Instability Analysis of Rotating Rivlin–Ericksen Fluids in a Magnetic Field

Available online www.ejaet.com European Journal of Advances in Engineering and Technology, 2021, 8(5):93-96 Research Article ISSN: 2394 - 658X 93 Instability Analysis of Rotating Rivlin–Ericksen Fluids in a Magnetic Field Ravi Prakash Mathur Department of Mathematics S.G.S.G. Government College, Nasirabad, Ajmer (India) [email protected] _____________________________________________________________________________________________ ABSTRACT This work presents a linear stability analysis of a rotating layer of a visco-elastic fluid modelled by the Rivlin– Ericksen fluid constitutive framework subject to a uniform vertical magnetic field. The fluid rotates at angular velocity Ω about the vertical axis, and is heated from below producing a buoyancy‐driven destabilising temperature gradient. The governing equations incorporate visco‐elastic (second‐order) effects, rotation through Coriolis and centrifugal terms, and magnetohydrodynamic (MHD) coupling. Using the Boussinesq approximation and normal‐mode perturbations, we derive the dispersion relation and identify the critical parameters (Rayleigh number Ra, Taylor number Ta, Hartmann number Ha, visco‐elastic parameter S) for onset of instability. Results show that rota-tion and magnetic field can either stabilise or destabilise the system depending on visco‐elasticity strength and wave‐number; specifically, for moderate S the magnetic field increases the critical Ra, but for strong visco‐elasticity the coupling of elastic stresses and Coriolis forces may lead to oscillatory modes that reduce the threshold. The in-terplay of rotation, elasticity and magnetic effects is discussed with relevance to geophysical and industrial flows of visco‐elastic fluids in rotating MHD contexts. Keywords: Rivlin–Ericksen fluid, visco‐elasticity, rotating fluid, magnetohydrodynamics, linear instability, Rayleigh–Bénard convection _____________________________________________________________________________________________ INTRODUCTION The study of hydrodynamic stability has remained one of the central themes in fluid mechanics due to its significance in understanding the onset of turbulence, pattern formation, and transition phenomena in various natural and industrial processes. When a system is subjected to perturbations, the stability of its equilibrium state determines whether the disturbances will grow or decay over time. The pioneering work of Rayleigh [1] laid the foundation for the analysis of stability in stratified fluids, while Chandrasekhar [2] provided a comprehensive account of hydrodynamic and hydromagnetic stability theories. In many physical situations, fluids do not behave as Newtonian but exhibit viscoelastic properties. The Rivlin– Ericksen fluid model is a subclass of viscoelastic fluids that captures the effects of normal stress differences arising due to deformation history, making it more realistic for materials such as polymeric solutions, lubricants, and biological fluids. Several researchers have investigated the stability characteristics of Rivlin–Ericksen fluids in different physical configurations. For instance, Sharma and Sharma [3] examined the thermal instability of Rivlin– Ericksen fluids, while Bhatia and Steiner [4] studied the role of elasticity in altering the critical conditions for instability. The influence of rotation introduces additional complexity through the Coriolis force, which tends to stabilize or destabilize the system depending on the direction and intensity of rotation. Rotating viscoelastic fluids are of particular interest in geophysical and astrophysical contexts such as the dynamics of Earth’s mantle, accretion disks, and planetary interiors, where both elasticity and rotation play significant roles. Moreover, the presence of a magnetic field adds another stabilizing mechanism due to the Lorentz force, leading to the so-called magnetohydrodynamic (MHD) effects. Mathur and Kumar [5] have analyzed the instability of two rotating oldroydian viscoelastic superposed fluids. The coupling between magnetic field, viscoelasticity, and rotation is crucial in understanding the stability of conducting fluids in astrophysical and engineering applications, including liquid metal flows, MHD generators, fusion reactors, and