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Strong suppression of electron convection in Dirac and Weyl semimetals

It is shown that the convective instability in electron fluids in three- and two-dimensional (3D and 2D) Dirac and Weyl semimetals is strongly inhibited. The major obstacles for electron convection are the effects of the Coulomb forces and the momentum relaxation related to the interaction with impurities and phonons. The effect of the Coulomb forces is less pronounced in 2D materials, such as graphene. However, momentum relaxation still noticeably inhibits convection making it very difficult to achieve in practice.

Introduction.-Electron hydrodynamics is an unusual transport regime that can be realized in clean crystals. As first conjectured by Gurzhi in the 1960s [1, 2], electrons could form a hydrodynamic fluid when the electronelectron scattering rate is larger than the scattering rates of electrons on impurities and phonons. In such a regime, the electron transport should reveal some conventional hydrodynamic effects, including the Poiseuille-like profile of the current in a wire, the formation of vortices, etc.
Guided by experience with conventional fluids, one may expect similar hydrodynamic effects to show up in the electron fluid. For example, the possibility of a preturbulent regime in graphene was proposed in Refs. [46,47]. (Kagome metals may provide a compelling platform for realizations of turbulence in electron fluids [48].) The formation of the Rayleigh-Bénard convective cells in graphene was suggested and numerically studied in Ref. [49]. Generally, the Rayleigh-Bénard instability occurs in fluids subject to a buoyancy force (e.g., caused by gravity) and temperature gradient that results in a local thermal expansion of fluid [50,51]. If the buoyancy force is strong enough, it becomes favorable to develop regular convective cells where the heat transfer is greatly assisted by the fluid motion. The observation of such cells would be a definitive signature of electron hydrodynamics. The possibility of electron convection is not only of academic interest but also can have important practical applications. Indeed, the convective regime may be invaluable for an effective heat transfer because the Nusselt number, which quantifies the ratio of convective to conductive heat transfer, is up to 10 (1000) times larger for a laminar (turbulent) flow. Since there are significant differences between conventional fluids and electron plasma in semimetals, it is necessary to investigate the onset conditions for electron convection in detail.
In this Letter, we show that the Rayleigh-Bénard instability in electron fluids in Dirac and Weyl semimetals is strongly inhibited. We identify two major obstacles for the electron fluid convection: (i) Coulomb forces and (ii) momentum-relaxation effects. Unlike many conventional fluids, the electron plasma is electrically charged. In a semimetal, the overall electric neutrality is preserved by the compensating charge of immovable lattice ions. Any deviation from local neutrality induces a strong electric field and, consequently, is energetically unfavorable. In other words, the effects of the Coulomb forces strongly suppress local expansion and compression of the electron fluid. For the same reason, a temperature gradient cannot easily induce sufficiently large local density devia-tions needed for triggering convection.
We support this qualitative argument by a quantitative estimate of the Rayleigh number. Due to the aforementioned obstacles, the latter is so large in 3D Dirac and Weyl semimetals that achieving convection becomes practically impossible under any realistic conditions. As one might expect, the effects of the Coulomb forces are less pronounced in 2D materials. However, due to ubiquitous disorder effects and interaction with phonons, the momentum relaxation also noticeably inhibits a convective flow in both 2D and 3D systems.
Model.-The starting point in our discussion of the electron fluid is the Navier-Stokes equation [13,14], (1) Here, w = ǫ + P is the enthalpy, ǫ is the energy density, P is the pressure, u is the electron fluid velocity, n is the electron number density, −e is the electron charge, and v F is the Fermi velocity. Further, η and ζ are the shear and bulk viscosities, respectively. In relativisticlike systems, ζ ≈ 0 [52] and η = η kin w/v 2 F , where η kin ∼ v 2 F τ ee is the kinematic shear viscosity and τ ee is the electronelectron interaction time (see, e.g., Refs. [53,54]). For a relativisticlike fluid, P = ǫ/d and w = (d + 1)ǫ/d, where d is the spatial dimension. The momentum relaxation, which is inevitable in real solids, is quantified by the relaxation time τ that contains contributions from scattering on impurities and phonons. For simplicity, we assume that the electron fluid is isotropic, which is sufficient for the purposes of this study. Finally, we account for an electric field E, which includes both external and induced contributions.
The electric and energy current densities are Here, µ is the chemical potential, T is temperature, and σ is the intrinsic conductivity [13]. The electric and energy currents satisfy the standard continuity relations − e∂ t n + (∇ · J) = 0 (4) and respectively. Since the electron fluid is electrically charged, the hydrodynamic equations should be supplemented by Maxwell's equations. By assuming a slow flow, we neglect the effects of dynamic magnetic fields on fluid motion. In such a quasistatic approximation, only the Gauss law is relevant. Here, −eδn is the deviation of the electron charge density from the background equilibrium value. As we will show below, electric fields induced by δn play a profound role in suppressing electron convection in 3D. The Coulomb forces also hinder convection in 2D systems, but their effect is less dramatic. It is worth noting that the Coulomb interactions are responsible for both viscosity of the electron fluid and screening effects. Indeed, it is the microscopic interparticle Coulomb force that governs the electron-electron scattering and the formation of electron fluid. On the other hand, the Gauss law in Eq. (6) determines a background electric field, which comes as an average uncompensated field over macroscopic distances. This field is induced when the electron fluid is compressed or expanded locally with respect to the ion lattice.
To investigate the possibility of electron convection, we follow the same conceptual approach as in conventional fluids (see, e.g., Refs. [55,56]), but amend the hydrodynamic equations with the Gauss law. As the first step, we find the steady-state solutions for the temperature profile and the electric field in the absence of hydrodynamic flow. Then, by using these solutions as a background, we derive the threshold criterion for convection.
Steady-state solution without flow.-Let us start by determining the steady-state solution for the electric field and temperature inside a slab of finite thickness L along the x direction. For simplicity, we assume that the slab is infinite along other directions.
Temperature T (x) and the electric potential ϕ(x) take different values on the opposite surfaces of the slab, i.e., We employ a perturbation scheme where deviations of all quantities (denoted by a tilde) are small compared to their global equilibrium values (denoted by subscript 0), e.g.,T /T 0 ≪ 1, etc. By setting u = 0 in Eqs. (1) through (6), one finds that the temperature functionT (x) is determined by the Laplace equation, ∆T = 0. Its solution that satisfies the boundary conditions (7) reads By taking into account this solution, Eqs. (4) and (6) can be rewritten as follows: where q TF = 4πe 2 (∂ µ n) is the Thomas-Fermi wave vector. Its explicit expression is given in the Supplemental Material [57].
By using the equation of motion (1) at u = 0 and requiring the overall charge neutrality of the sample, we obtain the steady-state expressions forμ andφ (see Supplemental Material [57]). They reveal that the electric field is nonvanishing but screened strongly inside the sample. Also, the electric charge deviations are small (ñ ≈ 0) in the bulk of the slab. However, both electric field and charge density are noticeably nonuniform near the surfaces. Since convection is determined largely by bulk properties, it is justified to ignore the inhomogeneity of the electric field and other variables near the surfaces. Such an approximation simplifies analytical calculations but should not affect the qualitative results for sufficiently thick samples.
Convection threshold.-To determine the threshold of convection, we consider a hydrodynamic flow in the background of the screened electric field and the temperature gradient obtained in the previous section. As in conventional fluids [55,56], the analysis is simplified by using an analog of the Oberbeck-Boussinesq approximation. The resulting system, which is linear in the flow velocity u and the flow-driven deviations P u , E u , n u , µ u , and T u (appearing on top of the steady-state solution), reads ∇ · E u = −4πen u .
Here, we assumed that the cross-terms containing fluid flow velocity and the temperature gradient are small compared to other terms. In addition, we neglected the terms that are of the second order in steady-state deviations (e.g.,ñẼ ≪ n 0Ẽ ). The last term on the right-hand side in Eq. (12) and the first term in Eq. (13) are crucial for driving convection and their analogs are included in the Oberbeck-Boussinesq approximation for conventional fluids. Indeed, the term −en uẼ is similar to the buoyancy force for regular fluids. Recall that buoyancy is the consequence of an external (e.g., gravitational) force exerted on a fluid with a density gradient, typically induced by a temperature difference between the top and bottom surfaces. In the case of electrons, gravitation has negligible effects and the role of buoyancy force is played by the electric force related to the in-medium electric fieldẼ. Therefore, another key requirement for achieving convection is the nonzero compressibility of the fluid. It is taken into account by the first term in Eq. (13).
It is very important for the problem under consideration that, unlike ordinary fluids, expansion and compres-sion of the electron fluid give rise to strong electric fields. Indeed, in view of the Gauss law (15), any change of the electric charge density leads to an electric field. As we will explicitly show below, this is one of the key factors inhibiting convection in electron fluids as the energy price for the appearance of electric fields is very high.
