Simulation of Static Flying Attitudes with Different Heat Transfer Models for a Flying-Height Control Slider with Thermal Protrusion

The HDI heat transfer model developed by Chen et al. [3] originates from the model by Zhang and Bogy [7]. Zhang’s model and Chen’s model both use the velocity slip and temperature jump boundary conditions. Both models have shown that the heat flux on the ABS has two contributions. One is the heat conduction, which transfers heat from the slider to the air film when the ABS has a higher temperature than the disk surface; the other is the viscous dissipation due to the air flow within the HDI. For a simplified situation with disk velocity U in the slider length direction (i.e., x-direction) and zero disk velocity in the slider width direction (i.e., y-direction), as shown in Fig. 1, the heat flux on the ABS in both models can be expressed in terms of,where In Zhang’s modelwhile Chen’s model hasHere, k is the thermal conductivity of air; μ the viscosity of air; T _s and T _d the temperatures of the slider and the disk, respectively; h the local slider–disk gap; λ the local mean free path of air molecules; p the local air bearing pressure. The parameter b in Eqs. 2 and 3 can be expressed in the form ofwhere σ_T is the thermal accommodation coefficient; Pr the Prandtl number of air, γ the ratio of specific heats. The parameter a is expressed as a = (2 − α)/α, where α is the momentum accommodation coefficient. For an isothermal air bearing film, the mean free path λ is inversely proportional to the gas pressure p. It should be pointed out that the first term in the expression of q _Poiseuille represents the coupling of Poiseuille flow and Couette flow. That term is included in the heat flux contributed by Poiseuille flow just for a notation simplification here. Zhang’s model and Chen’s model both show that the heat flux on the ABS is dominated by the heat conduction while viscous dissipation is only a second-order effect. Because of this, only the heat conduction on the ABS, i.e., the cooling effect of the air bearing, is considered in the static flying height simulation by Juang et al. [2].

Ju [4] proposed another heat transfer model for the HDI. His DSMC results validate the heat conduction part based on the temperature jump theory. However, the viscous dissipation in Ju’s model is different from that shown in Eq. 1c. The viscous dissipation due to Couette flow is derived from an approximate solution to the Boltzmann transport equation, while the viscous dissipation contributed by Poiseuille flow is not included in Ju’s model. The surface thermal accommodation coefficient is fixed as unity. The complete expression for heat flux on the ABS in Ju’s model is,where ρ is the air density and T the air bearing temperature. The derivation of q _Couette is based on an assumption that the air flow is isothermal. When the temperature difference between the slider and the disk is negligible, i.e., T _s − T _d ≪ T _s and T _s − T _d ≪ T _d, the isotheral flow assumption is still valid for the air bearing. Then, T can be approximated by (T _s + T _d)/2.

As an extension to Zhang’s and Chen’s models, Zhou et al. [5] took into consideration the temperature dependence of the air molecule’s mean free path and proposed a generalized heat transfer model. They replace the air mean free path λ in Eq. 1b with λ_T, the air mean free path at temperature T. The expression λ_T = ξ(T/T ₀)^ω+0.5(p ₀/p) λ₀ [8] is adopted in their new model with parameters ξ and ω determined by an air molecular model or experimental data of the air. Here, λ₀ denotes the reference mean free path of air molecules at the reference temperature T ₀ and pressure p ₀. Experimental data suggest ξ = 0.80–0.85 and ω = 0.75 for an air film [5], while the hard sphere gas molecular model, which is used in Zhang’s and Chen’s models, has ξ = 1 and ω = −0.5, i.e., λ_T = λ = λ₀ p ₀/p. So, it is apparent that Zhou’s model can be reduced to Zhang’s and Chen’s models.

Different from these models based on the velocity slip and temperature jump theory, a HDI heat transfer model recently proposed by Shen and Chen [6] is derived from the solution to the linearized Boltzmann equation for the air film. This model agrees with the conduction heat part in Zhang’s, Chen’s, and Ju’s models, but it has different viscous dissipations. However, the linearized Boltzmann equation itself predicts vanishing viscous dissipation for the air film in the HDI, which is thin and subsonic [9]. This contradicts the non-zero viscous dissipation in Shen’s model.

