Low Reynolds number viscous flow past a sphere and a circular cylinder has been analysed by [1] [27] etc. Low Reynolds number flow is characterised by low velocities and dominant viscosity effects. The case where the solution is stable and linear under low Reynolds number flow is governed by the so-called slow flow equations or the Stokes equations. The Stokes equation is a special case of the Navier Stokes equations. At low Reynolds numbers, inertial forces are negligible viscous forces tend to dominate. This is almost the case of zero Reynolds number [32]. The reduction of Navier Stokes equations to Stokes equations brings in time independence and there is really no difference between slow and quick swimming as demonstrated by [33]. It might also be noted that Stokes flow is also called the reciprocal theorem [26]. The governing equations of Stokes flow are Partial Differential Equations (PDE’s) of the elliptic type.

When we apply the velocity boundary condition on the surface of an ellipsoidal, spherical or any other experimental body, the body being in a viscous fluid tends to slow down. This is due to the effect of viscous forces, which impair the motion of the body. The system behavior is strongly viscous, and the so-called creeping motion or Stoke Equations [34] govern their behavior.

The Navier-Stokes equations describe flow in viscous fluids with momentum balances for each component of the momentum vector in all spatial dimensions. The study focus on the boundary movement and swimming at low Reynolds number flow that is either very viscous, very slow, or very small. Low Reynolds number flow for water is about 10 to 4 m²⁄s. Flows with small Re are also called creeping flows or Stokes slow and appear on small scales in porous media, in micro-fluidic devices, and flows with small bodies such as bacteria, amoeba, and innate particles such as blood corpuscles, droplets, bubbles, and mineral grains. However, the inertial forces are negligible also in very large-scale geological movements such as glacier flow and tectonic plate movement. Important thing is that the low Reynolds number flow plays an important role in the study or lubrication theory, moving micro-organism and many area of biophysical and geophysical interest [27].

The Stokes Flow equations for creeping flow are obtained as the R e = 0 limit of the Navier-Stokes equations for momentum balance augmented by the mass balance law [19], and the fluid flow has made slightly compressible, ∇ ⋅ u = ϵ p , where ϵ > 0 . The slightly compressible Stokes equation can be written as,

− ∇ p + μ ∇ 2 u = 0 ∇ ⋅ u = ϵ p } (1)

The Stokes Equation (1) with boundary conditions of specified body surface velocities. This is a linear system of PDE, and it follows that the velocity field and hence the forces exerted by the fluid on a moving body are linear and the body surface velocities are proportional to μ .

For the analysis of the thin layer fluid flow between cylindrical and spherical object and bottom wall of the tank, the Reynolds Lubrication theory is used which is generally called the Lubrication theory. It is a most important approximation of Stokes equation whenever one of the dimensions of the system is small compared to the others [35] [36].

A body with the surface z = ( x , y , t ) is considered close to a solid plane at z = 0 . One can regard the gap as “thin” provided that “h” is small, and the pressure z-gradient can be neglected as well as the momentum transport terms,

∇ p = μ ∂ 2 u ∂ z 2 , u ( x , y , 0 ) = u ( x , y , h ) = 0 : u = − 1 2 μ z ( h − z ) ∇ p

Integration across the layer of the mass balance results in the Reynolds lubrication equation,

1 12 μ ∇ ⋅ ( h 3 ∇ p ) = d h d t (2)

If the body is rigid and moving in the z-direction, we have d h d t = V .

1) For Vertical Cylindrical Model (RLT)

For the vertical cylinder, we have axial symmetric axis,

h ( r , t ) = h ( t ) (3)

2) For Spherical Model (RLT)

For the sphere there is axi-symmetry, so with r the radial polar coordinate,

h ( x , t ) = δ + R − R 2 − x 2 = δ + R 2 ( x R ) 2 + O ( ( x R ) 4 ) (4)

A long body performs a wiggling motion, a sinusoidal lateral (η) displacement of every point on its surface, as traveling waves,

η = υ 0 ⋅ sin ( k ξ − ω t ) (5)

This motion, combined with translation and rotation of the local coordinate system, keeps body volume constant, but the body perimeter is not constant. The dimension of the object chosen in this simulation is 0.002 m and the shape of the moving object is elliptical. The wiggling amplitude υ 0 is small, on the order of the worm thickness, and the worm length is a few wavelength λ = 2 π / κ . In Figure 2, a snapshot of an elliptical striped worm about two wavelengths long is shown.

