The calculations reveal that a bubble in a uniaxial tension flow extends to infinity if the Weber number is greater than a critical value (W > Wc). The steady-state solutions suggest the existence of a limit point at a critical value of the Weber number.
Introduction
Finally, in the present paper, we consider a third alternative, based on a finite-difference approximation of the equations of motion applied to a boundary-fit orthogonal curvilinear coordinate system. The fluid mechanics part of the problem is therefore to obtain solutions of the Navier-Stokes equations using a finite-difference approximation in the boundary-fit (ξ,q) coordinates.
Solution algorithm
Starting from this initial state, simultaneously solve the discretized version of the governing equations (1), (4) and (5) subject to the boundary conflict. Starting from the predicted initial guess for the solution at the (n+l)th time step, iteratively solve the discretized version of the governing equation.
Prediction of the (n+l)-st time-step solution
Predict the solution for t = (n + l)∆i, (n ≥ 1) using the previous two solutions of the time step (Predictive Step : It should be noted here that the predicted solution does not satisfy all equations and boundary conditions, and that this step does not change the final result at all for each time step).
Iterative ADI method for the corrector step
Application to the problem of unsteady bubble deformation
To illustrate, we drew a coordinate system for the entire bubble, but in the real calculation we only considered the quadrant x > 0, σ ≥ 0. For the discrete solution of the numerical analysis, we can define the remainder as (w denotes the discrete solution).
Introduction to unsteady bubble deformation in a uniaxial
We found that the oscillation frequency decreases with increasing Weber number, for example, in the case of the prince. In this paper, we study the same problem by means of a full and unsteady numerical analysis to obtain accurate results for higher Weber numbers and to find the zero frequency point.
Statement of the problem
Of course, the analysis of small amplitude fluctuations is based on small (or zero) W approximations of the stationary form, and the true zero frequency should be expected to be different from the value found by the perturbation analysis or, in the extreme, that the zero frequency point may not exist at all. We use the equivalent radius α of the bubble as a characteristic length scale, the product Ea as a characteristic velocity scale, and E~1 as a characteristic time scale.
Unsteady deformation of the bubble for slightly supercritical
3-5, the bubble returns to the steady state only if the initial deformation does not exceed a certain critical size. When the Weber number is reduced by f1∕2 = 1∙8, the bubble returns to the stable shape obtained by Ryskin and Leal15 via their stable code.
Unsteady deformation of a bubble in start-up from rest
If the bell shape is sufficiently deformed at some point, either due to stretching in a flow with W >. Waa, or due to the initial overshooting of the shape when starting a sphere, the bubble will stretch at lower values of W.
Sudden removal of external flow after very large deforma
This result for W = 0.02 is consistent with the common perception that an elongated bubble or inviscous droplet with a waist will extend indefinitely in an extensional flow at low Reynolds number (cf. Hinch2δ). Specifically, the existence of a waist in an initially thin bubble does not lead to the bursting of the very small Weber.
Oscillatory motion of a bubble in an inviscid straining flow
Conclusion
Unsteady bubble deformation near critical Weber number (half length and shape for R = 10) (asymptotic curve is given by dlχ∕i∣dt = Z1∕2 with a suitable initial condition). Steady and unsteady deformation of a bubble in a biaxial shear flow is considered for Reynolds numbers in the range 0 ≤ R ≤ 200.
Steady-state shape and oscillation frequency of a bubble in
The assumption of the existence of the stable form, ∕χ, implies the existence of a corresponding velocity potential, ≠ι, so that the pair (∕ι, ≠ι) represents a solution of (2) − (5) for the given W (in fact the velocity potential can be determined uniquely for a given shape from (2), (3) and (5), but the ιmiqueness is not needed in this proof). The oscillation of a bubble in a uniaxial voltage current was studied in detail by Kang and Leal.11 The main result in the paper is that the oscillation frequency of the primary mode (n = 2) decreases as the Weber number increases; i.e.,.
Estimation of the marginal Reynolds number .............................∙τ
The invariance of the steady-state shape in the potential flow boundary implies an interesting result in steady-state bubble deformation for a viscous fluid as the Reynolds number increases from zero to infinity, and this will be discussed in the following subsection. Since the vortex is solenoidal (V ∙ ω = 0), the creeping flow deforms to a prolate shape in the uniaxial extensional flow for arbitrary E > 0. It follows that a bubble in a biaxial extensional flow must deform to an oblate shape in the creeping flow limit.
As can be seen in (16), to accurately estimate the pressure correction on the surface of a spherical bubble, the vorticity distribution in the region outside the bubble is required, which requires us to solve the vorticity equation. To obtain the first-order strain in W, we assume that the surface of the bubble is described by the form
Vorticity stretching and the vorticity distribution in the wake
Numerical scheme
In this analysis, the transient algorithm described in I was used only by modifying the difference scheme for the vorticity equation as described below (note that the transient algorithm given in I with time step ∆f = oo degenerates into a steady-state algorithm Ryskina and Leala16). However, we found that this non-conservative difference scheme results in an unconditionally unstable nu.
