**Table 1.**

*Physical properties of silicone oil viscosities used.*

force that could resist motion. Viscous forces are forces acting due to the viscous nature of the liquid. Hence, from the plot, it can be inferred that

viscous forces have a domineering effect over inertia forces [41].

*The Effect of Liquid Viscosity on the Rise Velocity of Taylor Bubbles in Small Diameter Bubble…*

The forces acting on the Taylor bubble are shown in **Figure 12**. These forces have an influence on its rise velocity. Surface tension force helps to hold the bubbles together due to the cohesive force existing between them. This accounts for why 'surface tension force' was indicated at the centre of the bubble in **Figure 12**. Based on the proposition of White and Beardmore [22], the effect of surface tension force can be neglected when Eotvos number is greater than 70. So, since for all the viscosities considered, Eotvos number is greater than 70, its effect on the rise

The dominating effect of viscous forces over inertia forces can be further con-

From the various viscosities considered, the dimensionless property numbers are

It will be observed that as viscosity increases, Morton number, *Mo* increases while the Eotvos number, *Eo* decreases, which culminates in the decrease of the dimensionless inverse viscosity, *Nf*. This confirms the proposition of Fabre and Line [42]. As the dimensionless inverse viscosity decreases, viscous effect

White and Beardmore [22] proposed that viscous effects come into play when

*L gD***<sup>3</sup>** *μ***2** *L*

. i.e.

<sup>&</sup>lt; **<sup>3</sup>** � **<sup>10</sup><sup>5</sup>** (23)

firmed using the Inverse dimensionless viscosity according to White and

the square of dimensionless inverse viscosity, *Nf* is less than **<sup>3</sup>** � **<sup>10</sup><sup>5</sup>**

*Variation of slug Reynolds number with superficial gas velocity for various liquid viscosities.*

*Nf* **<sup>2</sup>** <sup>¼</sup> *<sup>ρ</sup>***<sup>2</sup>**

velocity of Taylor bubbles can be neglected.

*DOI: http://dx.doi.org/10.5772/intechopen.92754*

Beardmore [22].

dominates [31].

**Figure 11.**

**211**

given as follows (**Table 2**).

**Figure 9.** *Variation of structure velocity with superficial gas velocity at various viscosities.*

#### **Figure 10.**

*Periodic structures in the slug as obtained from ECT Plot3d image reconstruction software (image display in single axial slice mode) at a superficial gas velocity of 0.02 m/s.*

the slug Reynolds number for all the viscosities are small, viscous effect will be dominant.

ii. Dominating effect of viscous forces over inertia forces: Inertia forces are forces acting due to motion of bubbles through the liquid. It opposes any *The Effect of Liquid Viscosity on the Rise Velocity of Taylor Bubbles in Small Diameter Bubble… DOI: http://dx.doi.org/10.5772/intechopen.92754*

> force that could resist motion. Viscous forces are forces acting due to the viscous nature of the liquid. Hence, from the plot, it can be inferred that viscous forces have a domineering effect over inertia forces [41].

The forces acting on the Taylor bubble are shown in **Figure 12**. These forces have an influence on its rise velocity. Surface tension force helps to hold the bubbles together due to the cohesive force existing between them. This accounts for why 'surface tension force' was indicated at the centre of the bubble in **Figure 12**. Based on the proposition of White and Beardmore [22], the effect of surface tension force can be neglected when Eotvos number is greater than 70. So, since for all the viscosities considered, Eotvos number is greater than 70, its effect on the rise velocity of Taylor bubbles can be neglected.

The dominating effect of viscous forces over inertia forces can be further confirmed using the Inverse dimensionless viscosity according to White and Beardmore [22].

From the various viscosities considered, the dimensionless property numbers are given as follows (**Table 2**).

It will be observed that as viscosity increases, Morton number, *Mo* increases while the Eotvos number, *Eo* decreases, which culminates in the decrease of the dimensionless inverse viscosity, *Nf*. This confirms the proposition of Fabre and Line [42]. As the dimensionless inverse viscosity decreases, viscous effect dominates [31].