The angular frequency is immaterial in the sense that the wiggle motion is periodic in time and the body has moved a certain distance per period. Consider the calculation of the swimming velocities of the model object. This model is designed by small amplitude to deform the surface of the swimmer. Small amplitude body motion is periodic motion.

For deformation or swimming of the object, we have chosen non-reversible flow. This means flow is going forward and backward but moving object has no way to go back to the original position. When we have chosen the moving boundary condition, allow re-meshing and use time dependent solver then we see the mesh deform and the object move. This means that the fluid particle is swimming periodically on average.

When we differentiate Equation (5) with respect to t then we get the velocity in the moving frame

d ξ d t = 0 d η d t = − v 0 ⋅ ω ⋅ cos ( ω t − k ξ ) } (6)

For the forward swimming motion, we calculate the force and torque that is exerted on the object by the domain fluid and also calculate the unknown linear and angular velocity of the swimmer object whose net force and torque on the object are zero.

The transformation from local coordinates to the global coordinates (see Figure 2) is

x = X 0 + T ( θ ) ξ (7)

where, ( x y ) = x , ( ξ η ) = ξ , ( X 0 Y 0 ) = X 0 , T = ( cos θ − sin θ sin θ cos θ ) .

Then, on the boundary of the wiggling swimmer (see Figure 2),

x ˙ = ( u v ) = θ ˙ D ( x − X 0 ) + T ξ ˙ + X ˙ 0 , D = ( 0 − 1 1 0 ) = ( − θ ˙ ( y − Y 0 ) + v 0 ⋅ sin θ ⋅ sin ( ω t − k ξ ) + U 0 − θ ˙ ( x − X 0 ) + v 0 ⋅ cos θ ⋅ sin ( ω t − k ξ ) + V 0 ) (8)

where, ξ must be computed from global coordinates ( x , y ) ,

ξ = ( x − X 0 ) cos θ + ( y − Y 0 ) sin θ (9)

These are the velocities of a point on the swimmer surface, used as boundary condition for the Stokes equation. From that solution, the forces f on the surface S are calculated, and the net forces and moment on the body,

F x = ∫ S f ⋅ e x d S , F y = ∫ S f ⋅ e y d S , M = ∫ S ( f y ( x − X 0 ) − f x ( y − Y 0 ) ) d S (10)

The sum of internal forces of all bodies are zero, the balance of linear and angular momentum for the deforming body take the form,

∫ Ω d d t u d m = ∫ s f d S ∫ Ω d d t u × X d m = ∫ s f × X d S

If the body were rigid, these equations would determine the motion of its centre of gravity and the angle θ of rotation, the three 2D rigid-body degrees of freedom. For the deforming body there is in general no such convenient reduction.

However, in the limit of small inertial forces, the acceleration terms vanish and we are left with

0 = ∫ S f d S 0 = ∫ S f × X d S (11)

consistent with the neglecting of inertial forces as expressed by the Stokes flow model. For numerical reasons, we have implemented the singular perturbation model of (11)

m d d t U 0 = ∫ S f d S , d d t X 0 = U 0 J d d t W = ∫ S f × ( x − X 0 ) d S , d d t θ = W (12)

where “m” and “J” are “small”. Since inertia forces are neglected in the momentum balance for the fluid, this is an inconsistent approximation.

However, it shows that if the system is stable, i.e. that the forces tend to act against the velocities, then there should be a well-defined limit as “m” and “J” vanish and it should be the solution of the differential-algebraic system, Equation (12). The numerical implementation uses the Ordinary Differential Equations (ODE) system with small “m” and “J”.

The ODE is used for differential algebraic equations or to add degrees of freedom to a PDE (Partial Differential Equations) using the introduced states. First time derivative represents velocity and second time derivative represents force and moment in ODE setting. In ODE system, there is no need to find consistent initial values for U 0 , V 0 and W. “Smallness” was determined by trial and error in choosing smaller and smaller “m” and “J”.