Non-conservative difference scheme for the vorticity equa
Furthermore, let us assume that the velocity field is given by the potential flow solution around a spherical bubble, 426) where and of "ψ" refer to the uniaxial and biaxial deformation flows, respectively. The necessary condition for stability is satisfied for all R in the case of uniaxial deformation flow, because α⅛⅛ < 0 for all k as we can see in (47) of ψ is for the uniaxial deformation flow). However, the necessary condition for stability is violated as R → ∞ in the case of biaxial deformation flow, because.
Indeed, when a 40 × 40 grid system was used in our numerical calculation, the ADI scheme with a non-conservative difference scheme failed to converge for R > 5 in the case of biaxial deformation flow, while convergent solutions were obtained for all R up to the maximum value considered ( 100) in the uniaxial deformation flow problem. The question is how we can avoid this problem while maintaining second-order accuracy in the spatial discretization.
Modified difference scheme for the vorticity transport equa
Results of steady state analysis
So far we have discussed the limiting behavior of the solution to the biaxial deformation flow problem in the limit R. In general, if the effect of vorticity due to viscosity vanishes as R → ∞, then the potential flow solution corresponds to the actual flow in the limit. In contrast to the biaxial flow case, the potential flow solution is the limiting solution as R → oo in the uniaxial flow case.
Consequently, there is a smooth change of curvature as R → ∞, and this implies that the potential flow solution corresponds to the limiting solution of the real flow in the boundary R. These regions of high curvature with the increase of R may explain why the potential flow solution is not a uniformly valid constraint solution of the real flow for all W in the biaxial tension flow case.
Unsteady deformation in potential flow
81 - . of the Weber numbers considered in the present study, the present unsteady analysis is limited to very few cases. The oscillation of a bubble in a biaxial deformation flow was also calculated for the case of W = 1 in the potential flow boundary. In Fig. 10, the radius of a bubble in the axial direction is plotted as a function of surface tension-based time scale t, and it is compared with that of a bubble in a uniaxial deformation flow.
For the uniaxial flow problem, the initial condition is also the steady-state solution for W = 2.7. As we can see in figure 10, the phases are different, but the frequencies are almost the same for both cases.
Unsteady deformation in high Reynolds number flow
Perturbation solution up to 0(ε3)
Estimation of the critical Weber number
Strictly speaking, the spherical shape only represents the limit of the steady-state solution for zero Weber number. Therefore, there is no O(∖∕W) term in the asymptotic expression for the oscillation frequency as a function of the Weber number. Despite the fact that the steady-state normal stress balance is not ex.
In Fig. 8 , we plot the square of the frequency for the n = 2 (ω,22) mode as predicted by the perturbation analysis, together with the numerical results of Kang and Leal (1987) . The present study also shows that the drag coefficient up to O(R~1) depends only on the 0 (1) vorticity distribution at the bubble surface and is independent of the vorticity distribution in the liquid. Examination of the normal stress distribution on the bubble surface shows that the drag coefficient at 0 (P-1) is independent of the vorticity separation.
In (16e) R' is the shape function of the bubble surface, which is given by Eq.
The Ψ-perturbation solution as a limiting form of the P2-
Small amplitude oscillation about the steady-state shape
- Oscillatory motion of a bubble about the spherical shape
- Oscillation of a slightly deformed bubble for W«l
- The influence of weak viscous effects on bubble oscillation
Discussion of results
Conclusions
Viscous pressure correction and drag coefficient
Before moving on to the specific problem of using this result in equations. 4) and (6) to obtain an expression for the drag coefficient for a spherical bubble. Therefore pυ(l,0,i) = O(∆_1); that is, pv adds to the resistance on the bell in the same order as the viscous voltages. Returning to the problem of a bubble in a flowing stream, we have the basic irrotational flow solution as.
Rewriting (14) to exploit the fact that only the Pi(cos0) mode contributes to the feature we have. The fact that the integrand in (15) vanishes for all r means that the viscous drag is completely independent of the vorticity distribution in the region outside the bubble up to O(P-1), even though the vorticity distribution and viscous pressure corrections are both complicated functions of r and θ.
Comparison with numerical analysis results
Since the viscous drag is only due to the Pi (cos 0) components of the normal stress, the P1(cos θ) components obtained numerically are also compared with the theoretical prediction of the present work. The drag coefficient Cd = 48/R, which was obtained by the dispersion method of Levich,1 was re-derived via direct integration of the total normal stress over the bubble surface. This fact is completely consistent with the result obtained earlier by the dissipation method, where the vorticity in the region outside the bubble is neglected for the calculation of the drag coefficient for .
Jackson (1963) included cumulative acceleration terms in the equation for the particle motion and he further allowed for local variation of space in the visibility of the bubble. In (il) F is the function that describes the bubble shape by F(x — x0,i) = 0, where xo(i) denotes the position vector of the center of mass.
Conclusion
- Governing equations and boundary conditions
- Dimensionless equations with respect to the moving frame of
- Steady-state solution
- Time-dependent kinematics (unsteady deformation)
- Normal stress imbalance for a spherical voidage bubble at a
- Inertial effect on the unsteady deformation
Conclusion
