Dynamics of a thin film of viscoelastic fluid flowing down an inclined or vertical plane

IF 2.7 2区工程技术 Q2 MECHANICS

Journal of Non-Newtonian Fluid Mechanics Pub Date : 2024-04-22 DOI:10.1016/j.jnnfm.2024.105237

S. Dholey, S. Gorai

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引用次数: 0

Abstract

The stability characteristics of a thin film of viscoelastic (Walters’ $B^{'}$ model) fluid flowing down an inclined or vertical plane are analyzed under the combined influence of gravity and surface tension. A nonlinear free surface evolution equation is obtained by using the momentum-integral method. Normal mode technique and multiple scales method are used to obtain the results of linear and nonlinear stability analysis of this problem. The linear stability analysis gives the critical condition and critical wave number $k_{c}$ which include the viscoelastic parameter $Γ$ , angle of inclination of the plane $θ$ , Reynolds number $R e$ and Weber number $W e$ . The weakly nonlinear stability analysis that is based on the second Landau constant $J_{2}$ , reveals the condition for the existence of explosive unstable and supercritical stable zone along with the other two (unconditional stable and subcritical unstable) flow zones of this problem which is $3 (1 + 3 Γ) R e - 3 c o t θ - 4 R e W e k^{2}$ = 0. It is found that all the four distinct flow zones of this problem exist in $R e$ - $k$ -, $θ$ - $k$ - and $Γ$ - $k$ -plane after the critical value of $R e_{c},$ $θ_{c}$ and $Γ_{c}$ , respectively. A novel result of this analysis is that the film flow is stable (unstable) for a negative (positive) value of $Γ$ irrespective of the values of $R e$ and $θ$ , as for example, a solution of polyisobutylene in cetane, compared with the viscous $(Γ = 0)$ film flow case. Finally, we scrutinize the effect of $Γ$ on the threshold amplitude and nonlinear wave speed by depicting some numerical examples in supercritical stable as well as subcritical unstable zone of this problem.

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粘弹性流体薄膜在倾斜或垂直面上流动的动力学原理

分析了在重力和表面张力的共同作用下，粘弹性（Walters'B′模型）流体薄膜沿倾斜面或垂直面流动的稳定性特征。利用动量积分法得到了非线性自由表面演化方程。利用法向模态技术和多尺度法获得了该问题的线性和非线性稳定性分析结果。线性稳定性分析给出了临界条件和临界波数 kc，其中包括粘弹性参数 Γ、平面倾角 θ、雷诺数 Re 和韦伯数 We。基于第二朗道常数 J2 的弱非线性稳定性分析揭示了该问题的爆炸不稳定区和超临界稳定区以及其他两个（无条件稳定区和亚临界不稳定区）流动区的存在条件，即 3(1+3Γ)Re-3cotθ-4ReWek2 = 0。研究发现，在 Rec、θc 和 Γc 临界值之后，该问题的四个不同流动区域都分别存在于 Re-k、θ-k 和 Γ-k 平面上。这项分析的一个新结果是，与粘性（Γ=0）薄膜流动情况相比，无论 Re 和 θ 的值如何，当 Γ 为负值（正值）时，薄膜流动都是稳定（不稳定）的，例如聚异丁烯在十六烷中的溶液。最后，我们通过描述该问题超临界稳定区和亚临界不稳定区的一些数值示例，仔细研究了 Γ 对阈值振幅和非线性波速的影响。

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来源期刊

Journal of Non-Newtonian Fluid Mechanics 物理-力学

CiteScore

5.00

自引率

19.40%

发文量

109

审稿时长

61 days

期刊介绍： The Journal of Non-Newtonian Fluid Mechanics publishes research on flowing soft matter systems. Submissions in all areas of flowing complex fluids are welcomed, including polymer melts and solutions, suspensions, colloids, surfactant solutions, biological fluids, gels, liquid crystals and granular materials. Flow problems relevant to microfluidics, lab-on-a-chip, nanofluidics, biological flows, geophysical flows, industrial processes and other applications are of interest. Subjects considered suitable for the journal include the following (not necessarily in order of importance): Theoretical, computational and experimental studies of naturally or technologically relevant flow problems where the non-Newtonian nature of the fluid is important in determining the character of the flow. We seek in particular studies that lend mechanistic insight into flow behavior in complex fluids or highlight flow phenomena unique to complex fluids. Examples include Instabilities, unsteady and turbulent or chaotic flow characteristics in non-Newtonian fluids, Multiphase flows involving complex fluids, Problems involving transport phenomena such as heat and mass transfer and mixing, to the extent that the non-Newtonian flow behavior is central to the transport phenomena, Novel flow situations that suggest the need for further theoretical study, Practical situations of flow that are in need of systematic theoretical and experimental research. Such issues and developments commonly arise, for example, in the polymer processing, petroleum, pharmaceutical, biomedical and consumer product industries.