J. Cent. South Univ. (2020) 27: 824-841

DOI: https://doi.org/10.1007/s11771-020-4334-x

Rotational nanofluids for oxytactic microorganisms with convective boundary conditions using bivariate spectral quasi-linearization method

Mlamuli DHLAMINI¹, Hiranmoy MONDAL², Precious SIBANDA¹, Sandile MOTSA^{1, 3}

1. School of Mathematics, Statistics and Computer Science, University of Kwa Zulu-Natal, Private Bag X01,Scottsvile, Pietermaritzburg-3209, South Africa;

2. Department of Mathematics, Brainware University, 398 Ramkrishnapur Road, Barasat, North 24 Parganas,Kolkata-700125, India;

3. Department of Mathematics, University of Swaziland, Private Bag 4, Kwaluseni, Swaziland

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: In this study, we considered the three-dimensional flow of a rotating viscous, incompressible electrically conducting nanofluid with oxytactic microorganisms and an insulated plate floating in the fluid. Three scenarios were considered in this study. The first case is when the fluid drags the plate, the second is when the plate drags the fluid and the third is when the plate floats on the fluid at the same velocity. The denser microorganisms create the bioconvection as they swim to the top following an oxygen gradient within the fluid. The velocity ratio parameter plays a key role in the dynamics for this flow. Varying the parameter below and above a critical value alters the dynamics of the flow. The Hartmann number, buoyancy ratio and radiation parameter have a reverse effect on the secondary velocity for values of the velocity ratio above and below the critical value. The Hall parameter on the other hand has a reverse effect on the primary velocity for values of velocity ratio above and below the critical value. The bioconvection Rayleigh number decreases the primary velocity. The secondary velocity increases with increasing values of the bioconvection Rayleigh number and is positive for velocity ratio values below 0.5. For values of the velocity ratio parameter above 0.5, the secondary velocity is negative for small values of bioconvection Rayleigh number and as the values increase, the flow is reversed and becomes positive.

Key words: bioconvection; oxytactic microorganisms; velocity ratio; rotational nanofluid; bivariate spectral quasilinearization method (BSQLM)

Cite this article as: Mlamuli DHLAMINI, Hiranmoy MONDAL, Precious SIBANDA, Sandile MOTSA. Rotational nanofluids for oxytactic microorganisms with convective boundary conditions using bivariate spectral quasi- linearization method [J]. Journal of Central South University, 2020, 27(3): 824-841. DOI: https://doi.org/10.1007/ s11771-020-4334-x.

1 Introduction

The study of boundary layer flow has received a lot of attention since the ground-breaking study by Lewig Prandtl in 1904. Many flows of interest occur within a boundary layer, and often in conjunction with heat and/or mass transport.

In many industrial processes such as the cooling of electronic devices, solar energy collectors, thermal insulation, underground nuclear disposal and chemical processes, heat transfer enhancement is of paramount importance [1]. Of late nanofluids have been identified as the best choice of fluids for heat transfer processes. The term nanofluid refers to a colloidal suspension of submicronic solid particles. A typical nanoparticle is a stable metal (Al, Cu, Ag, Au, Fe), oxide (Al₂O₃,CuO, TiO₂, SiO₂), carbide (SiC), nitrate (AlN, SiN), or non-metal (graphite) with sizes ranging from 1 to 100 nm. The base fluid normally is a conductive fluid, such as water, ethylene glycol, oil, polymer or bio-fluids [1]. The flow of a rotating fluid is encountered in geothermal flows [2]. It is also the reason for the secular variations in geomagnetic fields [3]. In rotating fluids, the Coriolis force is more prominent than the viscous and inertial forces[2]. The viscous forces are balanced by the Coriolis forces instead of the inertial forces. According to HIDE [4, 5], in the study of geophysical flows a key parameter is the Rossby number. The influence of the rotation is said to be weak if the Rossby number is significantly less than unity, and dominant if the Rossby number is significantly larger than unity. The study of rotational viscous fluids sheds some light on the body forces that act on particles immersed in the fluid. This area of study continues to receive considerable attention because of a variety of industrial applications [6]. The motion of rigid particles or drops in rotating fluids is encountered in a number of industrial processes such as in the manufacturing of hallow shells, separation of minerals, extraction of proteins and in waste water treatment [7-11].