The final set of ODE then becomes,

Force F and torque M on the object is calculated [22] for a body in slow motion with a known velocity and rotational rate.

At low Reynolds number flow, swimming motion of a small object depends on different wave parameters.

Consider the axial symmetric axis and the dynamic simulation of thin rigid fiber body, the force distribution is at zero Reynolds number with slow flow. The vertical cylindrical pressure force on the circular area is calculated by using Reynolds

lubrication theory. In the minimum separation between particle and bottom wall tank, lubrication dominates the short-range interactions. This short-range hydrodynamic force is calculated using lubrication approximation. The lubrications formulas are derived by [37].

Suppose the fiber has a finite length, a short-range interaction between the fiber and fluid, the lubrication equations of [37] are implemented using a dimensional radius of curvature is equal to the particle length. For the vertical cylindrical case our Reynolds Lubrication Equation is

To solve the force distribution the pressure is derived in the fiber coordinates.

The far-field hydrodynamic interaction and the end interaction of thin fiber have no effect. However, the details must be included when calculating the lubrication interactions. The equations in Claeys and Brady [37] are handled in the same manner as above using the lubrication analysis. For the bottom wall tank interacting with a cylindrical body, the total pressure force on the circular area in z-direction becomes,

falling in an infinite tank filled with viscous fluid. The same boundary conditions (vertical cylindrical) have used for force and velocity calculation. The sphere in tank case has been used to assess the element mesh density necessary to obtain the force accurately and for sufficiently small radius to tank width ratio R⁄W can be compared to the Stokes force on a sphere in an infinite fluid.

The spherical polar coordinates are,.

In addition, the Reynolds lubrication equation becomes,

Pressure, , and total force,.

Consider the two-dimensional (2D) numerical simulations of Stokes flow for the small swimming object at low Reynolds number. We have chosen two reference frames, one is fixed (Global Coordinates), and another is the moving frames of reference (Local Coordinates). The local coordinates moves with fluid particle. This motion is realised when we apply the moving boundary condition on fluid particle. For the swimming object with moving mesh, we choose the 2D plane axis. For the moving mesh, we have allowed for re-meshing using the arbitrary Lagrangian-Eulerian (ALE) technique and have added Stokes equation from MEMS module in COMSOL Multiphysics software.

When we allow for re-meshing, we get three frames: The spatial frame, the reference frame, the mesh frame. The spatial frame coordinates are x and y by default, the reference frame coordinates are X and Y by default and the mesh frame coordinates are xm and ym by default. In the moving case, we have chosen

a large tank that is filled by viscous fluid and elliptical object shape that is moving or swimming in the fluid. In Subdomain Setting we use free displacement which means mesh displacement is controlled only by the boundary conditions on the surrounding boundaries. In boundary setting we use two kinds of setting. For tank boundary we add mesh displacement module and in x and y direction add zero because our global coordinate is fixed and there is no mesh displacement in here, and for object boundary we add mesh velocity u and v, which is the free boundary.

There are some necessary steps to be taken to get a complete solution of vertical cylindrical model in COMSOL. Select geometry, constant parameters, set-up equation system, boundary conditions (applied at the global and subdomain level), and refine mesh or non-uniform mesh is to avoid error in solution between two bottom wall. One tries to choose different orders of elements for finite element solutions like the Lagrange quadratic or the Lagrange cubic. If the difference in solution between using quadratic or cubic elements is negligible, one can retain the lower order element. It is seen in this case that there is practically no difference between solutions arising from the choice of quadratic (2^nd order) or cubic (3^rd order) Lagrange shape functions. Hence, quadratic Lagrange elements are used. Finally, the solution is sought and then post processing is carried out to visualize the solution fields of interest.

The velocity and pressure plots are near the bottom wall of the tank. The pressure plots show some spikes at the cylinder edges. This is owing to the singularities that arise at sharp corners or edges of the cylinders [38]. An option to avoid singularity is to refine the meshes at these locations. This could be done either by increasing the order (p-convergence) or by increasing the number of elements (h-convergence). However, either of these options didn’t yield the result and one is led to think that the alternative would only be to smoothen out the sharp edges.

