ms⁻²)
τ: heat capacity ratio (−)
D_B: brownian diffusion coefficient (m²s⁻¹)
D_T: thermophoretic diffusion coefficient (m²s⁻¹)
α_m: thermal diffusivity of nanofluid (m²s⁻¹)
T_$w$: surface temperature (K)
n_$w$: surface number of motile micro-organisms (−)
T₀: reference temperature (K)
n₀: reference number of motile micro-organisms (−)
η: dimensionless variable (−)
ϕ: dimensionless nanoparticles volume fraction (−)
β: dimensionless Deborah numbers (−)
Nr: buoyancy ratio parameter (−)
Pr: Prandtl number (−)
Nt: thermophoresis motion parameter (−)
Lb: bioconvection Lewis number (−)
Ω: micro-organisms concentration difference parameter (−)

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Author(s)

Three-dimensional magnetohydrodynamic (MHD) flow of Maxwell nanofluid containing gyrotactic micro-organisms with heat source/sink

I. INTRODUCTION

II. PROBLEM FORMULATION

III. HOMOTOPY ANALYSIS METHOD (HAM)

A. Problem of zeroth-order

B. mth-order problem

IV. RESULTS AND DISCUSSION

V. CONCLUDING REMARKS

ACKNOWLEDGMENTS

NOMENCLATURE

REFERENCES

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Three-dimensional magnetohydrodynamic (MHD) flow of Maxwell nanofluid containing gyrotactic micro-organisms with heat source/sink

I. INTRODUCTION

II. PROBLEM FORMULATION

III. HOMOTOPY ANALYSIS METHOD (HAM)

A. Problem of zeroth-order

B. mth-order problem

IV. RESULTS AND DISCUSSION

V. CONCLUDING REMARKS

ACKNOWLEDGMENTS

NOMENCLATURE

REFERENCES

Citing articles via

Related Content

Resources

Explore

pubs.aip.org

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