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ENQUIRE PROJECT DETAILS BY GENERAL PUBLIC |
| Project Details |
| Funding Scheme : | General Research Fund | ||||||||||||||||||||
| Project Number : | 14303420 | ||||||||||||||||||||
| Project Title(English) : | Modelling and analysis of diffuse interface models for two-phase micropolar fluid flows | ||||||||||||||||||||
| Project Title(Chinese) : | 建模和分析兩相微極性流體的擴散界面模型 | ||||||||||||||||||||
| Principal Investigator(English) : | Prof Lam, Kei Fong | ||||||||||||||||||||
| Principal Investigator(Chinese) : | |||||||||||||||||||||
| Department : | Department of Mathematics | ||||||||||||||||||||
| Institution : | Hong Kong Baptist University | ||||||||||||||||||||
| E-mail Address : | akflam@hkbu.edu.hk | ||||||||||||||||||||
| Tel : | |||||||||||||||||||||
| Co - Investigator(s) : |
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| Panel : | Physical Sciences | ||||||||||||||||||||
| Subject Area : | Mathematics | ||||||||||||||||||||
| Exercise Year : | 2020 / 21 | ||||||||||||||||||||
| Fund Approved : | 555,754 | ||||||||||||||||||||
| Project Status : | Completed | ||||||||||||||||||||
| Completion Date : | 30-6-2024 | ||||||||||||||||||||
| Project Objectives : |
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| Abstract as per original application (English/Chinese): |
Micropolar fluids are among the simplest cases of fluids with microstructures, where each fluid particle has its own internal rotations. Examples include ferrofluids, blood flows, bubbly liquids and liquid crystals, all of which play significant and important roles in various industries and also in the human body. Therefore, it becomes essential for scientists to effectively capture the behaviour of these kinds of fluids with the appropriate mathematical models, and to better analyse and simulate their flow dynamics.
The classical Navier-Stokes equations that are well-studied by the mathematical community is not a suitable description for micropolar fluids, as the internal rotations of each fluid particle are not accounted for. Through the works of A. Cemal Eringen and his coworkers, we now have mathematical models that extend the Navier-Stokes equations to capture these internal rotations. However, in many real world applications we often encounter mixtures of multiple kinds of fluids. In these situations we have to further extend previous models to account for the interactions between individual fluids that can influence the gross motion of the mixture.
The main aims of this proposal are to develop new mathematical models for mixtures of micropolar fluids, study their mathematical properties, and derive stable and efficient fully discrete numerical schemes for simulations. The key ingredient is the diffuse interface methodology which offers a model that is amenable to further mathematical analysis and allows us to address the existence of weak solutions and the stability of discrete solutions. Also included in the modelling is the dynamics of the contact line that is created when two or more fluids meet at the same solid boundary. This describes wetting and dewetting processes crucial in modern-day applications such as inkjet printing and lubrication. We believe the proposed model and the accompanying analytical and numerical ideas can provide a theoretical foundation for many industrial and biological processes that involve mixtures of non-standard fluids. 微極性流體是具有微觀結構的流體的最簡單情況，其中每個流體粒子都有自己的內部旋轉。例子包括鐵磁流體，血液流動，氣泡狀液體和液晶，它們在各個行業以及人體中都起著重要的作用。因此，對於科學家來說，用適當的數學模型有效地捕獲這些流體的行為，並更好地分析和模擬其流動動力學就變得至關重要。 數學界精心研究的經典Navier–Stokes方程不適用於微極性流體，因為它沒有考慮每個流體粒子的內部旋轉。通過A. Cemal Eringen和他的同事的工作，我們現在有了擴展Navier-Stokes方程以捕獲這些內部旋轉的數學模型。但是，在許多實際應用中，我們經常會遇到多種流體的混合物。在這些情況下，我們必須進一步擴展先前的模型，以考慮可能影響混合物總運動的各個流體之間的相互作用。 這個建議的主要目的是為微極性流體的混合物開發新的數學模型，研究其數學性質，並推導穩定有效的全離散數值模擬方案。關鍵因素是擴散界面方法，該模型提供了可用於進一步數學分析的模型，並使我們能夠解決弱解的存在和離散解的穩定性。建模中還包括當兩種或多種流體在同一固體邊界處相遇時創建的接觸線動力學。這描述了在現代應用中至關重要的潤濕和反潤濕過程，例如噴墨打印和潤滑。我們相信，提出的模型以及隨附的分析和數值思想可以為涉及非標準流體混合物的許多工業和生物過程提供理論基礎。 |
