El-Dib Yusry O, Matoog R T
Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis, Cairo, Egypt.
J Colloid Interface Sci. 2005 Sep 1;289(1):223-41. doi: 10.1016/j.jcis.2005.03.054.
The present work deals with the gravitational stability of an electrified Maxwellian fluid sheet shearing under the influence of a vertical periodic electric field. The field produces surface charges on the interfaces of the fluid sheet. Due to the rather complicated nature of the problem a mathematical simplification is considered where the weak effects of viscoelastic fluids are taken into account. The solutions of the linearized equations of motion and boundary conditions lead to two simultaneous Mathieu equations with damping terms and having complex coefficients. Stability criteria are discussed through the assumption of symmetric and anti-symmetric deformations. The disappearance of surface charges from the interfaces obeys a certain relation derived in the marginal state. Furthermore, the case dealing with general deformation is discussed through marginal state analysis. The stability behavior in resonant and nonresonant cases are studied. In addition, the stability picture in the case of absence of the field frequency is studied. The numerical examination for stability showed that the relaxation time ratio plays a destabilizing influence in the case of anti-symmetric deformation or in the general deformation. The stabilizing effect for the relaxation time ratio is saved in the case of general deformation in the presence of the field frequency. In the later case the viscosity, the velocity, and the thickness parameter play a stabilizing influence. A dual role is readied for these parameters in the absence of the field frequency or in the anti-symmetric deformation. The field frequency still plays a destabilizing role in both cases.
本文研究了在垂直周期性电场影响下，带电麦克斯韦流体薄片剪切时的引力稳定性。该电场在流体薄片的界面上产生表面电荷。由于问题性质相当复杂，考虑了一种数学简化，其中考虑了粘弹性流体的微弱影响。线性化运动方程和边界条件的解导致了两个带有阻尼项且系数为复数的联立马蒂厄方程。通过对称和反对称变形的假设来讨论稳定性准则。界面上表面电荷的消失遵循在临界状态下导出的某种关系。此外，通过临界状态分析讨论了一般变形的情况。研究了共振和非共振情况下的稳定性行为。此外，还研究了不存在场频率情况下的稳定性情况。稳定性的数值检验表明，在反对称变形或一般变形的情况下，弛豫时间比起到了不稳定作用。在场频率存在的情况下，在一般变形的情况下，弛豫时间比的稳定作用得以保留。在后一种情况下，粘度、速度和厚度参数起到了稳定作用。在不存在场频率或反对称变形的情况下，这些参数起到了双重作用。在场频率存在或不存在的两种情况下，场频率仍然起到不稳定作用。