Nanofluid bioconvection gives rise to spontaneous pattern formation and density stratification. This is a phenomenon that occurs when instability is induced by the interaction of the swimming of denser self-propelled microorganisms, nanoparticles and buoyancy forces [1, 12]. Bioconvection arises in biological systems, biotechnology such as the mass transport enhancement in microscale mixing and the synthesis of biosensors [13]. Some of microorganisms that normally give rise to bioconvection include gravitaxis, gyrotaxis and oxytaxis. This study considers the movement of oxytactic microorganisms. These are bacteria that consume oxygen such as Bacillussubtilis. They swim up the oxygen concentration gradient. KUZNETSOV [12] considered a novel nanofluid with oxytactic microorganisms. The bacteria are oxygen consumers and swim towards the top region when this is exposed to the elements. LEE et al [14] investigated falling bacteria plumes caused by bioconvection. Their study was an extension of work in Ref. [15]. In both studies the falling plumes are attributed to the instability of bacteria-rich boundary layer close to the surface that is denser than the rest of the fluid. The falling plumes transport bacteria and oxygen from the upper boundary layer to the lower region of the chamber, which is depleted of both bacteria and oxygen. The use of microorganisms in delivering oxygen is encountered in industrial processes such as aerobic fermentation [16]. Other uses of microorganisms in industrial processes are encountered in bio-reactors [17].

Bioconvection has been studied by several researchers [18-25]. The current study incorporates a rotational electrically conducting nanofluid with an applied magnetic field to the study of bioconvection induced by oxytactic microorganisms. We also consider the impact of varying the velocity of the plate relative to the fluid velocity and the influence of fluid and flow parameters.

2 Mathematical analysis

The physical model and coordinate system is given in Figure 1, where x, y and z are Cartesian coordinates and u, v and w are the velocity components in the corresponding directions. We consider an insulated flat plate which coincides with the plane z=0. The plate moves with a velocity U₁ in the x direction in a viscous, incompressible, electrically conducting nanofluid that is rotating with constant angular velocity Ω about the x-axis containing oxytaxis microorganisms. There is also a uniform free stream velocity U₂ parallel to the x-axis. A magnetic field B₀ is applied along the z-axis. The temperature and concentration of nanoparticles and free stream microorganisms are kept at constant T_w, C_w and n_w while in the free stream these are assumed to be T_∞, C_∞ and n_∞ respectively. The effects of the Coriolis force and Hall currents give rise to a force in the y-direction, which induces cross flow [2]. The set of equations modeling the flow in a rotating frame of reference with Maxwells electromagnetic equations are given by

                              (1)

                           (2)

                      (3)

             (4)

             (5)

         (6)

Figure 1 Flow configuration and coordinate system

The boundary conditions for Eqs. (1)-(6) are given in form:

                       (7)

where ρ is the density of the base fluid; υ is the kinematic viscosity; σ is the electrical conductivity of the fluid; N is the Hall parameter; g is the gravitational acceleration; β is the volumetric coefficient of expansion for the fluid; ρ_p is the density of the nanoparticles; ρ_f is the density of the nanofluid; γ is the average volume of the microorganisms; ρ_m is the density of microorganisms; α_m is the thermal diffusivity coefficient for microorganisms; τ is the ratio of heat capacitance of nanoparticles to the base fluid; D_B is the Brownian diffusion coefficient; D_T is the thermophoretic diffusion coefficient; σ* is Stephan Boltzman constant; C_p is the specific heat capacity of the fluid at constant pressure; b is the chemotaxis constant; W_c is the maximum cell swimming speed (bW_c is assumed to be constant); D_n is the microorganism diffusivity; k_f is the thermal conductivity of the solid and h_f is the convective heat transfer coefficient.