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| Realisation of objectives: | The first three objectives have been completely achieved, where we utilize a stable discretization scheme in Objective 3 and prove the convergence of discrete solutions to fulfil Objective 2. At present, the PI is studying various interesting asymptotic limits of the new diffuse interface model, such as the sharp interface limit mentioned in Objective 4, as well as the more interesting nonlocal-to-local asymptotics that connects nonlocal variants of the model derived in Objective 1 to their local counterpart. As the latter is more novel than the former, much of the PI's attention has been focused on this nonlocal-to-local aspects. Hence, our results summarized in the 3rd preprint can be regarded as 70% fulfillment of Objective 4 as stated in the original proposal. | ||||||||||||||||||||
| Summary of objectives addressed: |
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| Research Outcome | |||||||||||||||||||||
| Major findings and research outcome: | We extend the current state-of-the-art and developed new diffuse interface models describing the motion of a mixture of two incompressible and immiscible micropolar fluids. These differ from traditional mathematical descriptions of Newtonian fluids by the addition of a new set of equations for the micro-rotation. Utilizing recent results established by many authors, we study the solvability of the new model, and in particular, we include in our analysis the important case of moving contact lines, which appear in industrial applications such as inkjet printing and coating processes. Furthermore, our results pave the way for a theoretical comparison between micropolar and Newtonian fluid mixtures, where key estimates capturing the gap between these two different types of models are derived. | ||||||||||||||||||||
| Potential for further development of the research and the proposed course of action: |
An interesting direction is to consider optimal control of micropolar fluids. The micropolar description serves as a simplified setting for magnetic-influenced ferrofluids, which are applied in versatile settings, such as sealing, coolants, lubricants, and in display devices. Hence, the analytical results developed from the project provide a stepping stone in the study of optimal control involving micropolar fluid mixtures. Another direction worthy of pursuit is to develop effective models for micropolar flow in thin domains, which can be applied for example to curved electronic displays. The idea is to employ asymptotic expansions to derive the effective equations of a two dimensional model that captures the dynamics of the flow when the height of the domain is significantly smaller than all other length scales. Such models would also generalize the current state-of-the-art for Darcy--Cahn--Hilliard systems. These topics would make very good PhD research projects. | ||||||||||||||||||||
| Layman's Summary of Completion Report: | Micropolar fluids are fluids with internal rotations, and appear for instance in bubbly liquids, blood flow and ferrofluids. In industrial and biological settings, mixtures of micropolar fluids are often present and so it is essential to develop mathematical models that can accurately capture the relevant physical phenomena. In this project we developed new mathematical models for mixtures of micropolar fluids, and our theoretical investigations demonstrated that these models are mathematically sound. This opens up further studies in related optimal control problems and development of efficient numerical schemes. Our ideas developed in this project can also be used to tackle more challenging and interesting problems prevalent in other areas of scientific applications. | ||||||||||||||||||||
| Research Output | |||||||||||||||||||||
| Peer-reviewed journal publication(s) arising directly from this research project : (* denotes the corresponding author) |
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| Recognized international conference(s) in which paper(s) related to this research project was/were delivered : |
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| Other impact (e.g. award of patents or prizes, collaboration with other research institutions, technology transfer, etc.): |
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| SCREEN ID: SCRRM00542 |