The difference between these heat conduction models can be directly seen from the comparison between the non-dimensional conduction heat fluxes given by these models. When the slider-disk separation is much larger than the air mean free path, i.e., h ≫ λ, the rarefaction effect of air molecules can be neglected and the conduction heat flux is reduced to \( q_{\text{cont}} = - k{\frac{{T_{\text{s}} - T_{\text{d}} }}{h}}. \) Normalized by this heat flux, the expressions of non-dimensional conduction heat flux in Zhang’s and Chen’s models, Ju’s model and Zhou’s model can be written as, where D is the inverse Knudsen number equal to \( \sqrt \pi h/2\lambda. \) Figure 2 shows these non-dimensional conduction heat fluxes with air parameters in Table 1. The largest difference between the conduction heat flux predicted by Ju’s model and Chen’s is at an inverse Knudsen number of approximately 3.2; between Zhou’s model and Chen’s, it occurs when an inverse Knudsen number is about 3.0 and T is 0 °C. The overall relative difference between these three types of non-dimensional heat flux is less than 20%.

This article focuses on the numerical analysis of the air bearing cooling effect and the viscous dissipation effect on the slider’s static flying attitude, and numerical comparisons of different static flying attitudes obtained when different HDI heat transfer models are applied. First, Ju’s model is used to analyze the effect of viscous dissipation on the slider’s static attitude, as the heat conduction term and the viscous dissipation term in Ju’s model are both validated by Ju’s DSMC simulation results and our analytical analysis [9]. Second, the slider’s static simulation results obtained with Zhang’s and Chen’s models, Ju’s model and Zhou’s model are compared to determine the difference due to modeling the air bearing cooling.

In the numerical analysis, an INSIC pico slider (1.25 mm × 1 mm × 0.3 mm) [2] and a commercial femto slider (0.85 mm × 0.7 mm × 0.23 mm) are used. The pico slider’s ABS is shown in Fig. 3 and its heating-power-off static flying attitude is shown in Table 2. The static flying height of a slider with thermal protrusion is numerically obtained using the iterative loop code developed by Juang et al. [2]. This loop contains a Reynolds equation solver for the steady-state flying of an air bearing slider and a finite-element analysis program to calculate the thermal protrusion due to inside heating, heat convection on non-air-bearing surfaces and complex heat transfer at the ABS. Here, the CML static air bearing program is used to solve the generalized Reynolds lubrication equation for the slider’s static flying attitude. In the iteration, the ABS with an updated thermal protrusion profile is input into the CML program. The finite-element model for a pico slider with a giant magnetoresistive head and a micro heater developed in reference [2] is used here for the protrusion calculation in ANSYS. The heat conduction at the ABS, i.e., air bearing cooling, can be treated as heat convection at the ABS with given convection coefficients in ANSYS. As the boundary conditions of heat flux and heat convection cannot be applied to the same boundary in ANSYS, the viscous dissipation flux is treated as surface heating on the ABS, which has an areal heat generation rate twice the viscous heating flux.

The commercial femto slider’s ABS is shown in Fig. 4. Due to the lack of an accurate ANSYS model for the read/write transducer, heater, and other components in this femto slider, the structures of the read/write transducer and micro heater used in reference [2] are scaled down and adapted in the simulation of this femto slider. The same loop is used to obtain the femto slider’s static flying attitude with different HDI heat transfer models.

A heating power of 200 mW is used in the simulation of the commercial femto slider, whose ABS is shown in Fig. 4. Usually 200 mW is beyond its practical working power. Table 5 lists the static flying attitudes of the femto slider with different heat transfer models. It is obvious that the flying attitudes obtained with Ju’s model with and without Couette-flow-caused viscous heating, and Zhang’s and Chen’s models are almost the same. At the ambient temperature of 75 °C, the static flying attitudes obtained with Zhou’s model and Zhang’s and Chen’s model are also very close. The largest difference is not more than 2% when compared with the results obtained with Zhang’s and Chen’s models.