Far from the surface, the pressure gradients in the x and y directions, -ρ^-1ρ_x and -ρ^-1ρ_y, must balance the Lorentz and Coriolis forces [2] and are given by the relations:

                (8)

In this study we excluded the effect of viscous dissipation and Joule heating. Viscous dissipation is the heat generated in a fluid due to the frictional forces between fluid layers as they slide over each other. Joule/Ohmic heating is the heat generated due to resistance of a current as it passes through a medium. The effect of both viscous and Joule heating has been studied extensively and is well documented in Refs. [26, 27]. Since we assumed that we have a heated plate, we therefore assume that it is the major contributor of heat and all other heat sources are negligible.

3 Transformation of equations

The following variables are used to transform the system given by Eqs. (1)-(6),

                        (9)

The velocity componentsand are given as

          (10)

On using Eqs. (8)-(9), Eqs. (2) - (6) and the boundary conditions (7) are transformed into the following boundary value problem:

  (11)

    (12)

               (13)

 (14)

              (15)

where the prime denotes differentiation with respect to η. The corresponding boundary conditions (7) are transformed as

                    (16)

                   (17)

     (18)

          (19)

                   (20)

The parameters in the above differential equations and associated boundary conditions are defined as

                   (21)

where M is the modified Hartmann number; Gr is the Grashof number; Nr is the buoyancy ratio; Rb is the bioconvection Rayleigh number; Pr is the Prandtl number; S is the velocity ratio; Ra is the radiation parameter; Nb is the Brownian motion parameter; Nt is thermophoresis parameter; Pb is the bioconvection Peclet number; τ₀ is the constant microorganisms concentration difference parameter; Sc is the Schmidt number; Sb is the bioconvection Schmidt number; Re_x is the Reynolds number; Bi is the Biot number.

4 Momentum, heat and mass transfer coefficients

The parameters that are of interest are the skin friction coefficient, C_f, which measures the shear stress on the surface, the Nusselt number, Nu, which is the ratio of convective to conductive heat transfer across (normal to) the boundary, the Sherwood number, Sh, which is the ratio of the convective mass transfer to the rate of diffusive mass transport and the local density number of the motile microorganisms, Nn, which is the ratio of the convective microorganism transfer to the rate of diffusive microorganism transport.

The local skin friction coefficients in the x and y directions are given as

             (22)

The local Nusselt number, local Sherwood number and local density number of the motile microorganisms are given by

         (23)

where

.         

5 Numerical solution using bivariate spectral quasi-linearization method

The two-variable nonlinear boundary value problem given by Eqs. (11)-(15) together with the boundary conditions Eqs. (16)-(20) was solved numerically using the bivariate spectral quasilinearization method (BSQLM). The choice of using a spectral-based method is that they require less grid points yet giving accurate results and take less computational time compared to other methods [28, 29]. The flow domain and time interval given by η[0, L_x] and ξ[0, L_t] are transformed to x[-1, 1] and t[-1, 1] using the linear transformations ηL_x(x+1)/2 and ξL_t(x+1)/2 respectively. The solution is approximated using Lagrange interpolation polynomial of the form:

             (24)

where u(x, t) is interpolated at selected grid points both in the x and t directions. The grid points are defined as

   (25)

The functions L_i(x) are the Lagrange cardinal polynomials given by

                   (26)

where

                 (27)

L_j(t) is also defined in a similar manner. We begin by defining the functions F, G, Θ, Φ and X for Eqs. (10)-(14) as

(28)

    (29)

          (30)

 (31)

               (32)

We construct the solution using the iterative process given by Eqs. (28)-(32):

                     (33)

            (34)

      (35)

     (36)

 (37)

subjected to the boundary conditions

                  (38)

              (39)

                     (40)

                    (41)

              (42)

The coefficients in Eqs. (33)- (37) are given as

;

              (43)

The initial guess functions are selected in such a way that they satisfy the boundary conditions at ξ=0. These are chosen as

                  (44)

For a detailed explanation on how the resulting matrices and corresponding boundary conditions are implemented on a MATLAB code, one can refer to work by MOTSA et al [28, 29].

