Chandrasekhar Number (Q): Magnetohydrodynamic Stability

Definition, Units, and Measurement

Chandrasekhar number expresses the squared Hartmann number Ha² and is written as:

Q = B² L² / (ρ ν η).

Here B denotes magnetic flux density measured in tesla, L is a representative length scale in metres, ρ is fluid density in kg·m⁻³, ν is kinematic viscosity in m²·s⁻¹, and η is magnetic diffusivity with units m²·s⁻¹. Because ν and η appear in the denominator, Q grows when the fluid has low viscosity or low magnetic diffusivity, corresponding to stiff magnetic field lines that dominate momentum transport. Experimentalists deduce Q by measuring B with Hall probes or flux loops, determining material properties, and selecting a relevant L—such as the channel half-width in a duct or the radius of a stellar convection cell.

In practice, uncertainties in magnetic diffusivity often limit Q precision. Researchers therefore back-calculate η using conductivity measurements (η = 1/(μ₀σ)) where μ₀ is the vacuum permeability and σ the electrical conductivity. Accurate viscosity data require rheometry across expected temperature ranges because liquid metals like sodium or eutectic lead–bismuth exhibit strong temperature dependence.

Historical Development and Scientific Background

Subrahmanyan Chandrasekhar introduced the number during his investigations of magnetoconvection in the 1950s. Building upon the Hartmann flow solutions that considered steady motion of conducting fluids between charged plates, Chandrasekhar generalised the stability analysis to include buoyancy, rotation, and magnetic tension. His landmark monograph Hydrodynamic and Hydromagnetic Stability framed Q as a control parameter measuring magnetic suppression of convection. The work connected laboratory flows with solar observations, particularly explaining why sunspot umbrae exhibit reduced convective transport when strong fields increase Q.

Later, astrophysicists applied Q-based criteria to model magnetorotational instabilities in accretion disks and to study the onset of magnetoconvection in white dwarf envelopes. In terrestrial laboratories, the number became central to research at facilities like the Riga dynamo experiment, where liquid sodium flows under intense magnetic fields, and to design studies for liquid-metal blankets in tokamak fusion reactors.

Mathematical Relationships and Conceptual Insights

Chandrasekhar number is equivalent to Ha², with the Hartmann number defined as Ha = B L √(σ/(ρ ν)). Because magnetic diffusivity η equals 1/(μ₀σ), substituting yields Q = B² L² σ/(ρ ν / μ₀) = Ha². This relationship reveals that Q represents the squared ratio of Lorentz to viscous forces. When Q ≫ 1, magnetic tension imposes quasi-rigid constraints on velocity profiles, leading to Hartmann boundary layers of thickness L/Ha where shear localises.

Chandrasekhar’s linear stability analyses showed that the critical Rayleigh number for convection increases linearly with Q, meaning stronger fields suppress buoyant overturning. In rotating systems, Q also interacts with the Taylor number, producing magnetorotational thresholds central to geodynamo and astrophysical disk models. Numerical magnetohydrodynamics (MHD) codes solve the coupled Navier–Stokes and induction equations, often nondimensionalised using Q, magnetic Reynolds number Rm, and Prandtl number. Comparing these parameters clarifies whether magnetic field diffusion, viscous dissipation, or inertial effects dominate.

In engineering, duct flows under transverse magnetic fields are characterised by a dimensionless pressure drop proportional to Q. Designers must account for the induced Lorentz force −σ (v × B) × B, which scales with B² and thus with Q, to predict power requirements for pumping conducting coolants. Analytical solutions indicate that velocity profiles flatten as Q grows, improving heat transfer uniformity but increasing wall shear stress near electrodes.

Applications in Engineering, Astrophysics, and Emerging Technologies

Liquid-metal cooling systems in fast breeder reactors and advanced fission concepts rely on Q analysis to ensure stable flow under strong magnetic fields used for electromagnetic pumps and flow meters. Designers tune magnetic coil strength and duct dimensions to maintain Q within ranges that damp turbulence without causing excessive pressure drop. In fusion devices, blankets containing lithium–lead eutectic serve both as tritium breeders and as heat removal media; here, Q informs the design of swirl tapes, insulating coatings, and flow channel inserts.

Space physicists evaluate Q when interpreting solar magnetograms. Regions with high Q indicate field-aligned plasma motions, explaining features such as prominences and coronal loops where Lorentz forces govern stability. Similarly, planetary scientists use Q to model the damping of convection in the metallic hydrogen layer of Jupiter, connecting fluid dynamics to magnetic field observations from spacecraft missions.

Microfluidic engineers are adapting Chandrasekhar-based design to lab-on-a-chip platforms that use external magnets to manipulate ionic solutions. By increasing Q through strong permanent magnets or superconducting coils, designers can align particles, damp vortices, and achieve precise mixing ratios for biochemical assays. Emerging biomedical devices exploit magnetically actuated flow control to process blood with reduced hemolysis, benefiting from Q-informed channel geometries.

Additive manufacturing of conductive inks and metals also invokes Q. Electromagnetic stabilisation of melt pools in powder bed fusion relies on magnet coils that raise Q, suppressing spatter and improving layer uniformity. Process engineers integrate Q-based metrics into digital twins that monitor coil currents, melt pool velocities, and viscosity changes in real time.

Importance, Best Practices, and Future Directions

Understanding Chandrasekhar number is essential whenever magnetic fields interact with conductive fluids. Engineers should verify magnetic field uniformity, as spatial gradients in B lead to position-dependent Q values that can trigger local instabilities. Accurate property databases for conductivity and viscosity, validated over operational temperature ranges, are equally crucial. Coupling computational predictions with experimental diagnostics—such as ultrasonic velocimetry and magnetic probes—provides feedback that keeps Q within targeted bounds.

Future research aims to integrate Q into real-time control algorithms. For example, fusion reactor concepts propose adjusting magnet coil currents based on MHD sensors to maintain optimal Q, preventing flow stagnation. Astrophysical observatories combining helioseismology and magnetograms will refine Q-based models of stellar convection zones. As electrified transport and metallurgical processes adopt electromagnetic stirring, the Chandrasekhar number will remain a guiding metric for stability, energy efficiency, and product quality.