Rheological effects of micropolar slime on the gliding motility of bacteria with slip boundary condition
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Rheological effects of micropolar slime on
the gliding motility of bacteria with slip
boundary condition
Ali, N, Asghar, Z, Beg, OA and Javed, T
http://dx.doi.org/10.1016/j.rinp.2018.02.070
Title Rheological effects of micropolar slime on the gliding motility of bacteria
with slip boundary condition
Authors Ali, N, Asghar, Z, Beg, OA and Javed, T
Type Article
URL This version is available at: http://usir.salford.ac.uk/46156/
Published Date 2018
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Results in Physics 9 (2018) 682–691
Contents lists available at ScienceDirect
Results in Physics
journal homepage: www.journals.elsevier.com/results-in-physics
Rheological effects of micropolar slime on the gliding motility of bacteria
with slip boundary condition
Z. Asghar a,⇑, N. Ali a, O. Anwar Bég b, T. Javed a
a
Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan
b
Fluid Mechanics, Aeronautical and Mechanical Engineering Department, School of Computing, Science and Engineering (CSE), University of Salford, M54WT, UK
a r t i c l e i n f o a b s t r a c t
Article history: Gliding bacteria are virtually everywhere. These organisms are phylogenetically diverse with their hun-
Received 15 January 2018 dreds of types, different shapes and several modes of motility. One possible mode of gliding motility in
Received in revised form 12 February 2018 the rod shaped bacteria is that they propel themselves by producing undulating waves in their body. Few
Accepted 27 February 2018
bacteria glides near the solid surface over the slime without any aid of flagella so the classical Navier-
Available online 8 March 2018
Stokes equations are incapable of explaining the slime rheology at the microscopic level. Micropolar fluid
dynamics however provides a solid framework for mimicking bacterial physical phenomena at both
Keywords:
micro and nano-scales, and therefore we use the micropolar fluid to characterize the rheology of a thin
Gliding bacteria
Undulating surface model
layer of slime and its dominant microrotation effects. It is also assumed that there is a certain degree of
Micropolar fluid slip between slime and bacterial undulating surface and also between slime and solid substrate. The flow
Slip effects equations are formulated under long wavelength and low Reynolds number assumptions. Exact expres-
Exact solution sions for stream function and pressure gradient are obtained. The speed of the gliding bacteria is numer-
Modified Newton-Raphson method ically calculated by using a modified Newton-Raphson method. Slip effects and effects of non-Newtonian
slime parameters on bacterial speed and power are also quantified. In addition, when the glider is fixed,
the effects of slip and rheological properties of micropolar slime parameters on the velocity, micro-
rotation (angular velocity) of spherical slime particles, pressure rise per wavelength, pumping and trap-
ping phenomena are also shown graphically and discussed in detail. The study is relevant to emerging
biofuel cell technologies and also bacterial biophysics.
Ó 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND
license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction most of the gliding bacteria) plays an important role in enhancing
their ability to associate with macrophages, shape maintenance of
Prokaryotic microorganisms achieve locomotion by several pos- bacterium, molecular sieving and facilitating binding of bacterium
sible modes, for instance, swimming in a fluid medium, crawling, to the host molecules [1]. The role of S-layer in bacterial motion is
gliding, and skimming over solid substrates. Among these microor- very important. S-layer helps to maintain overall rigidity of the cell
ganisms, gliding is adopted by rod-shaped bacteria which do not pos- wall and surface layers, as well as cell shape, which are important for
sess flagella. Some pertinent examples include flexibacter stain BH self-propulsion. S-layers protect the cell from ion/pH changes,
3, cytophaga, oscillatoria and vitreoscilla. This gathering of life forms osmotic stress, detrimental enzymes, bacterial viruses, and predator
also incorporates the phototrophic cyanobacteria and a portion of bacteria. They can provide cell adhesion to other cells or surfaces. For
the phototrophic green microbes and additionally the chemo- pathogenic bacteria they can provide protection from phagocytosis.
heterotropic filamentous floating microscopic organisms, fruiting Bacterial gliding is not only limited to aqueous phases but
myxobacteria, and cytophagae. Generally, few of these bacteria glide adjusted to drier regions and to development inside of solid
on a thin layer of adhesive slime. The slime is basically a loose thin masses, for example, soil, silt, and decaying wood that are pene-
sheath, composed of exopolysaccharides, glycoproteins and glycol- trated by little channels [2]. The exact mechanism of gliding motil-
ipids secreted by the bacterium, and is distinct to the S-layer (surface ity is still in debate. Available literature supports the bacterial
layer). The S-layer consist of monomolecular layer composed of (5 to motility by rotating cell surface discs, electro-kinesis, slime secre-
10% of total cell protein) glycoprotein. Bacterial S-layers (present in tion, osmotic forces, generating peristaltic waves within their sur-
face, and contractile components [3,4]. Here, we focus our
attention in examining the bacterial gliding by means of undulatory
⇑ Corresponding author.