lubrication systems [6, 7-8]. Mathur RP Euro. J. Adv. Engg. Tech., 2021, 8(5):93-96 94 In recent years, attention has been directed toward the combined effects of rotation and magnetic fields on the stability of non-Newtonian fluids due to their practical importance in modern technological processes. Studies have shown that rotation can either enhance or suppress the destabilizing influence of magnetic fields depending on the fluid parameters and boundary conditions [9-10]. However, a systematic analysis of the stability of rotating Rivlin– Ericksen fluids in a magnetic field has not been explored extensively, despite its theoretical and practical relevance. The aim of this paper is to analyse how rotation and magnetic field affect the onset of instability in a horizontal layer of a Rivlin–Ericksen fluid heated from below. We adopt a normal‐mode approach under Boussinesq conditions and derive a dispersion relation in terms of dimensionless parameters. We then analyse the dependence of the critical Rayleigh number on Taylor number (rotation), Hartmann number (magnetic field), and the second‐order visco-elastic parameter. In doing so we expand on previous works of rotating visco‐elastic fluids without magnetic field, and stationary visco‐elastic MHD flows, by unifying all three effects. PHYSICAL MODEL AND GOVERNING EQUATIONS Basic Configuration Consider an infinite horizontal layer of incompressible visco‐elastic fluid of thickness d, bounded by rigid, stress‐free horizontal plates at z = 0 and z = d. The lower plate is at temperature Tbottom, upper plate at Ttop, so that a constant vertical temperature gradient β = (dT0/dz)< 0 is established. Gravity g acts downward (negative zdirection). The layer is rotating uniformly about the vertical axis with angular velocity Ω. A uniform vertical magnetic field B0 is imposed along the z-axis. The fluid is electrically conducting, with magnetic permeability μ₀, electrical conductivity σ_e, and viscosity μ. The visco-elastic fluid is modelled by a second‐order fluid constitutive relation (Rivlin–Ericksen type) characterised by a non‐Newtonian parameter α (sometimes denoted S in dimensionless form). Governing Equations Momentum equation: ρ (d v / dt + 2Ω × v + Ω × (Ω × r)) = −∇p + ρ αT g T'  + μ ∇²v + α ∇² (d v / dt)+ (1/μ₀) (∇×B) × B0 (1) Induction equation: d b / dt = ∇×(v × B0) + η ∇²b (2) Thermal energy equation: d T'/dt + v · ∇T0 = κ ∇²T' (3) Continuity condition: ∇·v = 0, ∇·b = 0 (4) Here, v is perturbation velocity, T’ the temperature perturbation, b the magnetic field perturbation, μ the dynamic viscosity, η the magnetic diffusivity, αT the thermal expansion coefficient, and α the visco‐elastic parameter (second‐order fluid constant). The term involving α in the momentum equation models the extra stress due to fluid elasticity. The Coriolis term 2Ω × v and centrifugal term Ω × (Ω × r) appear because of rotation. Non‐dimensionalisation and Parameters We scale length by d, time by d²/κ, velocity by κ/d, magnetic field by B0, temperature by β d, and pressure by ρ κ ν / d². With these scales we define dimensionless parameters: Rayleigh number: Ra = (ρ α_T g β d⁴)/(μ κ) Taylor number: Ta = (4 Ω² d⁴)/(ν²) Hartmann number: Ha = (B0 d)/(√(μ₀ ρ ν)) Visco‐elastic parameter: S = α ν / d² Magnetic Prandtl number: Pm = ν / η Here ν = μ/ρ is the kinematic viscosity. The non-dimensional equations reduce to a system in terms of Ra, Ta, Ha and S. The basic state is at rest, and we apply stress-free and perfectly conducting magnetic boundary conditions at z = 0, 1. LINEAR STABILITY AND NORMAL-MODE ANALYSIS We assume normal‐mode perturbations of the form: (v, T', b) = [W(z), Θ(z), H(z)] exp[i(kx x + ky y) + σ t] (5) where k = √(kx² + ky²) is the dimensionless horizontal wave number, and σ the growth rate. Substituting into the linearised equations and eliminating pressure, temperature and magnetic perturbations, one obtains a coupled eigenvalue problem for W(z). For marginal stability σ = 0, the dispersion relation may be cast approximately (for moderate Pm) as: Racrit(k) = (k⁴ + Ta k² + Ha² k² (1 + S)) / k² (6) Minimisation with respect to k yields the global critical Rayleigh number: Racrit ≈ 2 * (Ta + Ha² (1 + S))½ (7) Thus Racrit increases with rotation (Ta) and magnetic field (Ha²), but the visco‐elastic parameter S modulates the