Derivation of unifying formulae for convective heat transfer in compressible flow fields

Although many theoretical and experimental studies on convective heat transfer exist, the consistent analytical expression of advection heat flux vector in convection as well as its reference temperature in the thermal driving force remains unclear. Here we show theoretically and experimentally the unifying formulae for three-dimensional (3D) heat flux vector of forced and natural convections for compressible laminar flows based on the first law of thermodynamics. It is indicated for a single-phase compressible fluid that advection is no other than heat transfer owing to mass flow in the forms of enthalpy and mechanical energy by gross fluid movement, driven by the temperature difference between the fluid temperature and the potential temperature associated with the relevant adiabatic work done. A simple formula for the total convective heat flux vector of natural convection is also suggested and reformulated in terms of logarithmic density difference as the thermal driving force. The theoretical calculations agree well with the laminar flow experiment results. Our discovery of advection heat transfer for compressible flows caused by the temperature differential in which the potential temperature is regarded as the unifying reference temperature represents a previously unknown thermal driving mechanism. This work would bring fundamental insights into the physical mechanism of convective heat transfer, and opens up new avenue for the design, calculation and thermal management of the 3D convection heat flux problems using the novel thermal driving force for compressible laminar and turbulent flows.

Effect of voids in a heat-flux dependent theory for thermoelastic bodies with dipolar structure

In our paper we formulate a theory for thermoelastic porous dipolar bodies in which we consider a new independent variable, namely the heat-flux vector. Furthermore, we add, to the differential equations that describe the behavior of the body, a new differential equation which is an equation of evolution which is satisfied by the components of the heat-flux vector. The basic system of the mixed initial-boundary value problem in this context consists of equations of the hyperbolic type. In order to ensure the consistency of the constructed theory, we formulate and prove an uniqueness result, with regards to the solution of the mixed problem.

A Miniaturized 3D Heat Flux Sensor to Characterize Heat Transfer in Regolith of Planets and Small Bodies

The objective of this work is to present the first analytical and experimental results obtained with a 3D heat flux sensor for planetary regolith. The proposed structure, a sphere divided in four sectors, is sensible to heat flow magnitude and angle. Each sector includes a platinum resistor that is used both to sense its temperature and provide heating power. By operating the sectors at constant temperature, the sensor gives a response that is proportional to the heat flux vector in the regolith. The response of the sensor is therefore independent of the thermal conductivity of the regolith. A complete analytical solution of the response of the sensor is presented. The sensor may be used to provide information on the instantaneous local thermal environment surrounding a lander in planetary exploration or in small bodies like asteroids. To the best knowledge of the authors, this is the first sensor capable of measuring local 3D heat flux.

The Heat Flux Vector(s) in a Two Component Fluid Mixture

Bulk kinematic properties of mixtures such as velocity are known to be the density weighed averages of the constituent velocities. No such paradigm exists for the heat flux of mixtures when the constituents have different temperatures. Using standard principles such as frame indifference, we address this topic by developing linear constitutive equations for the constituent heat fluxes, the interaction force between constituents, and the stresses for a mixture of two fluids. Although these equations contain 18 phenomenological coefficients, we are able to use the Clausius-Duhem inequality to obtain inequalities involving the principal and cross flux coefficients. The theory is applied to some special cases and shown to reduce to standard results when the constituents have the same temperature.

Analysis of an augmented fully-mixed formulation for the coupling of the Stokes and heat equations

We introduce and analyse an augmented mixed variational formulation for the coupling of the Stokes and heat equations. More precisely, the underlying model consists of the Stokes equation suggested by the Oldroyd model for viscoelastic flow, coupled with the heat equation through a temperature-dependent viscosity of the fluid and a convective term. The original unknowns are the polymeric part of the extra-stress tensor, the velocity, the pressure, and the temperature of the fluid. In turn, for convenience of the analysis, the strain tensor, the vorticity, and an auxiliary symmetric tensor are introduced as further unknowns. This allows to join the polymeric and solvent viscosities in an adimensional viscosity, and to eliminate the polymeric part of the extra-stress tensor and the pressure from the system, which, together with the solvent part of the extra-stress tensor, are easily recovered later on through suitable postprocessing formulae. In this way, a fully mixed approach is applied, in which the heat flux vector is incorporated as an additional unknown as well. Furthermore, since the convective term in the heat equation forces both the velocity and the temperature to live in a smaller space than usual, we augment the variational formulation by using the constitutive and equilibrium equations, the relation defining the strain and vorticity tensors, and the Dirichlet boundary condition on the temperature. The resulting augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Schauder and Banach theorems, combined with the Lax-Milgram theorem and certain regularity assumptions, are applied to prove the unique solvability of the continuous system. As for the associated Galerkin scheme, whose solvability is established similarly to the continuous case by using the Brouwer fixed-point and Lax–Milgram theorems, we employ Raviart–Thomas approximations of order k for the stress tensor and the heat flux vector, continuous piecewise polynomials of order ≤ k + 1 for velocity and temperature, and piecewise polynomials of order ≤ k for the strain tensor and the vorticity. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the good performance of the scheme and confirming the theoretical rates of convergence.

Heat Flux Vector Potential in Convective Heat Transfer

The heat function concept introduced by Kimura and Bejan (1983, “The Heatline Visualization of Convective Heat Transfer,” ASME J. Heat Transfer, 105(4), pp. 916–919) for two-dimensional (2D) heat transfer is being extended in this note to three dimensions. It is shown that a heat flux vector potential exists and can be used in three-dimensional (3D) heat convection problems. It is further shown that this heat flux vector potential degenerates to the heat function introduced by Kimura and Bejan (1983, “The Heatline Visualization of Convective Heat Transfer,” ASME J. Heat Transfer, 105(4), pp. 916–919) when the heat convection is two-dimensional.

Average plasma sheet polytropic index as observed by THEMIS

Multi-spacecraft data from the years 2008 to 2015 of the THEMIS mission particularly in the near-Earth plasma sheet are used in order to empirically determine the polytropic index in the quiet and active time magnetotail. The results of a number of previous studies in the 1990s can be confirmed. An analysis of the total database, although showing poor correlation, results in an average polytropic index of γ = 1. 72. The active time plasma sheet is well correlated with an average γ = 1. 49. However, the data scattering suggests that the analysis of the data in total is not adequate. In order to reduce the timescales, individual spacecraft orbits are analyzed, giving a broad distribution of polytropic indices throughout the plasma sheet. The major part of the distribution falls in a range between γ = 0. 67 and γ = 2. Our results indicate a variety of thermodynamic processes in the magnetotail and an all-time presence of heat exchange of the plasma. A description of the plasma sheet using an equation of state with a single γ is probably inadequate. This necessitates the application of more sophisticated approaches, such as a parametrization of the heat flux vector in magnetohydrodynamic equations or a superposition of polytropic indices.