Non – Linear Chemotactic Hydromagnetic Bioconvection

Non – Linear Chemotactic Hydromagnetic Bioconvection

Prof. Dr. P. K. Srimani Mrs. Radha.D

R&D,Director (DSI) Assistant Professor, Dept of Mathematics,

Bangalore BMS College of Engineering, Bangalore

Abstract

The effect of external magnetic field on the chemotactic bacterial bioconvection by considering a Continuum model is investigated. Chemotaxis causes cells to swim out of the plume because the high concentration of the cells constituting the plumes leads to a lower concentration of oxygen in the surrounding fluid. Further worlds major portion consists of bio-mass; therefore it is of immense interest and at most importance to study bioconvection under different types of constraints. A similarity solution is found for the plume in which the cell flux and the volume flux could be matched to those in the boundary layer and also outside the suspension regions. Axisymmetric plumes is formed by applying two scales one with respect to the radial co-ordinate and the other with respect to the similarity variable. The effects of magnetic field are remarkable and encouraging and the computed results are in excellent agreement with those of hydrodynamic case in the limiting case.

1. Introduction

   The spontaneous formation of patterns in suspensions of swimming microorganisms due to their tactic nature viz. oxytactic, gyrotactic, chemotactic etc., is termed as bioconvection. The microorganisms exhibiting bioconvection have the following key features: (i) They are denser than water (ii) They swim upwards due to their tactic nature. This leads to an unstable situation in the system and thus an overturning instability develops leading to pattern formation [1][2][3]. Also Magnetic field has a strong influence on the system in many real time situations. Experiments on bioconvection containing suspensions of bacteria (Bacillus Subtilis) have revealed the formation of Falling plumes (Figure 1.) when the system becomes unstable.

   Cell rich upper boundary layer

   Outer region

   Plume

   Figure 1. Formation of Bioconvection Plume

   Some literatures pertaining to bioconvection in deep chambers are [4][5]. The study of such a phenomenon has a variety of applications in biological and physiological problems. Further, chemotaxis and oxygen consumption are important in setting up the basic state and soon after, the resulting plumes are entirely buoyancy driven and the cells are merely advected. In such cases, the velocity would vary across the plume [6][7] . The present work investigated the nonlinear Hydromagnetic bioconvection in order to study the effect of magnetic field on the formation of falling plumes (Axisymmetric) where the oxygen consumption and chemotaxis are important. The model constituted the quasi steady situation in which an upper boundary layer containing a high concentration of bacteria feeds a falling plume of cell-rich fluid. The suspension was divided into three separate regions as shown in Figure 1, a cell-rich upper boundary layer of known thickness , a falling plume of unknown width

   which also contained a high concentration of

   bacteria and the fluid outside the plume which had to circulate in order to conserve mass. Here, the assumption of the axisymmetric nature of the plume reduced the 3D-problem to 2D-problem [8]. No much literature is available in this direction. The solutions were obtained by a Fast Computational Technique.

2. Mathematical Formulation

   The bacterial suspension (Bacillus Subtilis) contained in a deep chamber reveal the development of a thin upper boundary layer of cell-rich saturated fluids which becomes unstable, leading to the formation of falling plumes which is a complex phenomenon. This was used as a basis for our mathematical model. The whole suspension was under the influence of uniform magnetic field .

   The dimensionless governing equations are: The equation of cell conservation

   : Strength of Oxygen consumption relative to i ts diffusion

   : Measures the relative strengths of

   directional and random swimming

   : Ratio of Oxygen diffusivity to cell diffusivity

   : Bio-Rayleigh number

   Sc : Schmidth number

   Pm : Magnetic Prandtl number

   : Kinematic viscosity of the fluid

   c ,w : Densities of cell and water g : Acceleration due to gravity

   : Volume of a cell

   N .HN UN HN

   (1)

   B : Modified Hotmann n umber

   t

   : Magnetic Pr eamibility

   The equation of oxygen concentration

   . U HN

   t

   (2)

   m : Magnetic viscosity of the fluid

   2.2 Boundary Conditions

   The Navier Stokes equation (with Boussinesq approximation)

   Sc1 U U.U P 2U N B H.H

   1. No slip condition at Z = 1(bottom of the chamber).

      t

      e

      (3)

   2. Stress free condition at the upper surface of the

      The conservation of mass

      .U 0

      The magnetic induction equation

      H U.H H.U P 2H

      (4)

      (5)

      chamber, i.e., at Z = 0.

