Abstract
This study explores the influence of a magnetic field that is perpendicular to the interface on the Rayleigh-Taylor instability. The analysis focuses on electrically conducting and incompressible fluids, specifically a viscoelastic liquid in the upper region and a viscous fluid in the lower region. To investigate the problem, the irrotational flow theory is used, which accounts for viscosity through normal stress balance and does not consider tangential stresses or impose no-slip conditions at the rigid boundaries. Using Rivlin-Ericksen’s model for the viscoelastic liquid, a second-order dispersion relation is derived and numerically analyzed to determine the growth rate of the instability. Several plots are generated using the dispersion relation, and the stability of the interface is discussed for various physical parameters. The study reveals that the inertial forces contribute to the instability of the interface, while surface forces tend to stabilize it.
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Abbreviations
- \(\rho _{t1}\):
-
Density of upper fluid layer
- \(\rho _{t2}\):
-
density of lower fluid layer
- \(\mu _{t1}\):
-
viscosity of upper fluid layer
- \(\mu _{t2}\):
-
viscosity of lower layer fluid
- \(\mu '_{t1}\):
-
viscoelasticity of viscoelastic fluid
- \(\sigma \):
-
surface tension
- \(\psi _{t1}\):
-
maganetic potential function for lower fluid layer
- \(p_{t1}\):
-
pressure in viscoelastic fluid layer
- \(p_{t2}\):
-
pressure in viscous fluid layer
- \(\ell \):
-
surface elevation
- \({\textrm{h}}_{{\textrm{t1}}}\)::
-
thikness of Rivilin - Ericksen fluid
- \({\textrm{h}}_{{\textrm{t2}}}\)::
-
thikness of viscous fluid
- \(\omega \):
-
growth rate of disturbances
- \(H_{t1} ,H_{t2}\)::
-
maganetic field intensity in upper layer and lower layer respectively
- \(\psi _{t2}\):
-
maganetic potential function for viscous fluid layer
- \({\textrm{k }}\):
-
wave number
- \(\varphi _{t2}\):
-
potential function for viscous fluid layer
- \(\varphi _{t1}\):
-
potential function for viscoelastic fluid layer
- \(\tau \):
-
stress tensor
- \({\textrm{D}}\):
-
deformation rate tensor
- \(D_T\):
-
upper convected derivative
- \(q_{t1}\):
-
velosity of upper fluid layer
- \(q_{t2}\):
-
velosity of lower fluid layer
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Shukla, A.K., Awasthi, M.K., Dharamendra (2024). Impact of Vertical Magnetic Field on the Rivlin-Ericksen Fluid Interface: An Irrotational Flow Approach. In: Singh, J., Anastassiou, G.A., Baleanu, D., Kumar, D. (eds) Advances in Mathematical Modelling, Applied Analysis and Computation . ICMMAAC 2023. Lecture Notes in Networks and Systems, vol 953. Springer, Cham. https://doi.org/10.1007/978-3-031-56304-1_4
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