Table 1 gives parameter values that were used in similar studies.

We validate the accuracy and convergence of our numerical scheme by performing an error analysis. We plot the norm of residual errors in the variables against the number of iterations. The norm of residual errors for all the variables was less than 10^-10 after 8 iterations. This shows that the BSQLM is an appropriate scheme for solving the resulting boundary value problem. Figure 2 shows the norm of residual errors for different variables at different number of iterations.

Table 1 Parameter values and their source

6 Discussion and results

In this section we discuss the results of our findings. We investigate the impact that certain key parameters have on the physical, thermal and concentration properties of the flow. We analyze the variation of the local skin friction coefficient, Nusselt number, Sherwood number and density of motile microorganisms in Table 2.

Figure 2 Norm of residual at different iterations

We investigate the influence of the velocity ratio parameter S in Figure 3. It gives the relative velocity of the plate to the velocity of the fluid. A value of zero corresponds to the case of a fluid flowing past a stationary plate. A value of 1 corresponds to the case of a plate moving in a stationary fluid. 0

Table 2 Variation of local skin friction coefficient

Figure 3 Influence of velocity ratio:

We analyze the effect of the Hartmann number on the fluid’s velocity in Figure 4. The Hartmann number is a dimensionless parameter defined as the ratio of electromagnetic force to the viscous force. An increase in the Hartmann number for the case when the plate is moving relatively faster than the fluid (S>0.5) results in a decrease in the fluid’s primary velocity in the boundary layer. This is due to an increase in the Lorentz force that tends to oppose the flow resulting in a frictional drag [30]. An opposite phenomenon is observed for the case when S<0.5, that is, an increase in the Hartmann number results in a velocity increase for the primary flow close to the surface. The Hartmann number causes decrease in the secondary velocity; however, the flow is reversed for values above and below the critical value of 0.5.

Figure 4 Effect of Hartmann number on fluid’s velocity:

The influence of the Hartmann number on other parameters is depicted in Figure 5. Our results for the case when S>0.5 are consistent with results found in Ref. [1] that it causes an increase in the temperature chemical species concentration and motile microorganisms. A reverse effect is observed for values of S<0.5.

The buoyancy ratio parameter Nr is analyzed in Figure 6. The buoyancy ratio parameter measures the influence of free and forced convection [31]. In free convection the flow is due to a temperature gradient whereas in forced convection an external device like a pump is used to generate or maintain the flow. It is noticed that for S>0.5 an increase in the buoyancy ratio parameter results in a decrease in both the primary and secondary velocities. An increase in the temperature, chemical species and microorganism concentration is also associated with increasing the buoyancy ratio parameter, results consistent with results in Ref. [32]. In fact, the increase in the chemical species volume fraction is attributed to decrease in the velocity of the fluid [20]. All other parameters behave in a similar manner for values of S<0.5 except the secondary velocity that is in the reverse direction.

The impact of Hall parameter N is assessed in Figure 7. The Hall effect is the production of a voltage difference across an electrical conductor to an applied magnetic field. The Hall parameter is defined to be the ratio of induced electrical field to the applied magnetic field. An increase in the Hall parameter results in an increase in the velocity profile [33, 34]. This is attributed to the fact that the magnetic field on the velocity damping reduces as the Hall parameter is increased. However, there are some studies that reported results are contrary to this [2, 35]. Our results show an increase in the velocity profile for the secondary flow and the primary velocity for the case when S<0.5. When S>0.5, the primary velocity decreases with increasing values of the Hall parameter. The phenomenon responsible for the observed trend is not fully understood.