   3. The vertical components of velocity vanish at both the boundaries.

   4. Zero cell flux at both the boundaries.

      t m

      The variables are non dimensionalized as :

   5. Zero oxygen flux at the bottom surface and C = Co at the free surface.

      1 , N

      N ,

      C Cmin , D

      D H,

   6. H = 0 at both the boundaries

      h N0 C0 Cmin

      N N0

      TD

   7. The vertical components of velocity vanish at both

      the boundaries.

      V b VsH, K K0H.b Vs , t N0 ,

      p

      U U h , H H

      D H

      Mathematically, At Z = 0,

      2

      N0 0

      U.Z 0, z2 U.Z 0, 1

      Where h is the Depth of the chamber

      N

      Vs : It has dimensions of velocity

      H Z NHZ 0, HZ 0 .

      N0 : Initial cell concentration

      U : Saturated fluid velocity DN0, K0 : Constants

      At Z = 1,

      N

      DN : The cell diffusivity

      : The oxygen concentration C0 : Initial Concentration

      H : The step function

      T : The time

      H0 : The constant magnetic field

      U.Z 0, U Z 0, Z 0, Z 0, H.Z 0

3. Axisymmetric Plumes using Radial Co-ordinates

In the plume, the radial co-ordinate is scaled as

2.1 Dimensionless Parameters

R r, 0 1

Rescaling:

(6)

K N p

b V

,

,

D

H2p

N NAn, 1 CAC, W WA w,

0 0

Dc C0 Cmin

s , DN0

c , B DN0

0

wDN0

U WA u, P PAp

(7)

N gp

where NA , C A, WA , and PA are scale factors. Then the

0 c w

,Sc , P

m

axisymmetric governing equations (Neglecting O ( 2 )

D D m D

N0 w N0 N0

terms) are,

1 n 2n 2

n

n

1 u w

2w w w

2w

r R r2

WA u r

w z

Sc r r u

r2 r

Z w rZ

n C n C

2C

n 3w 1 2w 1 w

(18)

CA r

r r

r n r2

(8)

r

3 r

2 2

r

2 2 2

r

• hx hz

r r

2hz

hz hz

2hz

WA u C w C NAn C 1 C

(9)

B r r hz 2 r Z hz rZ

r Z CA r r r

2

r

u u w 0

(10)

Differentiating (14) w.r.t r and substituting for HA,WA

r r Z

u w 2w w w 2w

W

W

2 1 u

u

P p

2u

r

r u

r2

• r

  Z w rZ

  1 u 2 B HA h

  1 u 2 B HA h

  W Sc

  A

  A

  u w A

  2

  2

  r

  Z

  r r

  A

  A

  (11)

  hx w

  2w

  hz w

  2w

  (19)

  2

  2

  x hx hz

  A

  A

  hx

  r

  r hx

  r2 r

  Z hz rZ

  A

  A

  r r W r Z

  2

  3

  w w P p N

  Pm 1 hz 1

  hz

  hz

  2W Sc1 u

  w 2 A 2 A n

  r2 r r

  r2

  r3

  A r

  Z

  W Z W

  A A (12)

  2w 1 w

  2 H 2 h

  h

  B A hx z hz z

  Now, imposing the boundary conditions:

  r2

  r r WA r

  Z

  n C w

  2 W

  u u

• w u

2H h

u h

u

r 0, r

0, u 0, r 0

A

H

r

1h

Z

2h

A

x r

z Z

(13)

Also, r , n 0, C

r

0, w 0

P A

x x

m W r r

r2

3.1 Similarity solution for axisymmetric case

A

2W u w w w 2H h

w h

w

A r

Z

A

x r

z Z

In order to obtain a similarity solution [9][10] for

HA 1hz

2hz

(16), (17), (18), (19) the solution was posed of the form

Pm W r

r 2

h Za , w Zb,n Zc,C Zd,u Zab1

(20)