waves in its outer surface. This approach has been supported by
E-mail address: zee.qau5@gmail.com (Z. Asghar).
https://doi.org/10.1016/j.rinp.2018.02.070
2211-3797/Ó 2018 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Z. Asghar et al. / Results in Physics 9 (2018) 682–691 683
numerous researchers who have confirmed the robustness of the diverse array of liquids including polymers, plastic sheets, ferroflu-
theory of bacterial gliding as a result of undulating waves in the ids, liquid crystals, biological fluids etc. Some applications of
external layer of their cell and due to the secretion of the parallel micropolar models in boundary layer flows, industrial flows, cavity
array of slime fibrils. This hypothesis was suggested by Costerton flows, stretching sheet problems, blood flows, swimming problems
et al. [5] as a possible mechanism of gliding motility adopted by etc. is documented in Refs. [29–32,37–47].
vitreoscilla. Halfen and Castenholz [6] suggested that gliding Most of the bacterial gliding studies were analyzed without
motility in blue-green algae (oscillatoria princeps) is due to its considering the slip boundary condition. But, in real biological sys-
wavy surface, which is nearly in contact with the solid boundary. tems there is always a certain amount of slip and no-slip boundary
Humphrey et al. [7] supported the hypothesis of flexibacter’s condition is no longer valid as the bacteria secretes the slime and
movement by dispatching waves in its own outer surface. Read glides due to a negligible resistance and considerable slip effects
et al. [8] accepted this theory as a mechanism responsible for gen- between its lower surface and slime and also present between
erating the necessary force for gliding in cyanobacteria. In his the slime and solid substrate. Considering such models along with
review article, Hoiczyk [9] reported on the generation of small sur- slip effects at the wall (specially their dependence on shear stress)
face waves, which travel down the filament surface, as one of the gives us more insight of the biophysical problem (see Refs. [32–
two possible mechanism adopted by cyanobacteria for motility. 34]). It is strongly anticipated that the glider’s speed, forces pro-
The micrographical results of the locomotion of cytophaga sp duced by the glider and power (required for propulsion) will be
strain RB1058 also confirms the same phenomenon of gliding significantly altered with the rheological parameters of the
motility (associated with undulating outer surface) on a trail of micropolar model and slip boundary condition at both ends. The
thin liquid over a solid wall [10]. These biological studies moti- choice of micropolar fluid with slip condition give us the exact
vated O’Brien [11] to develop a mathematical model to study the solution for stream function and pressure gradient which is not
gliding motion of bacteria. He calculated the power dissipation the case in our previous studies [26,35] with no slip condition. This
and maximum speed of the flexibacter using lubrication theory. new finding helps us to calculate the gliding speed for large values
Subsequently the locomotion of the flexibacter by the same mech- of the rheological parameter. Moreover, in the present case the dis-
anism was addressed by Wakabayashi et al. [12] and later by Ridg- tance between bacterial lower surface and solid substrate is so
way and Lewin [13]. Nakane et al. [14] observed a novel apparatus small (filled with a thin layer of non-Newtonian fluid) which give
composed of Gld and Spr proteins for Flavobacterium gliding. In rise to the dominant microrotation effects of the spherical slime
view of their results they proposed a model for Flavobacterium glid- particle.