The impact of thermal radiation is analyzed in Figure 8. The primary velocity increases with increasing values of the thermal radiation parameter for all values of S. This is because an increase in the thermal radiation parameter leads to conduction dominating absorption radiation, resulting in an increase in buoyancy force [33]. Also, an increase in thermal radiation parameter leads to an increase in the fluid’s temperature, causing a decrease of the fluid’s viscosity making it easier to flow. The secondary velocity increases with increasing values of thermal radiation for values of S>0.5. The result is consistent with result in Ref. [33]. However, for S<0.5, our results show a decrease in the secondary velocity for increasing values of radiation parameter in the reverse direction. Temperature, concentration of chemical species and microorganisms are plotted for the case when S=0.75. There are no differences in the graph of these variables for different values of S. From Figure 8 we notice that increasing the radiation parameter results in an increase in the temperature of the fluid. This is because increasing the radiation parameter increases the Rossland diffusion [36], resulting in an increase in the fluid’s temperature. Similar results are reported in Refs. [33, 37]. An increase in the radiation parameter results in an initial decrease of the chemical species. Far from the surface, the concentration of the chemical species increases with increasing values of thermal radiation. REDDY et al [37] also reported similar results; however, no physical explanation was given about the phenomenon of primary velocity for S=0.25. We postulate that this phenomenon is associated with primary velocity, secondary velocity for S=0.25 in the primary velocity profile for the thermal radiation parameter. An increase in the primary velocity in the boundary layer results in the chemical species being carried outside the boundary layer and deposited outside the boundary layer leading to the observed phenomenon. The concentration of the microorganisms decreases with increasing values of thermal radiation parameter.

Figure 5 Effect of Hartmann number:

Figure 6 Effect of buoyancy ratio:

Figure 7 Effect of Hall parameter:

We analyze the influence of the bioconvective Rayleigh number Rb in Figure 9. The Rayleigh number is defined as a dimensionless number that is associated with buoyancy-driven flow. Below a certain critical value, heat transfer is primarily due to conduction and above that critical value it is due to convection. The primary velocity decreases with increasing values of Rayleigh number for all values of S. Temperature, concentration of chemical species and microorganisms increase with increasing value of Rayleigh number. Similar results were obtained in Ref. [19]. Secondary velocity on the other hand increases with increasing values of Rayleigh number and is positive for S<0.5. For values of S>0.5, the secondary velocity decreases for small values of Rayleigh number and the flow is negative. For large values of Rayleigh number, the flow becomes positive and increases with increasing values of Rayleigh number.

In Figure 10, we analyze the impact of Brownian motion parameter Nb and the thermophoretic parameter Nt on the concentration of chemical species. Brownian motion is the random ‘indeterminate’ movement of microscopic particles in a fluid due to bombardment by molecules of surrounding medium or fluid. Our results show that an increase in the Brownian motion parameter leads to a decrease in the chemical species concentration in the boundary layer. Similar results for the parameter are reported in Refs. [1, 38, 39]. This is attributed to the fact that increasing the Brownian motion parameter leads to more solute being ‘pushed’ out of the boundary layer which is relatively ‘thin’ compared to the rest of the fluid. Thermophoresis, sometimes known as thermo-migration or thermo-diffusion is a phenomenon where microscopic molecules in a fluid exhibit a response to a force that is due to a temperature gradient. The force acts in the direction of a hot region to a cold region. An increase in the thermophoresis parameter leads to an increase in the concentration of chemical species. The result is in agreement with result in Ref. [38] that the fast flow from the stretching sheet carries the nanoparticles leading to an increase in the mass volume fraction boundary layer thickness. Similar conclusions about the parameter were also made in Refs. [1, 40].