To estimate the threshold for convection, we use the bulk steady-state solutions,ñ ≈ 0 andẼ =Ẽx, which include both external and induced fields. Then, (u · ∇)w = u x (∂ xw ) ∝ u x (T R − T L ). The explicit expression for (∂ xw ) as well as the general solutions to Eqs. (12) through (15) are given in the Supplemental Material [57].
We use a plane-wave ansatz for the hydrodynamic variables u x , T u , and µ u , e.g., where k ⊥ is the wave vector perpendicular to the surface normal. The characteristic equation for the system of Eqs. (12) through (15) reads [57] where k 2 = k 2 ⊥ + k 2 x . In Eq. (17), we used the following shorthand notations: Here, λ G is the Gurzhi length that quantifies the momentum relaxation and Ra is the Rayleigh number. In the limit λ G → ∞ and q TF → 0, the characteristic equation (17) coincides with the textbook result for conventional fluids (cf. Ref. [55]). Convective instability characterized by periodic spatial pattern of the fluid velocity is realized for real k x and k ⊥ . This requires Ra Ra min , where Ra min is determined from the characteristic equation (17) with wave vectors constrained by boundary conditions.
To derive the convection threshold Ra min , let us determine the allowed values of k x in Eq. (17). They follow from the boundary conditions for T u and u, i.e., For the perpendicular components of velocity u ⊥ , we employ the free-surface boundary conditions, which provide the most conservative estimate of the convection threshold. Indeed, the no-slip boundary conditions can be considered as a source of additional dissipation that further inhibits convection; see, e.g., Ref. [56] for neutral fluids. We found that the free-surface boundary conditions are satisfied for |k x | = πn/L with n = 1, 2, 3, . . . (see Supplemental Material [57] for details). It is worth noting, however, that the exact form of the boundary conditions is not important for our qualitative arguments because they only determine the allowed values of k x . The characteristic equation (17) gives the following relation between the Rayleigh number and wave vector: (21) This shows that the Coulomb forces and the momentum relaxation effects, quantified by q TF and λ G , respectively, increase the minimal value of the Rayleigh number Ra min needed to achieve convection. In order to determine Ra min , one should minimize Eq. (21) with respect to the wave vectors allowed by the boundary conditions, i.e., at k x = π/L. The general expression for the minimal Rayleigh number at finite λ G and nonzero q TF is cumbersome (see also Supplemental Material [57]). Since the Rayleigh number Ra increases with k x , we can estimate the corresponding lower bound Ra 0 (which is smaller than the actual Ra min ) by setting k x = 0 and neglecting k ⊥ compared to q TF and λ −1 G in Eq. (21), i.e., It is instructive to mention that q TF ≫ π/L and λ G ≪ L/π hold for realistic samples of 3D Dirac and Weyl semimetals and, therefore, the above estimate is indeed reasonable. To verify this, let us consider typical parameters for semimetals, e.g., µ 0 = 20 meV, T 0 = 25 K, and use the Fermi velocity v F ≈ 1.4 × 10 7 cm/s [58]. In this case, one obtains q TF ≈ 9.9 × 10 6 cm −1 by assuming quasiparticles with a 3D relativisticlike energy spectrum. For such a large Thomas-Fermi wave vector, one finds that q TF ≫ π/L in a wide range of experimentally achievable samples [59]. As for the momentum relaxation length, we estimate λ G ≈ 0.4 µm at τ = 0.1 ns and τ ee = 0.3 ps, which is quite small compared to a typical thickness of Dirac semimetal slabs. By using Eq. (22) and the same characteristic parameters as before, we estimate the lower bound for Ra min as It is instructive to compare this estimate with the benchmark result for conventional fluids. By taking λ G → ∞ and q TF = 0, we find that the critical value of the Rayleigh number is reached for k x = π/L and k ⊥,min = π/( √ 2L) giving Ra min = 27π 4 /4 ≈ 657.5. The enormous value of the lower bound for the minimal Rayleigh number (23) implies that convection in the electron fluid is strongly suppressed. Indeed, Eq. (19) gives the following estimate for the Rayleigh number in Dirac semimetals: whereẼ is the in-medium field that includes both external and induced components, and δT /T 0 is the normalized temperature difference between the slab surfaces. As is easy to see, the estimate in Eq. (24) is many orders of magnitude smaller than Ra 0 for any reasonable electric field and temperature gradient available in experiments. Thus, in agreement with the heuristic arguments, convection is ruled out for 3D Dirac semimetals. The minimal Rayleigh number Ra min as a function of the slab width L is presented by solid lines in Fig. 1 for the three fixed values of the chemical potential: µ 0 = 1 meV (red), µ 0 = 10 meV (blue), and µ 0 = 50 meV (green) [60]. As we see, the minimal Rayleigh number needed for convection is enormous for macroscopic samples. It approaches the value in conventional fluids, i.e., Ra min = 27π 4 /4, only for extremely thin slabs. In the same figure, the shaded regions show the ranges of realistic estimates for the Rayleigh number obtained from Eq. (24) by assuming rather conservative values δT ≤ T 0 andẼ ≤ 10 V/m. It is clear that the realistic values of the Rayleigh number are many orders of magnitude smaller than Ra min required for convection.
Graphene.-In view of the great interest in electron hydrodynamics in graphene, let us re-examine whether the convective instability is possible in the 2D case. While the electron convection in graphene was already studied in Ref. [49], the effects of the Coulomb forces and the momentum dissipation were not taken into account. The screening effects in gated graphene can be treated in the "gradual channel" approximation [61,62] (see also Ref. [21]). In this approximation, the induced electric field is given by where C = ε/(4πL g ) is the capacitance per unit area, ε is the dielectric constant, and L g is the distance to the gate. The minimum Rayleigh number is determined similarly to the 3D case considered above, but the Gauss law (15) is replaced by Eq. (25) (for details, see Supplemental Material [57]). We obtain , the boundary conditions are satisfied for |k x | = πn/L and n = 1, 2, 3, . . .. As in the 3D case, the Coulomb forces and momentum relaxation increase the minimal Rayleigh number needed to achieve convection.
Quantitatively, the effects of the Coulomb forces are much weaker in 2D. This is also supported by numerical estimates. Indeed, by using typical parameters for graphene, i.e., v F = 1.1 × 10 8 cm/s, µ 0 = 100 meV, T 0 = 100 K, L g = 100 nm, and ε = 1, we estimate Q ≈ 6.6 (see Supplemental Material [57] for details). Therefore, the Coulomb forces increase Ra min only by about an order of magnitude. The momentum relaxation effects, on the other hand, are very important. We estimate λ G ≈ 2.6 µm at τ = 0.1 ns and τ ee = ( 2 v 2 F /e 4 )( µ 0 /T 2 0 ) ≈ 0.2 ps. The corresponding minimal value of the Rayleigh number is at L = 100 µm. This is almost four orders of magnitude larger than the benchmark value 27π 4 /4. The Rayleigh number for graphene is estimated as Our estimate suggests that in order to exceed the minimal Rayleigh number (27), centimeter-sized graphene samples are needed when the total electric fields are of the order of 1 V/m. Thus, the momentum relaxation significantly inhibits convection in 2D systems too. The effect of the Coulomb forces, however, is less pronounced compared to the case of 3D Dirac semimetals. This situation resembles the role of the Coulomb forces in the spectrum of plasmons. Indeed, the plasmon dispersion relation is gapped in 3D due to the efficient screening of electric charge oscillations. On the other hand, in 2D systems the spectrum of plasmons remains gapless in the gradual channel approximation, where the Coulomb forces only enhance the plasma velocity.
Summary.-In this Letter, we showed that the convective instability is strongly inhibited in the electron fluid in 2D and 3D Dirac and Weyl semimetals. We identified the following two major inhibitors: (i) the Coulomb forces and (ii) the momentum relaxation effects due to scattering on impurities and phonons. Both are unavoidable and play a critical role in the charged electron fluid in semimetals. For realistic parameters in 3D Dirac and Weyl semimetals, the effect of the Coulomb forces dominates over the momentum relaxation and leads to an extremely large convection threshold. In 2D systems such as graphene, the key role is played by the momentum relaxation that increases the minimum Rayleigh number needed for convection. The corresponding threshold values are a few orders of magnitude larger than in conventional fluids. Our findings imply that the electron fluid convection is unlikely to be realized in Dirac and Weyl semimetals.
While we focused on the fluid of electron quasiparticles with an isotropic relativisticlike dispersion relation, similar arguments should apply to systems with anisotropic and even nonrelativisticlike dispersion relations, although the quantitative details will be different. For example, the critical value of the Rayleigh number may be different along different crystal directions for materials with anisotropic spectra. Because of the crucial role of the Coulomb forces, we speculate that fluids made of neutral quasiparticles such as magnons (if their hydrodynamic regime is realized [63,64]) promise to be better candidates for convection in solid-state materials. It is tempting to suggest that phonon fluids [2, 65-68] might also demonstrate convective instability. It is not clear, however, whether the analogy with conventional fluids is complete and the equivalent of buoyancy forces is present.  In this section, we present thermodynamic variables used in our analysis of electron convection. We employ a relativisticlike isotropic dispersion relation for the electron quasiparticles ǫ p = v F p in both 2D and 3D systems.
In 3D, the electron number density n equals where Li j (x) is the polylogarithm function. The energy density ǫ reads where the prefactor N g in Eqs. (S3) and (S4) is the degeneracy factor. We have N g = 2 in 3D Dirac semimetals with a single Dirac point and N g = 4, which accounts for the valley and spin degeneracy, in graphene. In addition, v F is the Fermi velocity, µ is the chemical potential, and T is temperature. The electron number density n and the energy density ǫ for a 2D relativisticlike spectrum are The pressure P and the enthalpy density w = ǫ + P are given by the standard expressions for a relativisticlike systems, P = ǫ/d and w = (d + 1)ǫ/d, where d = 2, 3 is the spatial dimension. Note that the following relations for the derivatives with respect to chemical potential and temperature are valid: where s = (w − µn)/T is the entropy density.