ing supported by mathematical analysis. A focal adhesion-based The present letter is arranged as follows: In Section ‘‘Micropolar
mechanism in Myxococcus xanthus is thoroughly analyzed by Islam slime model and mathematical analysis”, we present the geometry
and Mignot [15]. The study carried out by McBride and Nakane [16] of the problem along with the governing equations for micropolar
reveals that Flavobacterium johnsoniae glides due to motility fluid which is used to simulate the bacterial slime. The problem is
adhesins SprB and RemA which are delivered to the cell surface also formulated with lubrication theory i.e. long wavelength
type IX secretion system. Miyata and Hamaguchi [17] discussed approximation and low Reynolds number assumptions in Sectio
the prospects for gliding of Mycoplasma mobile. In their viewpoint n ‘‘Micropolar slime model and mathematical analysis”. Exact solu-
gliding machinery of this cell is composed of huge surface proteins tions are determined in Section ‘‘Exact solutions”. In Section ‘‘Forces
and internal jellyfish-like structure. The gliding mechanisms generated and power required for propulsion”, the forces gener-
reported in literature for myxobacteria, flavobacteria and ated by the organism and power required for propulsion are pre-
mycoplasmas are recently discussed in an excellent review article sented. Gliding speed is calculated in Section ‘‘Gliding speed of
by Nan and Zusman [18]. For myxobacteria they discussed two the organism”. The results obtained for a fixed glider are explained
possible mechanisms of gliding out of which helical roter model in Section ‘‘The pumping problem”. Conclusions are summarized in
is based on secretion of slime and generation of retrograde surface Section ‘‘Conclusions”. All the constants involved in the solution of
waves in the cell. the problem are listed in the Appendix.
The slime secreted by the bacteria is quite complex and cannot
be characterized by the Newtonian constitutive law [19–21]. Based
Micropolar slime model and mathematical analysis
on this observation, several non-Newtonian models have been
deployed in bacterial hydrodynamic gliding studies. These include
Let (U and V) are the velocity components along X- and Y-
the third grade, power law, Johnson-Segalman and Carreau model
directions (i.e. fixed frame OXYZ), respectively. Fig. 1 illustrates
which have been utilized by Siddiqui et al. [22], Hayat et al. [23],
the undulating surface model in which the upper peristaltic wall
Wang et al. [24], Mahomed et al. [25] and Ali et al. [26], respec-
represents the rightward moving waves generated in the outer sur-
tively. Most recently Asghar et al. [35] discussed the bacterial glid-
face of the bacteria. The velocity of the wave is c. Due to these con-
ing motility by assuming three viscoelastic models namely FENE-P,
tinuous moving waves bacteria pushes backward a thin layer of
SPTT and Rabinowitsch model simultaneously. In both studies
micropolar slime and thus, engaging itself in a self-powered motion
[26,35], they used perturbative approach to calculate the gliding
called gliding that permits the organism to propel with speed V g
speed which is valid for small values of rheological parameters
near the lower slippery solid boundary (substratum). So ðV g cÞ
with no slip boundary condition. However, no study is available
is the relative wave speed with respect to the fixed frame (OXYZ).
in the literature, in which the Eringen micropolar fluid model is
The profile of the sinusoidal wave within the glider’s outer surface
integrated in the undulating surface models with slip effects to
is described by the following expression [22–26].
describe the non-Newtonian behavior of the slime. Micropolar flu-
ids describe randomly oriented particles suspended in a viscous 2p
medium that represents the effects of microstructure. hðX; tÞ ¼ h0 þ a Sin ðX ðc V g ÞtÞ ; ð1Þ
k
The hypothesis of micropolar fluids was basically developed by
Eringen [27,28]. The characteristics of micropolar fluids such as where a is the amplitude, k is the wavelength and h0 is the mean
non-symmetric tensor, intrinsic motion and local microstructures distance of organism from the solid substrate. In the fixed frame,
makes them more sophisticated than other non-Newtonian fluid the flow is unsteady. However, it can be treated as steady in a frame
models. Due to these characteristics, micropolar fluid models have oxyz moving with the speed c. The frame oxyz is called the wave
been extensively used to simulate complex phenomena in a frame.
684 Z. Asghar et al. / Results in Physics 9 (2018) 682–691
Fig. 1. Schematic diagram of two-dimensional undulating surface model.
The dimensional slip conditions in fixed frame (OXYZ) of refer- The choice of micropolar model for slime is driven by the fact that
ence are the gap between substrate and glider is very small (of the order
l 105 m). The micro-rotation effect is of crucial importance for flows
U ¼ V g T XY at Y ¼ h; in narrow gapes (channel). It is expected that when fluid flow
g
ð2Þ through such gap the micro-rotation of the fluid particles (microp-
l
U¼ T XY at Y ¼ 0; olarity of the fluid) affect the important flow characteristics. This
g factor is investigated thoroughly by Pietal [31] for squeeze flow of
where l is the slip length. The choice of slip condition is motivated water, blood and electro-rheological suspension using micropolar
by the fact that the slime beneath glider is typically a non- fluid. In view of the above fact, it is quite natural to investigate
Newtonian fluid [26,35] and it is generally accepted that there is the effect of micropolarity of the fluid on gliding mechanism.