Figure 8 Effect of thermal radiation:

Figure 9 Effect of bioconvective Rayleigh number:

Figure 10 Effect of Brownian motion parameter Nb (a) and thermophoretic parameter Nt (b) on chemical species concentration

In Figure 11, we review the effect of the Schmdit number Sc on chemical species concentration, bioconvection Schmdit number and bioconvection Peclet number on microorganisms. An increase in the Schmdit number leads to a decrease in the chemical species concentration in the boundary layer. Similar results were obtained in Ref. [41]. Similarly, an increase in the bioconvection Peclet number causes a decrease in the microorganism’s concentration in the boundary layer. An increase in the bioconvection Peclet number increases the microorganism’s concentration in the boundary layer. The bioconvection Peclet number is defined as the ratio of swimming microorganisms to the speed of the bulk fluid motion. The bioconvection Peclet number is less than unity if the fluid speed is faster than the microorganisms and greater than unity otherwise. It is equal to unity if the fluid speed equals the speed of microorganisms. Increase in the bioconvection Peclet number increases the motile microorganisms in the boundary layer [42, 43]. This is so because increasing the Peclet number increases the advective transport rate compared to the rate of diffusion. This attracts microorganisms near the boundary and as a result, the microorganisms flux increases. An increase in the bioconvection Schmdit number leads to a decrease in the concentration of microorganisms in the boundary layer. A result similar to ones is reported in Ref. [43].

Figure 11 Effect of Schmdit number on chemical species concentration (a), bioconvective Schmidt number (b) and bioconvective Peclet number (c) on microorganism’s concentration

7 Conclusions

In this article we formulated and analyzed a model for a three-dimensional viscous incompressible rotational flow of an electrically conducting fluid with oxytactic microorganisms. The velocity ratio parameter is identified as a key parameter in this study. It has the potential of changing the dynamics of the flow. For values of this parameter above or below a critical value of 0.5 the flow reverses. In some cases, it alters the effect of other parameters, such as the Hartmann number on primary velocity. It is not clear at this point why/how these changes occur. Some of the key findings are:

The thermal radiation parameter increases the primary velocity while decreasing the secondary velocity for S<0.5. For S>0.5, there is a reverse effect for both the primary and secondary velocities.

Increasing the velocity ratio increases the temperature, concentration of chemical species and microorganisms.

Increasing the bioconvection Peclet number increases the microorganism’s concentration in the boundary layer.

Increasing the Schmidt number and bioconvection Schmidt number leads to a decrease in the chemical species concentration and microorganisms, respectively.

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(Edited by YANG Hua)

中文导读

二元谱拟线性化含嗜氧微生物旋转纳米流体的对流边界条件

摘要：在本研究中，考虑了旋转的含有嗜氧微生物的黏性、不可压缩的导电纳米流体和漂浮在流体中的绝缘板的三维流动动力学。研究考虑了三种情况：一种是流体拖动板，第二种是板拖动流体，第三种是板以与流体相同的速度漂浮在流体上。浓度较高的嗜氧微生物因流体中氧梯度的作用而上浮到顶部时会产生生物对流。速度比在这种流动的动力学中起关键作用。低于临界值或高于临界值时，改变参数会改变流动的动力学。改变速度比使其低于或高于临界值会改变流体流动的动力学，Hartmann数、浮力比和辐射参数对二次速度有反作用。另一方面，Hall参数对低于或高于临界值的速度比的初始速度有反作用。生物截面的Rayleigh数减慢了初始速度。当速度比低于0.5时，二次速度随着生物截面Rayleigh数的增大而加快。当速度比高于 0.5时，对于较小的生物截面Rayleigh数，二次速度起反作用，但随着数值的增加，对流动的作用发生被逆转，起正作用。

关键词：生物对流；嗜氧微生物；速度比；旋转纳米流体；二元光谱拟线性化

Received date: 2018-11-05; Accepted date: 2019-04-26

Corresponding author: Hiranmoy MONDAL, PhD, Associate Professor; Tel: +91-9434633158; E-mail: hiranmoymondal@yahoo.co.in; ORCID: 0000- 0002-9153-300X