S II. STEADY STATE
In this section, we discuss the temperature profile, the electric field, and the charge density for the steady state in the model defined in the main text. In particular, we consider a slab of a Dirac or Weyl semimetal with a finite thickness L along the x direction. For simplicity, we assume that the slab is infinite in other directions. In the absence of hydrodynamic flow, the local deviations of the chemical potentialμ and temperatureT from their values T 0 and µ 0 in global equilibrium are caused by an external electric field and a temperature gradient. In other words, where, by assumption, the deviationsμ andT are small compared to the equilibrium values. As we will show below, the temperature gradient and the external electric field are included via the appropriate boundary conditions. To the leading order in deviations, the equation of motion, the electric current continuity equation, and the Gauss law read respectively. Here, −e is the charge of the electron and σ is the intrinsic conductivity [S1]. For the intrinsic conductivity, we use the result obtained in a holographic approach [S2-S5], i.e., where τ ee is the electron-electron scattering time. The intrinsic conductivity in the hydrodynamic regime of graphene can be estimated as [S6] σ = 2e 2 π . (S13) By applying ∇ to Eq. (S9) and making use of Eqs. (S10) and (S11), we obtain the following equation for temperature deviations: In the derivation, we also took into account that ∇P = n 0 ∇μ + s 0 ∇T (S15) and neglected terms of higher order in deviations. In our setup, temperature is fixed at the surfaces of the slab Then the solution to Eq. (S14) readsT where δT = T R − T L . Clearly, it corresponds to a constant thermal gradient along the x direction. By using solution (S17), Eqs. (S10) and (S11) can be rewritten as where q TF = 4πe 2 (∂ µ n) is the Thomas-Fermi wave vector and we used n = (∂ µ n)μ + (∂ T n)T . (S19)

S3
The explicit relations for the square of the Thomas-Fermi wave vector are: Notice that q TF is calculated for equilibrium values of the chemical potential µ 0 and temperature T 0 . The general solution to Eq. (S18) isμ where C 1 and C 2 are integration constants.
Having determined the temperature and chemical potential profiles, let us find the electric fieldẼ = −∇φ inside the slab. We assume that the electric potential is fixed at the surfaces of the slab as follows: According to the Gauss law (S11), the electric potentialφ satisfies the following equation: Taking into account Eq. (S22), one finds where C 3 and C 4 are constants. From Eqs. (S9) and (S15), we see that the following equation should be satisfied as well: s 0 ∇T + n 0 ∇μ = en 0 ∇φ. Then we obtain where C δT = 2πe 2 δT [n 0 (∂ T n) − s 0 (∂ µ n)] n 0 q 2 TF (S32) and δϕ = ϕ R − ϕ L .

S4
The final solutions for the temperatureT , the chemical potentialμ, the electron densityñ, and the electric field E =Ẽx in the steady state readT It is easy to verify that the chemical potential, the electron density, and the electric field are nonuniform inside the sample. The numerical results for the steady-state solutions (S34) through (S36) are presented in Figs. S1(a), S1(b), and S1(c), respectively. To plot the results, we used the following representative values of parameters: µ 0 = 20 meV, T 0 = 25 K, and the Fermi velocity for WP 2 [S7] v F ≈ 1.4×10 7 cm/s. In addition, we set T R = 1.25×T 0 , T L = 0.75×T 0 , ϕ L = 0, and ϕ R = 1 mV. As illustrated in Fig. S1, the inclusion of the Gauss law strongly affects the solutions forμ andñ. First of all, the electron density deviations quickly vanish in the bulk of the slab. This is also confirmed by the dependence ofñ near one of the surfaces shown in Fig. S2(a). The electric fieldẼ is screened strongly but not perfectly inside the slab, see Fig. S2(b). A small nearly uniform value ofẼ survives inside the slab. Its value is determined by the thermoelectric effect, which is quantified by the first term in Eq. (S36). For the parameters at hand, we estimate the corresponding field asẼ Therefore, for the slab width L = 100 µm to L = 1 cm, the in-medium electric field ranges fromẼ ∼ 3 V/m tõ E ∼ 0.03 V/m. These results allow us to approximate the charge density and the electric field byñ ≈ 0 andẼ ≈ const in the bulk of the slab. This will be our starting point in determining the convection threshold in the next section.

S III. CONVECTION THRESHOLD
In this section, we investigate the threshold of convective instability of the electron fluid. The steady-state solution given in Eqs. (S33) through (S36) will be considered as a background for hydrodynamic flow, i.e., where the subscript u denotes the deviations related to the fluid flow u.

S III.A. 3D Dirac semimetals
Let us start with the case of 3D Dirac semimetals. As in the case of conventional fluids, we linearize the system of hydrodynamic equations in u, µ u , and T u . The resulting equations are In writing the above system, we took into account that the steady-state deviations are much smaller than the equilibrium values. Accordingly, we omitted terms quadratic in the steady-state deviations, e.g.,ñẼ. The cross-terms containing both hydrodynamic variables and steady-state deviations are also assumed to be small. However, as we discuss in the main text, there are two notable exceptions from this scheme. Similar to the Oberbeck-Boussinesq approximation for neutral fluids [S8, S9], we retain the buoyancy-like term −en uẼ in the Navier-Stokes equation. It plays a crucial role in convection by providing a connection between the fluid velocity and hydrodynamic deviations of temperature and chemical potential. Finally, since convection relies on the compressibility of a fluid, we retained (u · ∇)w in the energy conservation equation. This eventually allows for an advection-like term in the equation for T u .