magnetic term. Mathur RP Euro. J. Adv. Engg. Tech., 2021, 8(5):93-96 95 RESULTS AND DISCUSSION Influence of Rotation It is clear from equation (7), on increasing Ta (stronger rotation) value of Racrit raises, indicating that rotation stabilises the convective onset. Physically, the Coriolis force swirls the fluid, inhibiting vertical motions required for convection. For very large Ta the threshold becomes nearly linear in √Ta, consistent with classical rotating convection results. Magnetic Field and Visco‐Elastic Effect For S = 0 (Newtonian fluid), in equation (7) the Ha increases as Racrit ∼ 2 Ha, showing magnetic stabilisation. With finite S, the effective magnetic stabilisation term becomes Ha² (1 + S). Thus, for larger S the magnetic stabilising effect becomes stronger (higher Racrit). However, the presence of visco‐elasticity also introduces oscillatory modes: when S is large, the eigenvalue problem may yield complex σ, meaning overstability (oscillatory onset) rather than stationary onset. In oscillatory modes the true critical Ra may be slightly lower than predicted by the stationary formula, implying that elasticity can reduce the threshold under some conditions. Combined Effects and Critical Transitions For low S, the boundary between stable/unstable lies predominantly along Ha; for moderate S, rotation and magnetic field combine additively; for high S, there exists a regime where increasing Ha initially stabilises, but beyond a certain Ha the threshold decreases due to elastic oscillations dominating, producing a non‐monotonic behaviour. This result emphasises that in visco‐elastic rotating MHD flows, the magnetic field may either stabilise or destabilise depending on elasticity strength. Physical Implications and Parameter Regimes These findings have practical relevance. In industrial mixers where visco‐elastic fluids are stirred and subjected to magnetic fields (e.g., ferrofluids with polymer additives), the threshold for convective instability informs design for uniform mixing. In geophysical flows, e.g., molten polymer‐rich layers in planetary interiors with rotational and magnetic fields, the visco‐elasticity of silicate melts can modify stability constraints. The fact that elasticity can reduce the threshold under certain regimes implies that visco-elastic layers might become unstable more easily than Newtonian analogues despite strong rotation or magnetic fields [6, 10]. Limitations and future work The current analysis has several simplifying assumptions: incompressibility, linear visco‐elastic (second‐order) model, uniform material properties, uniform magnetic field and rigid boundaries. Real fluids may be compressible, have shear‐dependent viscosity, non‐uniform properties, and magnetic field gradients. Non‐linear effects, finite amplitude convection, and three‐dimensional models are also omitted. Future work should extend to more general visco-elastic constitutive laws (e.g., Oldroyd-B), include Hall currents or ambipolar diffusion in conducting visco‐elastic fluids, perform numerical simulations for finite amplitude responses, and validate the theory experimentally. CONCLUSION A linear stability analysis of a rotating basalt layer of Rivlin–Ericksen visco‐elastic fluid in a vertical magnetic field has been conducted. Key conclusions are: • Both rotation and magnetic field individually raise the critical Rayleigh number and hence stabilise the system. • The visco‐elastic parameter S enhances the magnetic stabilisation term (Ha²(1+S)), making magnetic stabilisation stronger for visco‐elastic fluids in the stationary regime. • For large S, oscillatory instabilities may dominate, potentially lowering the threshold relative to the stationary case—thus elasticity can paradoxically reduce stability in certain regimes. • The stability behaviour in the combined (Ta, Ha, S) parameter space is non‐trivial and shows competing effects of Coriolis, Lorentz and elastic forces. This work highlights the importance of visco‐elasticity in rotating magnetohydrodynamic settings, and suggests that classical Newtonian MHD stability results must be modified when elasticity is present. The theoretical outcomes provide a basis for further numerical and experimental investigations in visco-elastic rotating MHD systems. 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