a certain amount of slip present when non-Newtonian fluids flow The velocity field of the slime is given by v ¼ ðu; v ; 0Þ and its
in contact to a solid boundary. This behavior of non-Newtonian particle’s micro-rotation vector is w ¼ ð0; 0; wÞ. To facilitate
fluid is quite different from Newtonian fluids for which adherence solutions, the following dimensionless variable and numbers are
condition is valid. The transformations relating coordinates, veloci- introduced:
ties and pressures between the fixed frame (OXYZ) and the wave 2p y u v hðxÞ
frame (oxyz) are defined as: x ¼ x; y ¼ ; u ¼ ; v ¼ ;
h ¼ ;
k h0 c dc h0
x ¼ X ðc V g Þt; y ¼ Y; 2ph0
2
h0 1 2ph0 qch0
ð3Þ p ¼ p; T ij ¼ T ij ; j ¼ j; d¼ ; Re ¼ ;
u ¼ U ðc V g Þ; v ¼ V; P¼p kgc gc 2
h0 k g
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where u, v are the respective velocity components in the wave k Nð2 NÞ cg
frame along the x and y directions. The constitutive equations char- N¼ ; m¼ ; e¼ ð10Þ
gþk eð1 NÞ 2
h0 g
acterizing the steady flow of the micropolar slime in the wave frame
are given by Further we define a stream function by the relations
v i;i ¼ 0; ð4Þ
u ¼
@w
; v ¼
@w
; ð11Þ
@y @x
qv j v i;j ¼ T ij;j ; ð5Þ
Substituting Eqs. (7) and (8) into Eqs. (4)–(6), using relations (10)
the resulting dimensionless conservation equations for linear
qjv j wi;j ¼ Cij;j þ eijk T jk ; ð6Þ
momentum and angular momentum (micro-rotation) emerge as:
while the linear constitutive laws take the forms [27,32].
@w @ @w @ @w
d Re ¼
1 @y @x @x @y @y
T ij ¼ pdij þ ð2g þ kÞðv i;j þ v j;i Þ þ keijk ðxk wk Þ; ð7Þ " ! #
2 @p @w 2
@2
1 2 @ @w
þ N þ d þ ; ð12Þ
Cij ¼ awk;k dij þ bwi;j þ cwj;i ; ð8Þ @x 1 N @y @x2 @y2 @y
where T ij and Cij are Cauchy stress tensor and couple stress tensor,
@w @ @w @ @w
respectively. dij is the Kronecker delta, eijk is the alternating tensor, p d3 Re
@y @x @x @y @x
is the isotropic pressure, v i are the components of velocity vector, 2
" ! #
xi are the components of vorticity vector ðxi ¼ ð1=2Þeijk v k;j Þ and @p d @w @2 @2 @w
¼ þ N þ d2 2 þ 2 ; ð13Þ
wi are the components of micro-rotation vector w and q and j are @y 1 N @x @x @y @x
the fluid density and micro-gyration parameter and a comma after
a suffix denotes covariant differentiation. Furthermore, the coeffi- djReð1 NÞ @w @ @w @
w ¼
cients of viscosity g, k and the coefficients of gyroviscosity ag , bg N @y @x @x @y
!
and cg satisfy the following inequalities [27,32]: @2 @2 eð1 NÞ
2w þ d2 2 þ 2 ww : ð14Þ
2g þ k P 0; k P 0; 3ag þ bg þ cg P 0; cg P jbg j: ð9Þ @x @y N
Z. Asghar et al. / Results in Physics 9 (2018) 682–691 685
In the above equations, the superscript ⁄ is omitted for the brevity. @2w N
It is noted that due to the solenoidal nature of the microrotation T xy ¼ w; ð26Þ
@y 2 1N
vector, the coefficient of gyroviscosity ag and bg do not appear in
the governing equations. Moreover, these equations reduce to the
2
1 @ w N
classical Navier-Stokes equations (for the Newtonian case) when T yx ¼ þ w; ð27Þ
1 N @y2 1N
N, e and j vanish. Employing the long wavelength and low Reynolds
number assumption, Eqs. (12)–(14) reduce to [22–26]: T xx ¼ 0; T yy ¼ 0; ð28Þ
3
@ w @p @w In view of Eq. (25), the dimensionless pressure rise per wave length is
¼ ð1 NÞ N ; ð15Þ
@y3 @x @y given by:
Z 2p Z 2p
@p dp 3 sinh hmð2F þ h hV b Þ
¼ 0; ð16Þ Dpk ¼ dx ¼ dx: ð29Þ
@y 0 dx 0 hðN 1ÞðI23 þ I24 sinh hmÞ
eð1 NÞ @ 2 w @2w
2w ¼ : ð17Þ Forces generated and power required for propulsion
N @y2 @y2
Eqs. (15)–(17) are subject to following slip boundary conditions: The dimensionless horizontal and vertical forces on the bacte-
@w rial organism surface are given by
w ¼ 0; ¼ V b þ bT xy ; w ¼ 0; at y ¼ 0; Z 2p Z 2p
@y
ð18Þ Fx ¼ sx jy¼h dx; F y ¼ sy dx; ð30Þ
@w y¼h
w ¼ F; ¼ 1 bT xy ; w ¼ 0; at y ¼ h; 0 0
@y
Here sx and sy are components of the stress vectors s defined as
where b ¼ hl0 and / ¼ ha0 is the slip and occlusion parameter, follows:
!