Here,

A

r

(14)

(h: width of the plume, a = ½, b = 0, c = -1, d = 0) Since h :: Z1/2 , the similarity variable is defined as

WA = 2

(to retain advection terms)

r Z1/ 2

(21)

C A = 2/ (to retain chemotaxis term)

NA = / 2 (to retain the oxygen consumption term in 9)

Assuming the solution in the form

n Z 1H, C G , ZF,

= O ( 1 2 ) (to retain the buoyancy term in 12)

1 F F

(22)

PA = 4 (to retain the pressure term in 12)

u Z 2

, w F

2

2

A

A

H 2 (to retain the induction term in 14)

This leads to p = 0 hence p = p (Z) in (11). (15)

r

Substituting for CA , WA and NA in (8) and (9) :

( : Stream functions, Primes denote differentiation

w.r.t ). Substituting these into (16) and integrating once w.r.t with the boundary conditions at = 0, we

n n

n C n C

get the following: HF 2HG (23)

u r

w Z 2 r

r 2 r

r

(16)

Substituting into (17)

2n 2C 1 n 2n 0

G G 1 GF H 0

(24)

r2 r r r2

1 C C 1 C 2C

Substituting into (18)

u r w Z n r r r2 0

(17)

1 1 1

1 1 1

F F F Sc FF FF

Differentiating (12) w.r.t. r and substituting for NA WA ,

2 3

3

2

(25)

HA and :

H B* 1 MM 1 MM 0

3 2

Substituting into (19)

M F

The boundary conditions are ,

(26)

5. Results and Discussion

At 0

H G F F F F 0

2 2

At

H 0,G 0, F 0, M 0

1. Solution

   For 0 , CFD technique is employed. However for

   0 (i.e., when the chemotaxis is unimportant in the plume) analytical solutions are possible with = O (1),

   In this study, the deep chamber experiments of Figure.1 have been modelled in three separate regions:

   1. an upper boundary layer of depth R

   2. a falling plume of width

   3. the region outside the plume.

In the sections 3 and 4, solutions for the cell and the oxygen concentration and the fluid velocity in the upper boundary layer are determined under the influence of a uniform vertical magnetic field. The solutions are found to depend on the parameters like, Sc (Schmidth

number), Q ( the cell flux ), (Bio-Rayleigh number),

Co = 1, No

= 2

and = 2 . Following [4] [8] the

B* (Magnetic Parameter) and (diffusivity ratio). The

computations are performed using the MATLAB tool;

solutions for the equations (23, 24, 25, 26) are found to be ( see table 1)

B 12

1 Sc1 B

the computed results are presented through graphs in Figures 2 to 14.

The following observations are made: In Figures 2,3,4,5. the effect of variation in the magnetic parameter

192 A2 1

B* on the profiles of velocity w F1 , the cell

C 1 Sc 1 B

Table 1. Solutions for F, H, G'

At Sc 1, 1	At Sc 2, 1
12A2 F 2 B 1 A2	24A2 F 3 2B 1 A2
192A2 H 6 2 B 1 A2 2B	384A2 H 12 3 2B 1 A2 32B
G 96A 2 6 2 B 1 A2 2B	192A 2 G 12 3 2B 1 A2 32B

At Sc 1, 1	At Sc 2, 1
12A2 F 2 B 1 A2	24A2 F 3 2B 1 A2
192A2 H 6 2 B 1 A2 2B	384A2 H 12 3 2B 1 A2 32B
G 96A 2 6 2 B 1 A2 2B	192A 2 G 12 3 2B 1 A2 32B

concentration H and the oxygen concentration G is studied for the values, Sc = = = Q = 1.0. Here, the oxygen concentration is considered as

1 2 G where is assumed to be always

positive so that, all the bacteria are active. In Figure.2 the effect of similarity variable on the F profile are

shown. F decreases enormously and remains constant as

for all values of B* . Further, F is negative and

its value is highest in the hydrodynamic case. In other

words, F increases in absolute value as B* increases,

Also solutions satisfy the boundary conditions at 0

and .