S6
By using the explicit form of the steady-state solutions from Sec. S II, we obtain Notice that while we keep the spatial dimension d in the above equations, the system of equations (S41) through (S44) will be modified in 2D. We discuss the 2D case in Sec. S III.B. By using Eq. (S42), one can reexpress ∇ · u in terms of u x . Then Eq. (S43) can be brought in a form resembling the conventional heat transfer equation, i.e., The last two terms on the left-hand side of the above equation clearly distinguish it from the conventional heat transfer equation with an advection term. It is convenient to reexpress u x in terms of T u and µ u , i.e., To determine the relationship between temperature and chemical potential, we calculate the divergence of Eq. (S41) and obtain where is the Gurzhi length, η is the dynamic shear viscosity, and η kin = v 2 F η/w 0 is the kinematic viscosity. The Gurzhi length quantifies the momentum relaxation effects due to the scattering on impurities and phonons. Notice that the first and last terms on the right-hand side in Eq. (S50) can be omitted in the perturbation scheme used. Indeed, they contain small gradient terms multiplied by hydrodynamic variables. By neglecting these terms and rewriting ǫ u and n u in terms of the chemical potential µ u and temperature T u , we obtain Finally, applying curl twice to Eq. (S41) and taking the x-component, one obtains As before, the term ∝ (∂ xw ) could be neglected in the above equation leading to Equations (S49), (S52), and (S54) form the system of equations for u x , µ u , and T u . We search for the solutions of Eqs. (S49), (S52), and (S54) by employing the plane wave ansatz where k ⊥ is the wave vector perpendicular to the surface normal. The system has a nontrivial solution if the wave vector satisfies the following characteristic equation: where is the Rayleigh number. Besides the material parameters, it is determined by the electric field and the temperature gradient. The characteristic equation (S56) defines the relation between the wave vector k and the Rayleigh number. As is clear from the structure of this equation, depending on the sign of Ra, it could have real solutions for wave vector (Ra > 0) or only the imaginary ones (Ra < 0). The former type of solutions corresponds to periodic convective cells (known as the Bénard cells in hydrodynamics). Furthermore, a certain threshold value Ra min should be reached in order to realize convection. To find this threshold, one needs to know the allowed values of k x which follow from the boundary conditions. Before defining the boundary conditions, let us present the general expressions of the solutions for the hydrodynamic variables. The solution for T u reads as where k (j) x are six roots of Eq. (S56) and C (j) T are constants. The roots are complicated, therefore, we do not present their explicit form here. As to the x-component of the fluid velocity, it is given by and does not depend on the sign of k (j) x . The general solutions for the temperature (S58) and the x-component of velocity (S59) should satisfy the boundary conditions In addition, we employ the free-surface boundary conditions for u ⊥ , i.e., By using Eq. (S42), i.e., ∇ · u = −(∂ xw )/w 0 u x , we rewrite the above relations in a more compact form where we used the fact that Eqs. (S62) and (S63) hold for all r ⊥ . In addition, we neglected the term ∝ (∂ xw )(∂ x u x ) in Eq. (S64). Then one can apply ∂ z and ∂ y to Eqs. (S62) and (S63), respectively, and solve them in terms of u x . In the case of the no-slip boundary conditions we obtain While the analysis can be performed for both the free-surface (S64) and no-slip (S66) boundary conditions, we consider only the former case. Indeed, the free-surface boundary conditions allow for a much simpler equation for the wave vector k x whose solutions can be obtained analytically. On the other hand, the case of the no-slip boundary conditions requires numerical calculations (see also Ref. [S9]). While the choice of boundary conditions for the electron velocity results in different values of k x , the final conclusions are qualitatively similar. In addition, by using the arguments in the main text, the use of the precise value of k x could be avoided altogether.Since the no-slip boundary conditions can be considered as a source of an additional dissipation that only further inhibits convection, we focus only on the free-surface boundary conditions.
By substituting the expressions for T u and u x given in Eqs. (S58) and (S59) into the boundary conditions (S60), (S61), and (S64), we obtain the following characteristic equation: Here, we used the fact that k (1) x , and k (5) x = −k (6) x . It is clear that the above equation has a nontrivial solution for |k x | = πn/L with n = 1, 2, 3, . . ..
To make a connection with the literature, let us start with a clean electroneutral system, where λ G → ∞ and q TF = 0. This corresponds to the classical convection of a regular fluid. Then the characteristic equation (S56) reads The minimum value of the Rayleigh number Ra min needed to achieve convection is realized for k x = π/L and k ⊥,min = π/( √ 2L). It equals Let us now include the effects of the Coulomb forces quantified by q TF . For simplicity, we assume that λ G → ∞. Then the Rayleigh number Ra reaches its minimum at By using this result, the minimal value of the Rayleigh number needed to achieve convection reads As will be evident from the numerical estimates for realistic parameters, Ra min given in Eq. (S71) is a very large number for semimetals. The case of finite λ G and q TF = 0 can be studied along the same lines. The resulting minimal value of the Rayleigh number is given by Eq. (S71) with q TF → λ −1 G . In the general case of finite λ G and nonvanishing q TF , the expression for Ra min is bulky. Therefore, we do not present it here. Instead, in the main text, we derive the lower bound Ra 0 ≤ Ra min , which is more than sufficient to conclude that convection is strongly suppressed in Dirac and Weyl semimetals.
As evident from Eq. (S56), the Coulomb forces and momentum relaxation effects increase the minimal Rayleigh number. To quantify the role of these effects for electron convection, we provide numerical estimates of the minimal Rayleigh number in a few model cases. We notice that, for realistic parameters, q TF is usually much larger than L −1 . For example, for a 3D relativisticlike energy spectrum, q TF ≈ 9.9 × 10 6 cm −1 at µ 0 = 20 meV, T 0 = 25 K, and the Fermi velocity for WP 2 [S7] equals v F ≈ 1.4 × 10 7 cm/s. Therefore, Lq TF ≫ 1 for typical samples except for very thin films. In addition, although the effect of the Coulomb forces is reduced in a thin film L ≤ q −1 TF , a small thickness of the film is, in fact, unfavorable for convection. Indeed, according to Eq. (S57), the actual Rayleigh number Ra ∝ L 3 . Therefore, one would need much larger fields and/or temperature gradients to reach the convection threshold Ra min . As for the role of the momentum relaxation, we estimate λ G ≈ 0.4 µm at τ = 0.1 ns and τ ee ≈ 0.3 ps. In this case, q TF λ G ≈ 382 ≫ 1. Therefore, we conclude that the Coulomb forces are a dominant inhibitor of convection in 3D Dirac and Weyl semimetals.
We provide estimates for the minimal Rayleigh number Ra min in Fig. S3 in a few model cases: (i) q TF = 0 and finite λ G , (ii) q TF = 0 and λ G → ∞, (iii) q TF = 0 and finite λ G , (iv) q TF = 0 and λ G → ∞. As one can see, the minimal Rayleigh number remains much larger than the classical value 27π 4 /4 for reasonable values of temperature and chemical potential. This is the case even if the Coulomb forces are omitted. Notice that due to the increase of the electron-electron collision time as T 0 decreases (we assumed that τ ee ≃ /T 0 [S10]), the Gurzhi length rises. This leads to a decrease of Ra min with temperature at q TF = 0. We emphasize, however, that the realization of electron hydrodynamics under these conditions is highly unlikely. Ra min (a) In this section, we investigate the convection threshold in 2D systems focusing on graphene. We consider a gated sheet of graphene where the gradual channel approximation [S11, S12] (see also Ref. [S13] for graphene) can be used instead of the Gauss law (S44). In this case, the hydrodynamically induced electric field E u reads as Here, δn is the deviation of the charge density from its equilibrium value, C = ε/(4πL g ) is the capacitance per unit area, ε is the dielectric constant of the substrate, and L g is the distance to the gate. The rest of the analysis can be performed along the same line as in Sec. S III.A. In particular, we obtain the following velocity: The analog to Eq. (S52) reads n 0 1 + e 2 C (∂ µ n) ∆µ u + (∂ T ǫ) d ∆ + e 2 C n 0 (∂ T n) ∆T u = 0.
Finally, applying curl twice to Eq. (S41) and taking the x-component, one obtains the same relation as in the 3D case, see Eq. (S54). By using the plane wave ansatz similar to that in Eq. (S55), where k ⊥ and r ⊥ are vectors directed along the surface of the graphene ribbon, we obtain the following characteristic equation: where Q = e 2 (∂ µ n)/C. In addition, λ G = v F τ τ ee /4 and η = v 2 F τ ee /4 in graphene [S14]. By comparing Eqs. (S56) and (S75), we notice that the effect of the Coulomb forces quantified by Q is quantitatively different in 2D. However, qualitatively, its role is the same because nonzero Q increases the minimal Rayleigh number needed to achieve convection.
The boundary conditions in 2D have the same form given in Eqs. This expression is similar to that in the 3D case at q TF = 0 and finite λ G . It is clear that Coulomb forces quantified by Q and the momentum relaxation determined by λ G increase Ra min . Let us provide numerical estimates for the minimal Rayleigh number and illuminate the role of the Coulomb forces for electron convection in graphene. As we mentioned above, the role of the Coulomb forces is much weaker in 2D than in 3D. Indeed, by using the number density (S3), we estimate Q ≈ 6.6. Here, we used v F = 1.1 × 10 8 cm/s, µ 0 = 100 meV, T 0 = 100 K, L g = 100 nm, and ε = 1. Therefore, the Coulomb forces lead only to an order of magnitude enhancement of Ra min . The momentum relaxation is, however, more important. For example, we estimate λ G ≈ 2.6 µm at τ = 0.1 ns and τ ee = 1/(α 2 ) /T ≈ 2.2 × 10 −13 s with α = e 2 /( v F ). Therefore, L/λ G 1 for sufficiently large samples where convection is favorable.
The minimal Rayleigh number Ra min is shown in Fig. S4 for a few limit cases: (i) Q = 0 and finite λ G , (ii) Q = 0 and λ G → ∞, (iii) Q = 0 and finite λ G , (iv) Q = 0 and λ G → ∞. As in the 3D case, the minimal Rayleigh number remains much larger than the classical value 27π 4 /4 for reasonable values of chemical potential and temperature. However, the effect of the Coulomb forces is no longer dominant. Indeed, the momentum relaxation quantified by λ G plays an important role too. Because of the much weaker role of the Coulomb forces, the enhancement of the minimal Rayleigh number is more modest than in 3D. Instead of 8 to 10 orders of magnitude difference between Ra min and the classical value 27π 4 /4 in 3D Dirac semimetals, we obtain 2 to 8 orders of magnitude difference in graphene.


The use of modeling in the teaching of geometry and graphic disciplines at the University

1. Introduction: In connection with the introduction of new Federal state educational standards of higher education (FSES) to teachers of geometry graphic departments there was a problem with the revision of the contents and methodology of teaching disciplines “descriptive geometry”, “Engineering and computer graphics”, foundations of engineering education aimed at formation of basic knowledge for studying of the subsequent disciplines used graphic representation of information. The analysis of special literature has indicated that, despite the significant number of scientific publications on the research topic, the use of modeling in teaching geometric-graphic disciplines requires separate research. 2. Materials and methods: Study and analysis of scientific, educational and special literature on the research topic, comparison, systematization, synthesis, generalization. 3. Results: To implement modern teaching methods in the educational process in the geometric-graphic disciplines, the author applied the functionality of the system KOMPAS-3D and developed by: electronic slides based on the modeling included in the evaluation funds for engineering and computer graphics; set a clear three-dimensional (3D) models of twenty-assemblies products intended for assignments on practical and laboratory lessons on the computer in the system KOMPAS-3D to create an Assembly and working drawings; set of individual tasks designed to practical assignments on the computer in this system to create 2D-models of assemblies (Assembly drawings) and 2D models details (working drawings), associated with their three-dimensional models. 4. Discussion and Conclusions: Design and creation of didactic materials based on three-dimensional and flat modeling tools, methods and algorithms of computer graphics in the system KOMPAS-3D, their use in teaching geometric-graphic disciplines, the inclusion in the evaluation funds on descriptive geometry, engineering and computer graphics, contributes to the introduction of active learning methods to optimize the educational process of teaching of geometry and graphic disciplines.

