Effect of microorganisms on the stability analysis in magnetic nanofluids

A study of onset of convection of a new type of fluid, a fluid that contains both magnetic nanoparticles and microorganisms, is presented in this paper. We consider an infinite horizontal layer of water based magnetic nanofluids (MNF) containing gyrotactic microorganisms, heated from below, in the presence of uniform vertical magnetic field. Here we utilize the Chebyshev pseudospectral method to solve the eigen value problem in gravitational environment. The effect of various important parameters which are conducive for the stability of the system is shown graphically.


Introduction
The process when the spontaneous pattern is formed in the suspensions of swimming microorganisms is known as bioconvection [1]. Wager was the first to carry out the detailed observations of bio-convection at the beginning of 20th century. [2]. Then after a long gap of almost 50 years, the subject was taken up again by Platt [3], who apparently coined the term "bioconvection". Following common features are observed during the pattern formation due to the upswimming of microorganisms: they are little denser than the liquid they swim in, swimming is directed upwards on an average. Moreover, the patterns disappear when the microorganisms stop swimming. Due to the upswimming of microorganisms, the upper surface of the suspensions becomes too dense to be stable. If the density gradient is sufficiently large, overturning instability sets in that causes the cells to descend along thin lines known as plumes. The underlying bioconvection process is similar to that of Rayleigh-Bénard convection but, unlike the latter, it is driven solely by the swimming of microorganisms [4]. Kuznetsov and Avramenko [5] observed that the process of bioconvection intensifies the mixing process and helps in the settling of small particles. A numerical investigation had been conducted by Kuznetsov and Geng [6] to study the effect of bioconvection on mixing of small solid particles. The authors found that the small solid particles of ideal size "decelerate" the bioconvection process. Similarly very large particles having negligible diffusivity and very heavy particles with substantial settling velocity do not affect the bioconvection process. Nield and Kuznetsov [7] performed linear stability analysis for the onset of bioconvection in a thermally conducting fluid. They found that as the gyrotactic number and Péclet number increase the critical Rayleigh number decreases. Kuznetsov [8] reported that the bottom heavy nanoparticle distribution delays the onset of convection while the top heavy arrange-  [9] incorporated the gyrotactic effects in their analysis. Kuznetsov [10] considered horizontal porous layer of finite depth which consists both nanoparticles and gyrotactic microorganisms to study the onset of instability. The author demonstrated that the bioconvection Péclet number Q is the key factor in determining the role of microorganisms on the convective stability. Other experimental investigations were carried out by Loeffer and Mefferd [11], Wille and Ehret [12], Nultsch and Hoff [13], Kessler [14] and, Bees and Hill [15]. Some more aspects related to bioconvection are discussed in the references [16,17,18,19,20].
MNF are very useful in microfluidic devices but the problem of proper mixing is usually a matter of concern [21]. There are active mixers available for this purpose but most of them are very costly. The process of bio-convection helps in enhancing mixing and mass transfer in micro-devices. It also contributes in enhancing the stability of the nanofluids. Thus inclusion of microorganisms in MNF seems to be a way out to this problem. In order to use this new type of fluid in microsystems and microchannels, the behavior of such fluids must be comprehended at the primary level. In this paper, therefore, an attempt has been made in this direction. The onset of convection of a fluid which contains nanoparticles and microorganisms has been examined.

Formulation
Incompressible infinite horizontal layer of magnetic nanofluids containing gyrotactic microorganisms is considered. Two cases are considered: (i) both the boundaries are rigid (ii) upper boundary is stress-free while lower boundary is rigid.

Heated from below
W hen w = 0 and j.k = 0 In dimensionless form, equations (1)-(7) are: Here following are the non-dimensional parameters: where 3. The quiescent state solution Here all are functions of z only.

Linear analysis
To study the linear stability of the quiescent state, we now take infinitesimally small perturbations as This gives following set of linearized perturbation equations (dropping primes) where Here we assume Thus we have The eigen value problem: Equations (32)-(36) with boundary conditions (37) is solved by Chebyshev pseudospectral method [23].

Results and discussion
The numerical results are presented here for a 1 mm thick layer of water based MNF containing the gyrotactic microorganisms. Figure 1  gyrotactic microorganisms. The effect of Rn is to destabilize the system while α L stabilizes the system.
In Figure 2 (a), the plot depicts decrease in Ra c on increasing Rb, thus exhibiting the destabilizing tendency of Rb on the system. Increasing Rb means increasing the average concentration of microorganisms in MNF. Due to the up swimming tendency of microorganisms they produce top-heavy unstable density stratification which causes bio convection to occur. As the value of Rb increases, more and more microorganisms accumulate near the top surface. Thus an increase in the value of Rb causes destabilizing effect on the system. Figure 2 (b) displays the instability boundary in the (Ra c , Rn)-plane. The destabilizing effect of Rn is witnessed on the system because for any fixed value of Rb, the value of Ra c keeps on decreasing as Rn increases. It is also established that variation in the values of the Rn does not affect the propensity of Rb on the qualitative grounds.
Neutral curves are displayed for different values of the bioconvection Péclet number Q in Figure 3 (a). It is evident from the figure that an increase in the value of Q produces destabilizing effect on the system by reducing Ra c . Figure 3    To understand the effect of the gyrotaxis number G at the onset of convection, neutral curves are displayed for its different values in Figure 4 (a). The gyrotaxis number G is the deviation of the cell swimming direction from strictly vertical which causes bioconvection to occur and thus higher values of G makes the system unstable [19]. In Figure 4 (a) exactly same trend is observed: Ra c decreases with an increase in the value of the gyrotaxis number G depicting its destabilizing effect on the system. Figure 4 (b) shows Ra c as a function of Rb for different values of G. It is noted that Ra c decreases with an increase in the value of Rb for any particular value of G but it decreases sharply with larger values of G. Qualitatively, no difference has been found in the behavior of G with the variation in the Rb.   It is established from Figure 5 that Lb delays the convection process while Rb accelerates the same. It can be interpreted that qualitative behavior of Lb is not affected by the variation in the value of the Rb. In Table 1 three representative values of the volumetric fraction ∆φ of nanoparticles have been chosen to display the results on both type of boundary conditions.

Conclusions
Linear analysis theory has been applied to study the stabilizing and destabilizing effect of various parameters in a thin layer of water based MNF layer containing the gyrotactic microorganisms. Rigidrigid and rigid-free boundaries are considered. Combined effects of nanoparticles and microorganisms have been investigated in the gravity environment. In rigid-rigid boundary condition, destabilizing effect of microorganisms has been observed. The parameters, α L and Lb stabilize the system while Rn, Rb, Q and G advance the onset of convection.  Table 1. α L = 2, d = 0.001, ∆φ = 0.001, N A = 10, Le = 5000, Rb = 10, Q = 1.