Q 2 B 8 B

and the vertical fluid velocity w, at the center of the plume increases indicating that the horizontal fluid flow into the plume increases. From Figure.5, w0 as

as expected. In Figure.3 the effect of similarity

variable on H profile is shown, it reveals that, as B* decreases the cell concentration in the plume increases as expected. Physically it means that, the higher the concentration of the cells, the greater is the consumption of oxygen which means that the oxygen

A2

2304

for Sc 1

concentration at the center of the plume decreases. In Figure.4 the effect of similarity variable on G profile is

Q 3 2B 15 2B

shown. Clearly the width of th plume decreases as

A2

9216

for Sc 2

B* increases. The oxygen concentration is more in the

hydrodynamic case ( B* = 0) when compared to the hydro magnetic case ( B* 0).

Since HF d Q

0

Figure 2. F vs

Figure 5. F

vs

Figure 3. H vs

Figure 4. G vs

Figures 6,7, 8. reveal the effect of variation of Q (= 0.5, 3, 5) on the profiles of F, H, G'. As the cell flux increases, F slightly decreases and increases in absolute value. The values of F are considerably very large inside the plume and drastically decrease and become constant

for large values of and accordingly w0 for large .

Fig.7 reveals that the width of the plume drastically increases as the value of Q decreases and the plume becomes narrower for large values of Q. Thus, the high concentration of the cells leads to a greater consumption of oxygen which in turn reduces the oxygen concentration at the center of the plume. Thus, as the cell flux Q increases, the vertical fluid velocity w, at the center of the plume increases and the values of M increases, indicating the increase in the horizontal fluid flow into the plume.

Figure 6. F vs

Figure 7. H vs

Figure 8. G vs

From Figure 8., it is found that the oxygen concentration in the plume is high for large Q(= 5) and the width of the plume drastically increases as Q decreases. This clearly indicates that the oxygen concentration at the center of the plume is less since there is a greater consumption of oxygen in the plume for large Q.

Figures 9,10,11. represent the effect of variation of

on the profiles F, H, G' for fixed values of the parameters Sc B* 1. The values of

considered are 0.2, 2 and 5.

The effect of buoyancy becomes important when

is large. The cell concentration is more for small values of and the plume becomes narrower for large .

When the cell concentration in the centre of the plume increases, the plume becomes narrower, accordingly the oxygen profile becomes narrower and the oxygen concentration at the center of the plume increases. The consumption of oxygen will be more. Therefore, the velocity of the fluid in the center of the plume will be

larger when the buoyancy force is dominant. But w0 more rapidly than for small values of . The decrease in M for the increase in indicates that less fluid is

entrained by the plume.

Figure 9. F vs

Figure 10. H vs

Figure 11. G vs

Figures 12,13,14. present the graph of the profiles F, H, G' when the values of Sc (= 1, 2) are varied. The other

parameters have fixed values viz., = 1, Q = 1 B* = 1.

It is observed that as in the hydrodynamic case ( B* = 0), the variation in Sc has a significant effect on the behaviour of the profiles. There is a drastic change in the values of F for Sc = 1 and 2. As Sc increased, F decreases rapidly and F a constant value as

as expected. The cell concentration will be

more for Sc = 2 and accordingly the oxygen consumption in the plume will be more and there will be a reduction in the oxygen concentration in the plume.

Figure 12. F vs

Figure 13. H vs

Figure 14. G vs

Finally, it is concluded that (i) the governing dimensionless parameters have a remarkable effect in the hydrodynamic as well as in the hydromagnetic cases

(ii) the qualitative nature of the profiles is almost the same in both the cases but there is a drastic difference in the quantitative nature of the profiles. Figure 1. clearly indicates the strong influence of the magnetic parameter on the present bioconvective system, these clearly suggest that the plume convection could be suppressed or enhanced through the proper choice of the magnetic parameter. The results are in excellent agreement with the hydrodynamic case.

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