Abstract. 1. Introduction: In connection with the introduction of new Federal state educational standards of higher education (FSES) to teachers of geometry graphic departments there was a problem with the revision of the contents and methodology of teaching disciplines "descriptive geometry", "Engineering and computer graphics", foundations of engineering education aimed at formation of basic knowledge for studying of the subsequent disciplines used graphic representation of information. The analysis of special literature has indicated that, despite the significant number of scientific publications on the research topic, the use of modeling in teaching geometric-graphic disciplines requires separate research. 2. Materials and methods: Study and analysis of scientific, educational and special literature on the research topic, comparison, systematization, synthesis, generalization. 3. Results: To implement modern teaching methods in the educational process in the geometric-graphic disciplines, the author applied the functionality of the system KOMPAS-3D and developed by: electronic slides based on the modeling included in the evaluation funds for engineering and computer graphics; set a clear three-dimensional (3D) models of twenty-assemblies products intended for assignments on practical and laboratory lessons on the computer in the system KOMPAS-3D to create an Assembly and working drawings; set of individual tasks designed to practical assignments on the computer in this system to create 2D-models of assemblies (Assembly drawings) and 2D models details (working drawings), associated with their three-dimensional models. 4. Discussion and Conclusions: Design and creation of didactic materials based on threedimensional and flat modeling tools, methods and algorithms of computer graphics in the system KOMPAS-3D, their use in teaching geometric-graphic disciplines, the inclusion in the evaluation funds on descriptive geometry, engineering and computer graphics, contributes to the introduction of active learning methods to optimize the educational process of teaching of geometry and graphic disciplines.

Introduction
Creating high-quality products in modern conditions of industrial development implies the use of high-tech complexes and systems requiring qualified maintenance. Accordingly, special requirements are imposed on training of the engineering and technical staff. They should be capable to perform effective work in their specialty, to make high-quality decisions based on operational information, communicative and responsible.
The developed teaching aids, that are required for an effective educational process aimed at new computer technology development, should include modern software and technical means for industrial use [2].
The adoption of the new FSES, that has been driven by natural changes in industry, education system, and public life, requires updating of the educational process and introducing new special disciplines. There is a need to adjust the allocation of time for the educational process due to a decrease in classroom study hours and, accordingly, an increase in hours for independent study of the material by students. In this context, it is impossible, without the use of information technology, to train a highly qualified specialist who can work independently and apply the acquired skills.
Thus, the demand for a detailed development of the scientific methods for teaching geometricgraphic disciplines, focused on the development of the student's personality and on the modern production needs, was disclosed in [28].
The analysis of scientific and pedagogical publications on the topic of our study suggests that, despite a great number of studies on the development of methods, didactic materials, techniques for teaching graphic disciplines, it is important to sever the use of modeling in teaching as an independent area of scientific research.
The use of computer-aided design systems in the educational process is a prerequisite for implementing the educational standards for the geometric-graphic disciplines, mastering of which requires spatial thinking and the ability to understand and to reproduce graphic information [12].
In descriptive geometry, drawing up images is based on the projection method, which allows displaying volumetric elements on a plane. Images on a plane are usually called projections used to solve spatial engineering and geometric problems. A comprehensive perception of the original object and its projections is the main difficulty for students who need a lot of effort, ability, and time to overcome it.
Undeniable is the need to develop spatial thinking as the main characteristic of the engineer's technical intelligence [29]. The use of volumetric computer graphics, animation, and electronic slides for teaching geometric-graphic disciplines greatly simplifies the task of visual representation of geometric objects. In our opinion, the development of spatial imagination is facilitated by the use of modeling-based didactic materials.
The purpose of this study is to develop didactic materials based on modeling and to test them in the university's educational process.
The objectives of the study are:  developing and creating didactic materials (electronic slides, sets of problems) in geometricgraphic disciplines using the volumetric and plane modeling principles;  testing the didactic materials in the university's educational process;  applying the didactic materials in the evaluation funds in geometric-graphic disciplines.
The scientific novelty of the study lies in the development of modeling-based didactic materials (electronic slides, sets of tasks for practical exercises) and their implementation in the educational process, educational publications, and evaluation funds in geometric-graphic disciplines.
The reliability and scientific validity of the results obtained are provided through a comprehensive use of modern research methods, the comparison of research results with the studies and conclusions of domestic and foreign authors, an analysis and generalization of scientific publications on the didactic aspects of teaching geometric-graphic disciplines, and the latest domestic and international research on the problem under consideration. In addition, we reviewed publications on the development and implementation of computer graphics in the educational process, the use of innovative technology based on 3D modeling in teaching geometric-graphic disciplines.

Materials and methods
The paper analyzes the service functions and capabilities of KOMPAS-3D, a modern system of 3D solid-state modeling (ASCON, CJSC, http://www.ascon.ru) in the development of didactic materials for geometric-graphic disciplines based on volumetric and plane modeling.

Results
Modern CAD systems, including KOMPAS-3D, actively use methods of volumetric and plane modeling to create geometric models of designed objects.
Computer graphics makes it possible to automate the design document management, the tasks of which include presentation, preparation for visualization, creation, and implementation of actions with an image.
Today, many software programs are systems for interactive computer graphics, where the user can control the object properties.
CAD uses wireframe, polygonal, and volumetric models, which are the main types of mathematical models of geometric objects.
It is known that structural elements of a wireframe model are a face and a point. This model is used to describe engineering objects with flat approximating surfaces. Geometric objects on a plane can be represented using wireframe models. Such images are ambiguous and do not allow for automating the processes of removing invisible lines and obtaining sections.
Complex curved faces of engineering objects can be described using polygonal models. To build such a model, you have to involve the algorithm for creating an object by performing logical operations of intersecting or touching surfaces.
When creating a polygonal model, any surface is approximated by a polyhedron; all its faces are the simplest flat polygons; this process also uses quadrics and curved surfaces, allowing to carry out complex calculations and to work interactively with the model (examples are the Koons and Bezier surfaces).
An advantage of replacing complex surfaces with flat faces is the simplicity of mathematical methods when working with such models. A drawback of the method is an increased number of inmachine calculations and the memory used upon complicating the object shape or increasing the design object size through increasing the number of its faces.
Volumetric models allow rendering complex engineering objects with a high accuracy. The structural elements of such a model are a point, a contour element, and a surface. Structural elements form nodes of a network structure in the process of synthesizing a volumetric model. Using such a model, points can be divided into internal and external ones in relation to a real object at any time.
Requirements for high-level geometric modeling (Y. Gardan, M. Lucas) are: model regularity; ability to comprehensively create a complex object; ability to calculate engineering and geometric parameters. A model must be uniform, finite, rigid, compact, and open.
When modeling complex solid-state objects, the generatrices of which are curved lines, it is most expedient to use the basic types of mathematical models for geometric objects. This makes the algorithm for solving engineering-geometric problems of computer-aided design extremely simple.
Modeling is an essential element of any applied research or engineering development. According to the Big Soviet Encyclopedia, modeling is "a study of real objects of knowledge on their models; creating and studying models of objects and phenomena (living and inanimate systems, engineering structures...) and constructed objects (to determine, identify their parameters, to rationalize their creation methods…)". Computer geometric modeling associated with computer graphics includes methods and algorithms for the internal representation and transformation of geometric models on a computer. Models in computer graphics can be 2D: two-dimensional or flat or 3D: three-dimensional or volumetric. The use of applied software for the development and execution of design technical documents implies the use of flat 2D vector modeling in two ways:  the user sequentially sets the primitive to be created, and a list of relations and primitives where these relations are defined;  first, the object to be converted is created; then, the conversion command is executed, which is followed by analytical calculations and operations with a vector mathematical model.
The versatile computer technology includes volumetric computer geometric modeling, which is divided into the following types of modeling, depending on how the geometric body is described: wireframe modeling (set of vertices and edges); surface modeling (set of limiting surfaces); solid-state modeling (it consists of many simple volumetric elements using Boolean operations).
There are two main engineering tasks of computer modeling for volumetric bodies: creating a computer model of an existing product or its mathematical model; synthesis of the shape of a designed product that never existed (even in the form of material models).
Modern CAD systems deal with all types of geometric models and can even create combined models that include various element types. The geometric modeling of products uses a limited set of basic elements -objects or primitives: 2D objects, surfaces, and volumetric primitives.When performing logical operations of "intersection", "combination", and "subtraction" with simple volumetric elements using a certain algorithm, you can create a complex 3D solid-state model.To create a volumetric element in space, it is enough to shift a plane form (known as a sketch in the KOMPAS-3D system), relative to itself using the "Movement" operation. A sketch is created using 2D graphics tools and can consist of one or several contours; it also can be situated on one of the projection planes or on some auxiliary surface or on a plane face of the existing element.
Volumetric elements in the KOMPAS-3D system are created using basic operations:  Extrusion: the sketch moves perpendicular to its own plane.  Rotation: a sketch drawn by the base line rotates around its axis (an axial line segment). In this case, it is important for both elements to be in the same plane and for the sketch to be located only on one side of the axial line.  Kinematic modeling: the sketch as a generatrix is moved along a guide line created.  Lofted modeling: several different sketches act as guide lines.
To modify a model by adding or removing some elements, there are combination and subtraction operations. Such an operation includes additional options for changing or refining the rules for creating a volumetric element. Multiple cyclical repetitions of logical operations allow you to create a volumetric model of any shape and complexity.
The use of modeling in teaching geometric-graphic disciplines is as follows: Development of electronic slides. Didactic material means a kind of teaching visual aids for training sessions. As you know, pictures, tables, and maps, that are handed out to students in classroom lessons and for independent work; the materials developed on the basis of computer technology or demonstrated by the teacher in the classroom during the learning process contribute to:  developing the cognitive independence;  enhancing the cognitive activity of students;  enhancing their creativity;  reflection;  activating the both hemispheres of the brain;  reducing the time for presenting new material to students;  improving the motivational component of education;  developing the ability for cultural self-realization [4].
Based on the KOMPAS-3D computer modeling system, the authors has developed teaching aids and added them to the evaluation funds for the following disciplines: Engineering and Computer Graphics, Computer Graphics, and Computer Modeling for students in engineering specialties. Figures  1-2 shows examples of slides with a frame-by-frame breakdown of the educational material of the topic "Images: Views, Profiles, Sections" in engineering graphics.  The basic conceptual idea for developing the structure and content of classes in university geometric-graphic disciplines was creating visual electronic slides that break down educational material by elements. The use of volumetric and plane modeling, as an active tool for solving various engineering and graphic problems, provides for the development of professional competencies of a future engineer.In the process of training, it becomes clear to students that a volumetric model describes the geometry of all surfaces of complex shapes and contains all the necessary information about the configuration, shape, and dimensions of the product. Geometric modeling applications are usually Windows-based systems. They provide for volumetric parametric modeling, execution of various design documents in accordance with modern requirements of the standards set by the Unified System for Design Documents; extensive libraries of standard products are integrated into their shells.