White and Beardmore [22] proposed that viscous effects come into play when the square of dimensionless inverse viscosity, *Nf* is less than **<sup>3</sup>** � **<sup>10</sup><sup>5</sup>** . i.e.

$$\left(\mathbf{N}\_f\right)^2 = \frac{\left(\rho\_L^2\right)\mathbf{g}\mathbf{D}^3}{\mu\_L^2} < 3 \times 10^5\tag{23}$$

**Figure 11.** *Variation of slug Reynolds number with superficial gas velocity for various liquid viscosities.*

the slug Reynolds number for all the viscosities are small, viscous effect will

ii. Dominating effect of viscous forces over inertia forces: Inertia forces are forces acting due to motion of bubbles through the liquid. It opposes any

*Periodic structures in the slug as obtained from ECT Plot3d image reconstruction software (image display in*

be dominant.

*single axial slice mode) at a superficial gas velocity of 0.02 m/s.*

**Figure 10.**

**210**

**Figure 9.**

*Variation of structure velocity with superficial gas velocity at various viscosities.*

*Vortex Dynamics Theories and Applications*

*4.1.1 Drag force, FD*

of a body through it.

interpreted as follows:

**Figure 13(b)**.

**Figure 13.**

**213**

From Stokes law, drag force is given as:

*DOI: http://dx.doi.org/10.5772/intechopen.92754*

based on the fact that the flow is laminar.

Drag coefficient, *CD* is given as:

increase in drag force.

decrease in the rise velocity [42].

*Drag force versus (a) superficial gas velocity, (b) void fraction.*

*FD* ¼ **3***πμLVd* (24)

*Re* (25)

(26)

Drag force is the force due to the resistance provided by the fluid to the motion

From Eq. (24), Drag force is directly proportional to viscosity of the fluid.

*The Effect of Liquid Viscosity on the Rise Velocity of Taylor Bubbles in Small Diameter Bubble…*

*CD* <sup>¼</sup> **<sup>24</sup>**

*FD* <sup>¼</sup> **<sup>72</sup>***πμ***<sup>2</sup>**

Eq. (26) is used to compute drag force, *FD* which is plotted against superficial gas velocity for all viscosities considered as shown in **Figure 13(a)**. This can be

i. Drag force is directly proportional to viscosity, which confirms Eq. (26) according to Stokes law. Drag force is the force due to the resistance provided by the viscous nature of the silicone oil fluids to the motion of the large bubble through them. So, as viscosity increases, there is a linear

ii. The increase in drag force shows an opposition to the motion of the bubbles through the liquid acting parallel to direction of relative motion, hence causing the bubbles to rise at much slower rate. This obviously leads to a

To further confirm the effect of drag force, drag force was plotted against void fraction, which is the volume fraction of gas in the gas–liquid mixture as shown in

*L CDρ<sup>L</sup>*

On substituting Eq. (25) into (24) given that *Re* ¼ *ρLVd=μL*,

**Figure 12.**

*Forces acting on Taylor bubble (surface tension force helps to hold the bubbles together).*


#### **Table 2.**

*Morton, Eotvos and inverse dimensionless numbers.*

The square of the inverse square dimensionless viscosity for 5, 100, 1000 and 5000 mPa s are 41,012,747, 114051.4, 1152.37 and 46.09481, respectively. Since 100 mPa s, 1000 mPa s and 5000 mPa s satisfy the condition of *Nf* **<sup>2</sup>** <sup>&</sup>lt; **<sup>3</sup> <sup>10</sup><sup>5</sup>** , viscous effect dominates. This dominating effect of viscous force over inertia force possibly causes a decrease in structure velocity with an increase in viscosity. This can be further confirmed by obtaining a relationship between drag force and viscosity taking superficial gas velocity as a parameter.

*The Effect of Liquid Viscosity on the Rise Velocity of Taylor Bubbles in Small Diameter Bubble… DOI: http://dx.doi.org/10.5772/intechopen.92754*