V
respectively with V b ¼ cg 1. Here the flow rates in fixed (OXYZ) sx p @h þ T xy
s¼ ¼ @x
: ð31Þ
and wave (Oxyz) frames are denoted by H and F, respectively. They sy 1d p þ T yy
are related to each other according to [26]:
For brevity, here we omit the detailed derivation of Eq. (32), which
H ¼ F V b: ð19Þ can be found in Ref. [22]. From, Eqs. (26)–(28) and (31), Eq. (30)
gives:
Exact solutions Z 2p
3 sinhhmð2F þ h hV b Þ mðN 2ÞðI25 þ sinhhmI26 Þ
Fx ¼ þ dx;
0 ð1 NÞðI23 þ I24 sinhhmÞ 2I27 I28
In view of Eq. (16), Eq. (15) can be written as:
ð32Þ
@3w @ dp
¼ ð1 NÞ y Nw : ð20Þ F y ¼ 0: ð33Þ
@y3 @y dx
Integration of Eq. (20) yields: The vertical force is zero, due to the choice of symmetric wave form
on the glider surface. In the moving frame of reference this wave
@2w dp causes the glider surface to move to and fro in y-direction. Therefore
¼ ð1 NÞ y Nw þ C 1 : ð21Þ
@y2 dx the horizontal component of s does not play any role in the power
Using Eq. (21) in Eq. (17), we get propulsion. The power required for propulsion in non-dimensional
2
form [11] is:
d w dp Z 2p
2
m2 w ¼ ð1 NÞv2 y þ v2 C 1 : ð22Þ dh
dy dx P¼ p dx ð34Þ
0 dx
The general solution of Eq. (22) is:
Integration of (34) yields:
v2 dp Z 2p Z 2p
w ¼ C 2 cosh my þ C 3 sinh my þ ð1 NÞ y þ C 1 : ð23Þ dp 3 sinh hmð2F þ h hV b Þ
m2 dx P¼ h dx ¼ dx: ð35Þ
0 dx 0 ðN 1ÞðI23 þ I24 sinh hmÞ
Replacing the value of w in Eq. (21) and integrating twice gives:
dp Nv2 y3 N v2 y2
w ¼ ð1 NÞ 1þ 2 þ C1 1 þ 2 Gliding speed of the organism
dx m 6 m 2
NC 2 NC 3 Bacterium steadily glides over a solid wall only when the
2 cosh my 2 sinh my þ C 4 y þ C 5 ; ð24Þ
m m equilibrium condition (i.e. F x , F y and DP k must be zero) is satisfied.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
in which C 15 are constant of integration, m ¼ ð2 NÞ v and As in the Newtonian case, the lift force ðF y Þ generated by the
v ¼ N=eð1 NÞ.