LOCAL PROFILES
Local profile is a profile used to clarify the object structure in some limited area The KOMPAS-3D system is actively used for training students in engineering specialties at the Orenburg State University, the Tyumen Industrial University, and other Russian high schools. This a versatile system for development and designing engineering design and process documents in many areas of the Russian industry, allowing to create volumetric models of parts and their combinations at different complexity levels, as well as associatively related drawings -plane models.
Teachers of departments of graphic disciplines develop and create electronic slides in the KOMPAS-3D system based on the volumetric solid-state geometric modeling technology. Volumetric models of parts featuring complex geometric shapes shown on slides are created through a step-bystep execution of Boolean operations of mathematical logic on volumetric elements.
In addition, these slides are used as part of evaluation funds in geometric-graphic disciplines. A clear structure, integrity, and interdisciplinarity of the educational material contribute to its highquality perception. It is important that the educational material for lectures and practical classes should be developed taking into account the effective visual perception of information. At their lectures, teachers demonstrate visual computer slides and other visual aids that contribute to the development of student's geometric-graphic competencies.
Developing the set of tasks "3D models of Parts and Assembly Drawings of Products" intended to perform the task "Creation of Assembly Drawings and Working Drawings of Parts" in engineering and computer graphics in the KOMPAS-3D graphic editor [1,3].
"Assembly" is an electronic document (*.a3d extension) to create assembly units from solid parts in perspective view; "part" is electronic document for creating solid-state models (a file with *.m3d extension).
In the graphic KOMPAS-3D system, the authors have developed a package of 3D models (20 sets) of products, where each part, including standard products, is implemented in its own color scheme.The procedure for creating a 3D "Assembly" model includes the following steps, corresponding to the product specification: creating volumetric models of parts; 3D models of assembly units (if any); 3D models of standard products and other elements. The last step is to build a 3D model of the entire product -"Assembly".In general, solid-state modeling means creating objects using Boolean operations (addition, subtraction, and conjunction. The shape of the designed volumetric element is defined according to the rule of descriptive geometry -by moving a flat figure along a guide line. Each "Assembly" in the proposed task is a solid-state model, which includes volumetric models of all parts, as well as assembly units and standard products. In addition, the system contains information about the relative position of components and relations between the parameters of their elements. Models of individual parts are independent files, whereas the assembly file contains only links to them and a description of the relative position or creation of parametric links between their elements (edges, faces, and surfaces). Removing or adding new parts defines the "Assembly" composition.
For more clarity, the tasks are executed with a quarter profile by cutting off with planes. A flow chart for creating a 3D model of an assembly using the example of the Roller product is shown in figure 3; examples of some 3D models of product assemblies are shown in figure 4.
Thus, the use of teaching aids, that include a set of 3D models of product assemblies created in the KOMPAS-3D system based on volumetric modeling, helps optimize the educational process in geometric-graphic disciplines and allows creating a set of design documents based on the same: working drawings of parts, assembly drawings, and specifications.
The commands for creating associative views are in the menu "Insert" -"View from the Model" and "Insert" -"Auxiliary View". The "Views" panel contains command buttons. In the standard mode, KOMPAS-3D generates projection views, local views, arrow views, profiles, sections, detail elements, and local profiles. When creating an associative drawing, an associative relationship with the model is preserved: if any changes are made to the model, its associative projections are automatically changed too. An associative relationship between views and the model can be also deleted, if necessary. It is also possible to disable rendering of the elements that do not need to be shown in the drawing, if required.  Figure 3. The scheme of eating a 3D model of the product Assembly "Roller".
In one drawing, there may be several different associative views of different models. When creating a profile or a section, you can select "uncut" elements of the product. The "drawing tree" is used to manage the views.
To create arrow views, profile, sections, or detail elements, commands are used that are enabled automatically, after the image of the corresponding elements is shown in the drawing. These commands can be also activated manually. Designations and inscriptions in a drawing of views, profile, sections, and detail elements are automatically enabled and match each other. In the "Materials and Assortment" reference book, you can assign attributes with information about the model and material weights; they are automatically assigned also to the drawing with associative views.
The procedure for creating associative drawings of 2D models of product assemblies and 2D models of parts included in the relevant product is described below: Creating 3D assembly model components b) c) Figure 4. Examples of 3D models of assemblies a) backup the built-in screw; b) door; с) surface gauge.
 Open or create a model; using the "Hide" command, disable rendering the model elements that should not be shown in the drawing; add dimensions and designations, if you need to transfer them from the model; define the standard model orientation, for the front view to be fully shown; a custom orientation can be added, if necessary.  The commands for creating an arbitrary view in the KOMPAS-3D system are used to create assembly drawings and drawings of complex parts. Using the command for generating standard views, you can create a drawing of a simple model with all necessary projections.  Create projection images based on standard and arbitrary views created in the drawing.  In associative views, you need to perform image editing; to do so, customize the display settings in the views of objects and design elements available in the model; select "uncut" elements; disable the display of elements that should not be shown in the drawing.  Before generating the drawing: apply the required dimensions; add designations and inscriptions; you need to enable associability and parameterization in order to associate the constructed objects with the model and to track its changes. A profile or section line, the direction of the gaze arrow, and the designation of the detail element are marked automatically and have an associative relationship. Drawing elements with the same functional significance are recommended to be performed in separate layers for the convenience of generating and editing the drawing.  Disable the projection link between the views, if necessary, and compose the drawing on the sheet.  The set of individual tasks contains technical descriptions of the product designs.

Discussion and Conclusions
The use in the educational process of educational and methodological publications, including a set of 2D models of product assemblies (assembly drawings) and their 2D models of parts (working drawings), that are associatively related to their 3D models created in the KOMPAS-3D system based on plane modeling, allows to significantly improve the quality of the educational process in the study of geometric-graphic disciplines.
The use of volumetric and plane modeling in teaching geometric-graphic disciplines contributes to the visual perception of a complex geometric object by students, interiorization, and "objectification" of a virtual model. The ability to analyze a complex drawing, to select simple geometric components from an object of complex geometric shape allows them to learn how to pass from volumetric models to their flat counterparts (associative drawings), which greatly simplifies the drawing editing process. Therefore, the development and creating of didactic materials (visual electronic slides with a frame-by-frame breakdown of the educational material from the section "Images: Views, Profiles, Sections", set of individual tasks "3D Models of Parts and Product Assemblies" and "Assembly Drawing. Detailing") based on volumetric (3D) and plane (2D) modeling using modern tools, methods, and algorithms of computer graphics in the KOMPAS-3D system, their application in teaching geometric-graphic disciplines, and including them in the evaluation funds of geometricgraphic disciplines allow to implement active teaching methods to improve the quality of the educational process in geometric-graphic disciplines.
The scientific and practical significance of this study is due to the topic relevance and the research novelty. Its results can be used by university teachers during the training in descriptive geometry, engineering, and computer graphics. Our materials and findings can also be a basis for further developments in the application of volumetric and plane modeling in teaching general professional and special disciplines.


