Localised radial spots on the free surface of a ferrofluid

铁磁流体是载体流体中的铁颗粒的混合物，当施加磁场时，导致该流体的自由表面被操纵。模式出现，如斑点或六边形阵列的脊和证明这些模式的存在数学是一个复杂的过程。局部径向斑点，单一的轴对称峰，出现从平坦的铁磁流体，已被证明存在实验，通过局部增加流体中的磁场。这些斑点在电场减小后仍然存在，因此这些斑点的存在是否与它们可以自发地从平坦状态出现的潜在机制有关仍然是一个谜。该项目的主要目标是严格证明局部径向斑点的存在，在铁磁流体的问题，使用数学分析，以确定是否出现斑点从平面状态或另一种状态。本项目的新奇在于使用目前开发的技术，如径向中心流形减少，渐近技术和线性稳定性理论来回答上述问题。然而，这是不平凡的，因为上述技术中的一些需要进一步开发以结合问题的几何形状。一旦回答了上述问题，该项目的进一步扩展可能包括研究这些斑点中的两个或更多个在形成后如何相互作用。本研究项目符合EPSRCs数学科学主题下的数学分析和连续介质力学研究领域

Ferrofluids are a mixture of iron particles within a carrier fluid, causing the free surface of this fluid to be manipulated when an magnetic field is applied. Patterns emerge, such as spots or a hexagonal array of ridges and proving the existence of these patterns mathematically is a complex process. Localised radial spots, single axisymmetric peaks that emerge from the flat ferrofluid, have been shown to exist experimentally, by locally increasing the magnetic field in the fluid. These spots persist after the field is reduced, and so it remains a mystery if the existence of these spots is related to an underlying mechanism where they can spontaneously emerge from the flat state. The main objectives of the project areto rigorously prove the existence of localised radial spots in the ferrofluid problem using mathematical analysisto determine whether the spot emerges from the flat surface state or another state. The novelty of the present project lies in the using of currently developed techniques, such as radial centre-manifold reduction, asymptotic techniques and linear stability theory to answer the above questions. However, this is non-trivial as some of the above mentioned techniques need to be developed further to incorporate the geometry of the problem. Further extensions of the project once the above questions have been answered could include looking at how two or more of these spots interact with each other once they form. This project fits into the EPSRCs Mathematical Sciences Theme under the research areas of Mathematical Analysis and Continuum Mechanics