2 micropolar slime is also zero. Therefore, we are left with two non-
The constants C 15 can be determine by using boundary condi- linear equations as a result of equating to zero the expressions of
tions (18) and are given by: Dpk and F x . These equations are:
The axial pressure gradient is given by:
Z 2p
dp
dp 3 sinh hmð2F þ h hV b Þ Dpk ðV b ; FÞ ¼ ðV b ; F; xÞ dx ¼ 0; ð36Þ
¼ : ð25Þ dx
dx hðN 1ÞðI23 þ I24 sinh hmÞ 0 y¼h
For the micropolar slime, the stress tensor is non-symmetric. The Z 2p
dp
dimensionless forms of the shear and normal stresses for the prob- F x ðV b ; FÞ ¼ sxy ðV b ; F; xÞ h ðV b ; F; xÞ dx ¼ 0: ð37Þ
0 dx y¼h
lem under consideration are given by:
686 Z. Asghar et al. / Results in Physics 9 (2018) 682–691
From (36) and (37), it is difficult to find the closed form expressions This figure reveals the asymptotic behavior of the bacterial
of V b and F in term of other parameters. Hence, for the fixed values velocity profile for the large values of e. It is revealed that for fixed
of /, m, b and N, one can calculate the plausible numerical values of occlusion ratio ð/Þ and coupling number ðNÞ, the organism
F and V b using the modified Newton-Raphson method. The numer- achieves maximum velocity at e ¼ 0:1. Fig. 3(a) also reveals that
ical solution obtained via this procedure must satisfy physically the presence of slip (between bacterium and slime along with
realistic conditions such that 1 < V b < 0 and F < 0. To employ the slime and solid substrate) tends to slow down the gliding
modified Newton-Raphson method, we set speed. The bacterial speed as a function of coupling number is
depicted in Fig. 3(b) for different values of /. It is observed that
Vb Dpk
r¼ and g ¼ : ð38Þ bacteria glides faster over a slime with large coupling numbers as
F Fx compared to the slime for which N ¼ 0 (Newtonian case). More
Starting with some initial guess, the refined values of V b and F can efficient bacterial propulsion is therefore achieved in micropolar
be obtained using the formula: slime rather than Newtonian slime. Rheology of the bacterium
environment therefore contributes to improving propulsive
gðri Þ
riþ1 ¼ r i ; where i ¼ 0; 1; 2 . . . ; ð39Þ efficiency. Newtonian models under-predict the propulsive perfor-
Jðr i Þ mance. The plot also confirms that the organism achieves more
where the Jacobian matrix J is defined as: speed as the amplitude of wave with its outer surface gets larger.
" # Our results also estimate the power required for 0:5 lm wide
@ @
ðDpk Þ ðDpk Þ
J¼
@V b @F
; ð40Þ and 10 lm long glider as 5:54 1010 erg=s. Humphrey and
@ @
@V b
ðF x Þ @F
ðF x Þ Marshall [7] also reported the experimental value of power
required to sustain such motion given as 5 107 erg=s. Our calcu-
The iteration procedure described above is implemented in the
lated value is much less than the experimental value given by
symbolic code Mathematica and results obtained are shown graph-
Humphrey and Marshall, which confirms the feasibility of our
ically in Figs. 2–4.
analysis.
To develop a compatibility between the experimental values i.e.
The variation in power propulsion is depicted in Fig. 4(a) and
h0 ¼ 0:025 lm, k ¼ 0:07 lm, a ¼ 0:015 lm, g ¼ 1P and (b). Fig. 4(a) is in correspondence with Fig. 3(a). It is noted that
V g ¼ 1:5 lm=s (gliding speed of Flexibactar sp. Strain BH3) and
power required for propulsion decreases in magnitude with
our theoretical results Fig. 2 is plotted where, the value of wave
increasing slip effects.
speed can be approximated as c ¼ 2:5 lm=s (see Refs. [22,35]).
The power increases initially with increasing values of e and for
Hence the non-dimensional gliding speed can be approximated
large values of e it (power) exhibit asymptotic behavior. While
as V g =c ¼ 0:6.
Fig. 4(b) (is in correspondence with Fig. 3(b)) shows that the larger
Fig. 2 clearly testifies our model as it indicates that the plot amplitude waves therefore necessitate greater power for
reaches the experimental value of the bacterial speed with propelling the bacterium. Moreover, a slight increase in power is
considerable slip effects. also noted when slime rheology changes from Newtonian to
In order to illustrate the general trend, gliding speed of the bac- micropolar.
terium as a function of micropolar parameter ðeÞ is displayed in
Fig. 3(a) for different values of slip parameter ðbÞ.
The pumping problem
The closed form solution of Eqs. (15) and (17) along with the
boundary conditions given in Eq. (18) is obtained and the influence
of coupling number ðNÞ, micropolar parameter ðeÞ, slip parameter
ðbÞ, flux rate ðFÞ and bacteria wave amplitude ð/Þ on velocity com-
ponent ðuÞ microrotation ðwÞ, pressure rise ðDpÞ and stream func-
tion ðwÞ are graphically shown in Figs. 5–9 for V b ¼ 1. The case
V b ¼ 1 corresponds to the pumping scenario, in which bacterium
is stationary i.e., V g ¼ 0. Indeed, this case represents the peristaltic
flow of micropolar fluid between a substrate and a wavy wall.