Pedagogical Counselors’ Self-exploration of Writing Feedback on Preservice Teachers’ Teaching Journals

The study examines types of feedback pedagogical counselors write in response to preservice teachers’ reflective writing in Teaching Journals during their practicum. The study also examines variables that can affect written feedback on reflective writing: year in the college, preservice teachers’ levels, changes in feedback over time and personal differences among counselors. 689 comments written by five counselors from 74 Teaching Journal entries were explored. The results showed eight feedback types, in which asking for clarifications and leading preservice teachers to infer from pupils’ responses, were the most frequent. Differences of feedback types written to student teachers was affected by year in the college, by levels of the student teachers and by the context of the practicum. The discussion will focus on the eight feedback types and the linguistic aspect of the feedback.

Published by SCHOLINK INC. synergy created by the preservice teachers. It encourages preservice teachers' responsibility for their learning by sharing ideas and developing insights among colleagues, skills that will benefit them throughout their careers. All forms of journals benefit from feedback, which deepens self-exploration.
Feedback: Feedback can be a powerful tool if performed well. Its power lies in addressing both cognitive and motivational factors, giving learners information they need so they can understand how to react and what to do next. Hattie and Timperley (2007) proposed a model describing the complexity of feedback in three dimensions: feed up, feedback, and feed forward, by asking three questions: Where am I going? How am I going? and Where to next? Each question can address four levels of feedback: (1) about the task, (2) about the process (strategies that could be used), (3) about self-regulation (preservice teachers are informed how to continue), (4) about the preservice teacher as a person.
In an instructional setting, feedback is described as one of the many procedures that inform a student whether his/her response is right or wrong. Additionally, feedback can also provide instructional information to the student that gives a reason for why a response is correct or incorrect. The meta-analysis of Hattie's database of 7000 studies (Hattie, 2015) point at feedback as one of the most effective variables in improving learning.
Researchers in teacher education realized that feedback messages are filtered through the preservice teacher's perception, influenced by their prior knowledge, experiences, and motivation (Kagan, 1992). The research review of Butler and Winne (1995) revealed that external feedback (such as teacher feedback) and internal feedback (such as preservice teacher self-evaluation) affected preservice teacher knowledge and beliefs. Kluger and DeNisi (1996) found that the effects of feedback depend on its nature. For example, feedback that includes what to do and why, helps preservice teachers develop a feeling of control over their learning, thus acting as a motivational factor (Brookhart, 2008). Preservice teachers must be engaged with the feedback process in order to develop self-regulated skills (Tunstall & Gipps, 1996).
Feedback to Writing Journals: The purpose of feedback in higher education is to help preservice teachers monitor, evaluate and regulate their own learning (Nicol & Macfarlane-Dick, 2013). Although reflection is an act of self-exploration, researchers claim that reflective writing without feedback does not improve reflective thinking (Bain, Ballantyne, Packer, & Mills, 2002;Davis, 2006;Cohen-Sayag & Fischl, 2012). Thomas and Packer (2013) claim that preservice teachers need help in their reflective processes otherwise they will focus on "thinking about the tasks" and "thinking about the pupils" and will leave "thinking about myself as a teacher" behind. Consequently, pedagogical counselors should encourage preservice teachers to rethink their values and recognize prejudices and contradictions between values and actions. Descriptions of the dialogue during fieldwork in Nickle's (2013) study gave us a glance on feedback of the instructor and its importance on navigating students to delve into their reflection, but this study was among student teachers in their first semester of university and was not focused on teaching. As we inquire the research on reflective writing and the feedback of supervisors, we recognized two problems, which we experienced in our work as pedagogical counselors: First, Dissatisfactions from the feedback on reflective writing: Nicol (2010) described the dissatisfactions and difficulties of the feedback on the reflective writing from both sides, students claim on unclear feedback and Pedagogical counselors' report on misunderstandings, describing feedback as useless act which the student teachers did not change/improve their acts as suggested in the feedback. Second, a contradicting point on writing feedback: Immediate corrective feedback was found to be more effective than delayed feedback (Gürkan Serkan, 2018), without immediate feedback might leave the internal dialogue of the preservice teacher at a low and immature level of reflection. Yet, feedback from the pedagogical counselor can be interpreted as an act of assessment and might limit the self-exploration process (Ditchburn, 2015).
Considering these problems and the claim that reflective writing can be effective if preservice teachers take an active role in the process (Nicol, 2010), we decided to explore our feedback in an open platform, which will not entail remarks on the quality of reflection but only on submitting Teaching Journals.
Preservice teachers were also encouraged to respond to our questions and were not evaluated on their level of reflective writing, but were rather credited for the act of submitting a reflective journal.
Creating the right environment of reflection on teaching we decided to delve into our written feedback in order to understand how we encourage students to take a critical position and what affect our feedback.
Our main question was: What are the characteristics of counselors' feedback to reflective writing and which variables shape feedback to reflective writing?

Method
The study used a mixed method array regarding the following research questions: 1) What types of comments did we, the pedagogical counselors, write in response to the Teaching Journals?
2) To what extent did our comments on reflective Teaching Journals differ by year of learning in the college, iteration (first, second, third time of reflective writing), by teaching level of the preservice teachers in the practicum, and by individual differences among the counselors?
Questions 1, was analyzed qualitatively, while question 2 was explored quantitatively.

Participants
25 preservice teachers and five pedagogical counselors from the special education department in a teacher education college in the south of Israel participated in the study. Two supervisors were relatively inexperienced (four years or less) and three had an average of 20 years of experience. Three pedagogical counselors were experts in Learning Disabilities (LD) and two were experts in Complex Developmental Disorders (CDD). Three were supervisors in the second year and two in the third year.
At the end of the year, each supervisor chose 5 preservice teachers from her group who displayed three levels of performance in field experience (low, moderate, high), in terms of planning and teaching in class, which were classified with the consultant of the cooperative teacher. Scores of 88-100 in field experience were considered "high level", scores of 75-87 were considered "moderate level" and scores below that indicated "low-level" (Table 1). These 25 preservice teachers submitted 74 Teaching Journals divided between second/third year and three levels of preservice teachers as follows: The Context-Practicum in Special Education Department: The special education department enables the student to choose between focusing on Learning Disabilities (LD) practicum or Complex Developmental Disorders (CDD). This focus starts in the second and continued in the third year of their learning. Its meaning is mainly in the practicum but also in courses they can choose. LD needed to take 6 hours on reading, and language teaching and diagnosis and a course in emotional disturbances, while the focus on CDD courses on complex disability suggest, autism, social development issues of CDD children and a course on learning to use technology for the benefit of improving communication and learning.
The practicum was two days a week, LD students in regular schools, small classes or individual teaching with children in the Inclusive education program. The focus was on teaching reading and writing strategies at the different stages, alphabetic phase, solid alphabetic phase and fluency and reading comprehension. The student also practiced subject matter teaching history mathematics and other subjects according to their decision with the cooperative teacher in the classroom. Student teachers in CDD focused in their teaching on Literacy, communication capabilities and subjects related to daily life, Holidays and seasons. We decided to use these question because we recognized in the questions potential to lead the students to the five levels of reflection which was described in Bain, Ballantyne, Parker and Mills (1999):

Research Tools and
Reporting (simple descriptions of the lesson), Responding (descriptions of writers' responses to the occurrences in the lesson), Relating (explanations of the occurrences), Reasoning (analysis of the occurrences in the lesson based on Educational Literature) and Reconstruction (integration of theoretical concepts, self -experience and knowledge).
The reflective writing guide according to the five levels: 1) Explain the objectives of this lesson and the activities that you planned to achieve them.
2) What is the topic? Why did you choose to teach this topic and what is its importance?

Writing Feedback
Pedagogical counselors wrote their feedback on the reflective writing of student teachers. In order to start a dialogue with the preservice teachers, each supervisor wrote comments on the preservice teachers' reflective writing. The comments were intuitive and had no guide, but since starting a dialogue was in our mind we used a lots of questions which the students had to respond. We read the reflections and responded in various ways, relating to the lesson plan, its implementation, pupils' responses and the preservice teachers' critical thinking. We also responded to linguistic errors. This correspondence was conducted via Google Drive which enable us to create a follow up on the students reflective writing and on our responses on time.

Research Procedure
As part of their practicum assignments, the preservice teachers reflected on their teaching experiences in Teaching Journals three times each semester. We, as pedagogical counselors, wrote feedback comments in order to deepen their critical thinking. The process involved five steps: 1) Each pedagogical counselor presented the reflective writing guide, explaining its importance to their practicum and to teacher education development in general. Examples of reflective writing were exhibited in order to demonstrate different levels of writing. Discussion was held in order to enable the students to ask questions.
2) The preservice teachers wrote Teaching Journals about a lesson we had not observed, since our presence ended with face-to-face feedback which gave the preservice teacher insights and ideas on the lesson and would affect their reflective writing. Each reflection was written 1-4 days after the teaching lesson took place.
3) Reflective writing was sent to the Pedagogical counselor within a week of the lesson. The preservice teacher wrote: It may be that the pupils perceive us now as an authority more than at the beginning of the semester.
The pedagogical counselor's comment: I think this is not a matter of authority but of your caring attitude towards them [pupils], they feel that you try very hard to promote their learning abilities.
(Preservice teacher did not respond).
2) Context: A preservice teacher asked the pupils to discuss questions about a story they read in small groups and encountered pupil's difficulties in maintaining the discussion in groups.
The counselor's comment: If you had to perform the activity again what change would you make?
The preservice teacher wrote: If I had to teach the whole class again I would change the activity from a group activity to a whole class activity.
The counselor's response: Why are you giving up on group work?
The preservice teacher's response: It is important to mention that I did not give up, and I'm a great believer in group work. But, because the pupils were asked to discuss in groups, I understood that the difficulty lay there. Therefore, looking back, I would give them pair assignments. I think it would be easier for them and it could be a step towards group work.
3) Context: A preservice teacher expresses her opinion on the importance of the subject she is teaching: The preservice teacher wrote: It is important to teach the students about the "seven species of fruits" (Shiv'at Haminim). It is a Jewish tradition that the students should be aware of. Preservice teacher's response: Thank you for this comment honestly I did not think of it at all. It is part of the annual program.