The numerical (PDE) and the theoretical forces (LRT) are calculated, the x-axis corresponds to the ratio describing the motion of the cylinder towards the bottom wall of the tank and plotted the along y-coordinate are the forces on the cylinder calculated numerically and theoretically. The force grows quickly when is below 0.05. For values of greater than this the value of force asymptotically approaches the value of zero (0).

Logarithmic plots are used to find the differences in growth rates for the force in the numerical and the theoretical case. The logarithmic plot also shows that the force in the low ranges of is proportional to. The force plot seems to show a favorable agreement between the numerical and theoretical calculations

for the vertical cylinder. It must however be kept in mind that we have introduced the slightly compressibility as opposed to the incompressibility that is otherwise assumed.

Consider the sphere is close to the bottom wall boundary. The pressure field results shown in Figure 8(b) shows no spikes and this seems to indicate that the horizontal and spherical model are best used to avoid unsavoury spikes in the pressure fields.

The forces have been computed for the cases of sphere close to the bottom wall of the tank. We examine the force when the spherical model reaches near the bottom wall of the tank. The plots in Figure 9 seem to produce a good agreement for the force fields between the theoretical and the numerical simulations.

The six different images of mesh displacement is at different time steps (Figure 12). The first image corresponds to a time step of t = 0. In this the mesh is equally distributed around the corners of the swimming object. However in the third, fourth and fifth images one can see the mesh in its deformed form. The object deformation is identical to the way the mesh has deformed. On the left hand side there is compression in the negative x-direction and the right mesh elements are stretched. The meshes in the first image are structured while the remaining meshes are seen to be unstructured.

Force and torque are applied on the moving object from the global coordinate system and we get the linear and angular velocity of local variables. The slow motion of the worm is noticed by the change in velocities or motion of the local co-ordinate system.

The result shows that the x-direction mean displacement profile for global coordinate system and its derivatives displacement with respect to time profile of same coordinates system. The x-direction displacement refers to the moving object displacement per time period.

Imagine the object to be an ellipsoid moving by combinations of horizontal wave pattern travelling from one place to another with small amplitude and wave length. By using a combination of such oscillations for the motion, the swimmer can move in the opposite direction to that of the surface waves [34].

Figure 13 is a plot of the fluid velocity field directions near the swimming object. Here we see that the domain velocity direction has changed periodically with the object motion, except that they would be translated in the direction of the object wave motion.

Velocity profile observation for different motion cases

It has already been stated that the body motion is periodic if amplitude of motion is small [24] [39]. It’s a good approximation for periodic motion of ellipsoid when amplitude and wave length are both small. We use identical idea to observe the mean velocity profile and mean displacement in a periodic motion of the ellipsoid body.

The periodic motion is observed for different amplitudes and wave length to see mean velocity profile for different waveforms. Results from the comparison of these with the calculated velocity profile are shown in below. Velocity profile amplitude and wave length are default values in this observation. There seems to be a good agreement with the calculated values. The results are summarised in Table 1 below.

Observation:

From this velocity profile observation it is seen that the default mean displacement and mean velocity values agree very well. One, two and four time periods mean velocity values are almost the same implying that the wiggle motion is small and its moving very slowly in every period.

In the first observation, amplitude is reduced by 10% to and wave length remains fixed at. The x-direction mean velocity profile and mean displacement value are verified. They are almost identical to the mean velocity and mean displacement Figure 13 that are obtained in default calculation in Table 1.

In second observation, the amplitude has been reduced by 50% to but wavelength remains fixed at. This experiment results shows only for one and two time periods respectively. But the fourth time period motion could not be obtained with consistent initial values.

This simulation result is less efficient from the first observation mean velocity result and the mean velocity value decreases at small amplitudes. But at amplitude below 0.003 then there is an error suggesting a failure to find consistent initial values. In the third observation, we take the default amplitude value, and increase 75% the wave length to. One gets inefficient wiggling at the change of wave length alone and when the mean velocity value is very small. When we increase wave length over 1.5 then the solvers returns an error.

In fourth observation, we choose a smaller value for amplitude, and a larger value for wavelength. The velocity profiles remain insufficient and the mean displacement and mean velocity of wiggling object is not smooth. In this case we have changed both wave length and amplitude, and it is seen that in every period of this observation mean velocity value is very smaller than previous observations.