Fig. 5(a) and (b) have been plotted to illustrate the slip effects at
two different flow rates along with the effects of micropolar
parameter and coupling number on the velocity profile respec-
Fig. 2. Plot of non-dimensional gliding speed as a function of N. tively. Both figures show parabolic velocity profiles indicating non-
Fig. 3. Variation of the gliding speed with respect to the different rheological parameter.Z. Asghar et al. / Results in Physics 9 (2018) 682–691 687
Fig. 4. Variation of power propulsion with respect to the different rheological parameter.
Fig. 5. Variation of velocity profile with respect to the different rheological parameter. Panel (a) N ¼ 0:8 and panel (b) e ¼ 4:8, with / ¼ 0:6 and V b ¼ 1.
Fig. 6. Variation of microrotation profile with respect to the different rheological parameter with / ¼ 0:4 and V b ¼ 1. Panel (a) N ¼ 0:8, b ¼ 0 and F ¼ 1, panel (b) e ¼ 4:8,
b ¼ 0 and F ¼ 1, and panel (c) F ¼ 1; N ¼ 0:8 and e ¼ 4:8.
linear velocity variations at the cross-section x ¼ p=2. It is further flow of slime adjacent to it and as the distance from these slippery
observed that with increasing the coupling number (N) or decreas- boundaries are increased (i.e. in the central part) the situation
ing the micropolar parameter ðeÞ, the slime velocity is increased become opposite.
(i.e. slime acceleration occurs) in the central part of the domain Fig. 6(a)–(d) illustrate the influence of micropolar parameter ðeÞ,
y 2 ½0; h while the opposite trend prevails in the vicinity of the coupling number ðNÞ, flow rate ðFÞ and slip parameter ðbÞ on the
walls. Fig. 5 also depicts that the slippery boundaries assist the microrotation of the spherical slime particles. It is clear from688 Z. Asghar et al. / Results in Physics 9 (2018) 682–691
Fig. 7. Plots of pressure rise per wavelength Dp versus flow rate H for various values of e, N, b and /. Panel (a) / ¼ 0:3, b ¼ 0 and N ¼ 0:85, panel (b) / ¼ 0:3, b ¼ 0 and e ¼ 4:8
and panel (c) / ¼ 0:3, N ¼ 0:85 and e ¼ 4:8. Here V b ¼ 1.
Fig. 8. The pattern of streamline for different values of b when / ¼ 0:4, N ¼ 0:8, F ¼ 0:2, V b ¼ 1 and e ¼ 4:8.
Fig. 6 that the behavior of microrotation is symmetric about the In fact, in such case the micro elements of the slime exhibit
center of the channel. The micro-rotation increases uniformly in strong rotation near the walls. In contrast, no such boundary layer
magnitude with decreasing e in both channel halves (Fig. 6(a)). is formed with increasing N and b although the microrotation
For small values of e, microrotation varies sharply near the walls increases in magnitude in both the channel halves with increasing
and behaves linearly over the remainder of the cross section. This N (Fig. 6(b)). In contrast Fig. 6(c) shows that the increase in the slip
clearly indicates the formation of a thin boundary layer at both the effects causes a reduction in the microrotation of the spherical
walls for small values of e. slime particles.Z. Asghar et al. / Results in Physics 9 (2018) 682–691 689
Fig. 9. The pattern of streamlines for different values of N when / ¼ 0:4, b ¼ 0:25, F ¼ 0:2, V b ¼ 1 and e ¼ 4:8.
The plots of pressure rise per wavelength against time aver- that greater mixing is realized with increasing micropolarity of the
age flux H for different values of rheological parameters and slip slime.
parameter is illustrated in Fig. 7(a)–(c). It is demonstrated in
Fig. 7(a) that pressure rise decreases in the pumping region Conclusions
ðDp > 0; H > 0Þ with decreasing micropolar parameter ðeÞ. The
free pumping flux i.e., H for Dp ¼ 0 is found to decrease with The gliding motility of bacteria on a thin layer of micropolar
increasing m. In the co-pumping region ðDp < 0; H > 0Þ, the slime is investigated employing an undulating surface model.