Data Gathering
The overall data of reflective writing included 74 Teaching Journals written by preservice teachers, 44 of which were written by second-year preservice teachers, and 30 by third-year preservice teachers.
Our data of comment types was based on 689 comments which we, the pedagogical counselors, had written as feedback to the Teaching Journals submitted by the preservice teachers across two semesters.
The large number of comments enabled us to perform statistical analysis of the feedback with different variables as an answer to the second research question regarding differences of supervisors' feedback by year of learning in the college, iteration (first, second, third time of reflective writing), by teaching level of the preservice teachers in the practicum, and by individual differences among the counselors.

Data Analysis
Feedback writing of the supervisors was analyzed qualitatively by grounded theory process: 1) Analysis of comment type: We started by reading the data in pairs suggesting categories of comment types. The classification of the comments was analyzed in the group of all five researchers, in order to achieve consent. This process was repeated several times until the final version reported here was reached ( Table 2).

Results
The results will be presented in two parts: frequencies of feedback types, and differences that affect feedback types.

Frequencies of Feedback Type
The graph below shows that one quarter of all the comments appeared in the asking for clarifications category. The other six comment types were quite evenly distributed and ranged from 15% to 8%:   to syntax and spelling than to paraphrasing. g. The most frequent form of connecting theory to practice was asking to connect theory presented in the courses to their practice.
h. Directing preservice teachers to observe the cooperative teacher was more common than consulting with the cooperative teacher or other staff members. (Table 4) A Pearson t-test between second-and third-year preservice teachers indicates statistically significant differences for three comment types: 1) Suggestions for instructional moves were more frequent in the third year. 2) Asking for conclusions was more frequent in the second year. 3) Asking preservice teachers to consult cooperative teachers and other staff members were more frequent in the third year.

Teachers
Two of these differences between writing comment types and preservice teachers' levels were statistically significant: Expression of opinions and feelings were more frequent for low-level preservice teachers, and asking preservice teachers to connect theory to practice was twice as high for high-level ones than for moderate ones and six times higher than for low-level ones.

Differences by Pedagogical Counselors as Individual
Three pedagogical counselors (A; G; E;) were experts in Learning Disabilities (LD) and (D; N) two were experts in Complex Developmental Disorders (CDD). Pedagogical counselors differed in writing feedback types as presented in Table 6. Post hoc Duncan test indicates the source of these significant differences: 1) Counselors A, D and N wrote more comments expressing opinions and feelings.
2) Counselors G and E wrote more linguistic comments.
3) Counselors D and N wrote more comments asking for clarifications. They also wrote more suggestions for instructional moves, fewer requests to connect theory to practice and more comments directing the preservice teachers to consult with school staff. It seemed as if N and D supervisors of the CDD had more in common than the supervisors of the LD.

Summary of results:
The results presented a broad description of feedback types on reflective writing and several significant differences regarding the variables that we explored: 1) Feedback comment types-Eight feedback types aimed to encourage a high level of reflectivity by asking for more information about what happened in the class, leading preservice teachers to infer from pupils' responses, urging the preservice teachers to express their opinions, asking them to link theory to practice and directing them to revise their writing. Asking for clarifications was the most frequent comment type; counselors needed these clarifications in order to understand the procedures of instructional acts and the preservice teachers' responses. We will discuss this result.
2) Changes of feedback over time-Feedback changed over time but only linguistic comments lessened significantly over time. We do not think that the preservice teachers improved their writing through such short experience of reflective writing, but perhaps they became more aware of their writing after reading the linguistic comments.
3) Writing feedback to second or third year students-The pedagogical counselors wrote more comments with practical suggestions and comments about turning to the cooperative teacher and the staff to linguistic corrections that we focused on was partly due to their poor level of articulation in Hebrew.
But the explanation for the second comment type we used might be that with low-level preservice teachers we did not felt free to raise doubts in order not to undermine their confidence; instead we asked them to explain their feelings and express their opinions.

5)
Writing feedback and the context of the practicum-All comment types were affected by the complex developmental disorders context (second and third year). The Complex Developmental Disorders CDD counselors made more suggestions for instructional moves and directing preservice teachers to consult with cooperative teachers than other counselors. This result might be due to the complex difficulties dealing with Individuals complex difficulties.

Research limitations:
This study has two limitations: conclusions from these findings need to be handled with caution due to the small sample of counselors all from one college in one special education department. These might significantly limit the generalizability of the results.

Discussion
The discussion will focus on the eight feedback types and the context of feedback. The second part will discuss on asking for clarifications as major result and as linguistic aspect of the feedback.
Our motive to portray the types of written feedback was to reduce the gaps on the interpretation of teaching lessons between preservice teachers and their pedagogical counselors. The process we created in five steps invited the students to respond to feedback they received. This was an act of changing view on their teaching lessons, not an act of evaluation as Dylan (2016) suggest. In order to achieve this goal our feedback was delivered without evaluation of the reflective writing of each student.
Our self-exploration pointed out on eight feedback types which can be concluded to four levels directing student teacher's to professional development: (a) empowerment of the preservice teachers-by praising, sharing opinions and feelings; (b) improving teaching skills by explicit and implicit suggestions for instructional moves; (c) reconstructing instructional knowledge-by asking them to connect theory to practice, read up on the matter they were writing about and suggesting they consult with staff; (d) encouraging critical thinking-by asking for conclusions and clarifications on teaching moves and on pupils' behaviors.
Our findings on eight types have some similarity to late publication we found of Novillo Paula and Pujolà (2019)  well as consulting with cooperative teachers than in the learning disabilities field experience. This result was found in our previous study (Cohen-Sayag & Fischl, 2012) indicating the context of field experience affecting the type of comments pedagogical counselors make on reflective writing.
Since our main feedback was asking for clarifications we wondered to what extent our feedback was mainly linguistic. Although asking for clarification might appear to be a merely linguistic matter, we distinguished them from the linguistic comments because this was characterized by focusing on the content of the lesson-asking questions about class activities, plans, interactions and navigation. These questions usually caused rethinking the teaching moves and interactions as demonstrated in the example dialogues. We can conclude that, the pedagogical counselors' feedback, demanded clarity from student teachers' reflective writing above all by asking for clarifications more than any other feedback type.
Clarity is one of the main demand of writing in general, but here our clarity demands was in context of teaching and thus lead students to rethink and rephrase their teaching descriptions. The very act of writing can support and enhance critical thinking as part of the writing processes (Monyanont, 2014), by our demands for clarity we widened their critical point of view.
Most of the pedagogical counselors were convinced that linguistic comments are important.
Nevertheless, the common thinking was that linguistic comments might hurt the writer's feelings.
Furthermore, focusing on linguistic comments is not the main aim of reflective writing in teaching journal. In order to overcome this difficulty, we asked the preservice teachers whether they would like us to give feedback with linguistic comments or to ignore the linguistic aspect. All the preservice teachers were enthusiastic about improving their writing, some of them set a meeting to discuss their writing difficulties. A discussion like this is a strategy recommended by Kohn (1992), who argues that advancing reflective thinking in reflective writing depends on the comment type and context. He suggests encouraging discourse between the writer and the commenter.

Implications for teacher education:
The implications of this study to teacher education are four: a. Self-exploration of pedagogical counselors' feedback was our main derive of this research and we think it should be an important act of teacher education staff.
This research experience is vital to the development of pedagogical counselors in their efforts to create significant field experience for their preservice teachers. We recommend this self-exploration process for beginner pedagogical counselors, to explore their own feedback through such process. We think this point of observation can pave the way to reduce gaps and tensions between preservice teachers and their counselors that have been reported in several studies (Frankel, 2002;Cohen-Sayag, Hoz & Kaplan, 2012).
b. The use of comment type that we found can be used as indicators for self-exploration of pedagogical counselors to look at their feedback from different point of view. Further studies are needed to give a broad perspective on feedback types to reflective writing from different teaching education programs.
c. Our main suggestion is to promote preservice teachers' writing abilities through reflective writing.
Although it is not the aim of reflection, but since reflection is vital in the process of writing, it seemed Published by SCHOLINK INC.
that reflective writing is a good platform to enhance writing abilities in the context of professional environment.
d. The research process not only revealed differences between us, but also raised questions about the differences in the feedback given to second/third year preservice teachers and to the varying performance levels of the preservice teachers. We recommend that counselors be aware of these variables, mainly of the type of feedback we write to low-level preservice teachers in their field experience.