magnitude of pressure rise per wavelength decreases by reduc- The lubrication approximation theory (LAT) is utilized to simplify
ing the values of e. the flow equations of micropolar slime. The problem is solved
Fig. 7(b) illustrates the effect of coupling number N on Dp. In the analytically. The exact expression for the stream function w
pumping region, larger values of coupling number (N), enhance the and pressure gradient dp=dx are also obtained. The Modified
magnitude of Dp indicating that greater effort is rendered by the Newton Raphson method is used to calculate steady gliding
wavy wall (organism) to maintain the prescribed flux of a microp- speed of the organism. The pumping problem is also considered
olar slime in comparison to the Newtonian slime scenario. The free when the bacterium is held fixed and the associated undulating
pumping flux is nearly independent of the coupling parameter N. motion generates a net flow in the positive x-direction. Velocity,
However, larger values of N in the co-pumping region enhance micro-rotation, pressure rise and streamlines for different values
the magnitude of Dp. Fig. 7(c) shows a decrease in Dp with increas- of rheological parameters are also plotted to observe the slime
ing b. Free pumping flux also decreases with increasing slip effects behavior. The principal observations from the current study
between the fixed glider and secreted slime and also between the can be summarized as:
moving slime and fixed substrate.
Fig. 8 displays the streamlines representing the slime flow pat- The bacterium glides faster on the micropolar slime as com-
tern for four different values of the coupling number b. In general, pared to the Newtonian slime.
it is noted that the streamlines resemble in shape the wavy wall The speed of the organism increases with increasing amplitude
boundary (shown with solid pink wave). However, some recirculat- of the undulating wave on the organism surface, while slip
ing zones can be identified in the central region of the flow domain. resist the gliding motility.
These recirculating zones are largely dependent on the values of the The glider achieves maximum speed at e ¼ 0:1 irrespective of
involved parameters. Flow patterns of the slime for four different the choice of other parameters. These results are appropriate
values of slip parameter are represented in Fig. 8. It is evident that to use in Bio-mechatronics engineering to adjust the slime
the size of circulating bolus decreases with increasing b. Finally rheology with certain amount of slip in such a way that an
the bolus disappear at a particular value of slip parameter b ¼ 0:6 artificial mechanical glider can attain its maximum speed.
as shown in Fig. 8. Finally, the streamlines patterns for different cou- The gliding speed and power propulsion achieves asymptotic
pling numbers are shown in Fig. 9. The size of trapped bolus is behaviors for large values of rheological parameters. These
enhanced with increasing coupling number. This further confirms new results are quite interesting and give us more insight690 Z. Asghar et al. / Results in Physics 9 (2018) 682–691
to bacterial hydrodynamics, since our previous study [26,35] hm
I22 ¼ hI4 þ 2I4 b I2 I5 m2 b I2 I5 m tanh ;
predicts the monotonic behavior for gliding speed and power 2
at small values of rheological parameters. I23 ¼ 3hI2 I5 mð1 þ coshhmÞ;
For the case when the organism is held fixed and its undulating I24 ¼ 6I2 I5 þ hðhI4 þ 6I4 b 3I2 I5 m2 bÞ;
motion generate a net flow in the positive x-direction, velocity
I25 ¼ 3ðF þ hÞmN coshhm þ 3mNðF hV b Þ;
of the slime follows an increasing trend at the channel center
2
with increasing N (or decreasing e). I26 ¼ 3M þ m2 ðhð3F þ 2hÞ 3ðF þ hÞð2 þ MÞbÞ þ ðh m2 þ 3MÞV b ;
The microrotation profile is symmetric about the center of the hm hm
channel and slime micro-elements rotate with the greater I27 ¼ 2mðh ð2 þ NÞbÞcosh N sinh ;
2 2
speed as the coupling number and micropolar parameter are hm hm
2
increased but increasing slip result in reduction of microrota- I28 ¼ 3hmNcosh þ ð6N þ hm ðh 3ð2 þ MÞbÞÞsinh :
2 2
tion of the slime particles.
The pressure rise per wavelength in the pumping region
increases with a departure in the slime rheology from
Newtonian to micropolar. References
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Further reading
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nanoparticles (Ag and Al2O3-blood) impact on hemodynamics of overlapped theory. VIIEuropean congress on computational methods in applied sciences
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[41] Ijaz S, Shahzadi I, Nadeem S, Saleem A. A clot model examination: with [48] Zubair M, Waqas M, Hayat T, Ayub M, Alsaedi A. The onset of modified Fourier
impulsion of nanoparticles under the influence of variable viscosity and slip and Fick’s theories in temperature-dependent conductivity flow of
effects. Commun Theor Phys 2017;68. 676(5). micropolar liquid. Results Phys 2017;7:3145–52.
[42] Nazeer M, Ali N, Javed T. Numerical simulation of MHD flow of micropolar
fluid inside a porous inclined cavity with uniform/non-uniform heated bottom
wall. Can J Phys 2017. https://doi.org/10.1139/cjp-2017-0639.You can also read
