Thermal Energy EquationΒΆ

\(\mathrm{f}_1\mathrm{f}_4\boldsymbol{v}\cdot\boldsymbol{\nabla}\Theta\)

1401

rhotv_grad_s

\(\mathrm{f}_1\mathrm{f}_4\boldsymbol{v'}\cdot\boldsymbol{\nabla}\Theta'\)

1402

rhotvp_grad_sp

\(\mathrm{f}_1\mathrm{f}_4\boldsymbol{v'}\cdot\boldsymbol{\nabla}\overline{\Theta}\)

1403

rhotvp_grad_sm

\(\mathrm{f}_1\mathrm{f}_4\overline{\boldsymbol{v}}\cdot\boldsymbol{\nabla}\overline{\Theta}\)

1404

rhotvm_grad_sm

\(\mathrm{f}_1\mathrm{f}_4\overline{\boldsymbol{v}}\cdot\boldsymbol{\nabla}\Theta'\)

1405

rhotvm_grad_sp

\(\mathrm{f}_1\mathrm{f}_4v_r\frac{\partial s}{\partial r}\)

1406

rhotvr_grad_s

\(\mathrm{f}_1\mathrm{f}_4v_r'\frac{\partial \Theta'}{\partial r}\)

1407

rhotvpr_grad_sp

\(\mathrm{f}_1\mathrm{f}_4v_r'\frac{\partial \overline{\Theta}}{\partial r}\)

1408

rhotvpr_grad_sm

\(\mathrm{f}_1\mathrm{f}_4\overline{v_r}\frac{\partial \overline{\Theta}}{\partial r}\)

1409

rhotvmr_grad_sm

\(\mathrm{f}_1\mathrm{f}_4\overline{v_r}\frac{\partial \Theta'}{\partial r}\)

1410

rhotvmr_grad_sp

\(\mathrm{f}_1\mathrm{f}_4\frac{v_\theta}{r}\frac{\partial \Theta}{\partial \theta}\)

1411

rhotvt_grad_s

\(\mathrm{f}_1\mathrm{f}_4\frac{v_\theta'}{r}\frac{\partial \Theta'}{\partial \theta}\)

1412

rhotvpt_grad_sp

\(\mathrm{f}_1\mathrm{f}_4\frac{v_\theta'}{r}\frac{\partial \overline{\Theta}}{\partial \theta}\)

1413

rhotvpt_grad_sm

\(\mathrm{f}_1\mathrm{f}_4\frac{\overline{v_\theta}}{r}\frac{\partial \overline{\Theta}}{\partial \theta}\)

1414

rhotvmt_grad_sm

\(\mathrm{f}_1\mathrm{f}_4\frac{\overline{v_\theta}}{r}\frac{\partial \Theta'}{\partial \theta}\)

1415

rhotvmt_grad_sp

\(\mathrm{f}_1\mathrm{f}_4\frac{v_\phi}{r \mathrm{sin}\theta}\frac{\partial \Theta}{\partial \phi}\)

1416

rhotvp_grad_s

\(\mathrm{f}_1\mathrm{f}_4\frac{v_\phi'}{r \mathrm{sin}\theta}\frac{\partial \Theta'}{\partial \phi}\)

1417

rhotvpp_grad_sp

\(\mathrm{f}_1\mathrm{f}_4\frac{v_\phi'}{r \mathrm{sin}\theta}\frac{\partial \overline{\Theta}}{\partial \phi}\)

1418

rhotvpp_grad_sm

\(\mathrm{f}_1\mathrm{f}_4\frac{\overline{v_\phi}}{r \mathrm{sin}\theta}\frac{\partial \overline{\Theta}}{\partial \phi}\)

1419

rhotvmp_grad_sm

\(\mathrm{f}_1\mathrm{f}_4\frac{\overline{v_\phi}}{r \mathrm{sin}\theta}\frac{\partial \Theta'}{\partial \phi}\)

1420

rhotvmp_grad_sp

\(-c_6\boldsymbol{\nabla}\cdot\boldsymbol{F}_{cond}\)

1421

s_diff

\(-c_6\boldsymbol{\nabla}\cdot\boldsymbol{F'}_{cond}\)

1422

sp_diff

\(-c_6\boldsymbol{\nabla}\cdot\boldsymbol{\overline{F}}_{cond}\)

1423

sm_diff

\(c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\left(\frac{\partial^2 \Theta}{\partial r^2}+\frac{\partial \Theta}{\partial r}\left[\frac{2}{r}+\frac{\mathrm{d}}{\mathrm{dr}}\mathrm{ln}\left\{\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\right\} \right]\right)\)

1424

s_diff_r

\(c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\left(\frac{\partial^2 \Theta'}{\partial r^2}+\frac{\partial \Theta'}{\partial r}\left[\frac{2}{r}+\frac{\mathrm{d}}{\mathrm{dr}}\mathrm{ln}\left\{\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\right\} \right]\right)\)

1425

sp_diff_r

\(c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\left(\frac{\partial^2 \overline{\Theta}}{\partial r^2}+\frac{\partial \overline{\Theta}}{\partial r}\left[\frac{2}{r}+\frac{\mathrm{d}}{\mathrm{dr}}\mathrm{ln}\left\{\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\right\} \right]\right)\)

1426

sm_diff_r

\(c_6\frac{\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5}{r^2}\left(\frac{\partial^2 \Theta}{\partial \theta^2}+\mathrm{cot}\theta\frac{\partial s}{\partial\theta}\right)\)

1427

s_diff_theta

\(c_6\frac{\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5}{r^2}\left(\frac{\partial^2 \Theta'}{\partial \theta^2}+\mathrm{cot}\theta\frac{\partial \Theta'}{\partial\theta}\right)\)

1428

sp_diff_theta

\(c_6\frac{\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5}{r^2}\left(\frac{\partial^2 \overline{\Theta}}{\partial \theta^2}+\mathrm{cot}\theta\frac{\partial \overline{\Theta}}{\partial\theta}\right)\)

1429

sm_diff_theta

\(c_6\frac{\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5}{r^2\mathrm{sin^2}\theta}\frac{\partial^2 \Theta}{\partial \phi^2}\)

1430

s_diff_phi

\(c_6\frac{\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5}{r^2\mathrm{sin^2}\theta}\frac{\partial^2 \Theta'}{\partial \phi^2}\)

1431

sp_diff_phi

\(c_6\frac{\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5}{r^2\mathrm{sin^2}\theta}\frac{\partial^2 \overline{\Theta}}{\partial \phi^2}\)

1432

sm_diff_phi

\(F_Q(r)\)

1433

vol_heat_flux

\(\mathrm{f}_6(r)\)

1434

vol_heating

\(c_5\Phi(r,\theta,\phi)\)

1435

visc_heating

\(\mathrm{f}_7c_4\left( {\boldsymbol{\mathcal{J}'}}\cdot{\boldsymbol{\mathcal{J}'}} \right)\)

1436

ohmic_heat

\(\mathrm{f}_7c_4\left( {\boldsymbol{\mathcal{J}'}}\cdot{\boldsymbol{\mathcal{J}'}} \right)\)

1437

ohmic_heat_pp

\(\mathrm{f}_7c_4\left( \overline{\boldsymbol{\mathcal{J}}}\cdot\overline{\boldsymbol{\mathcal{J}}} \right)\)

1438

ohmic_heat_pm

\(\mathrm{f}_7c_4\left( \overline{\boldsymbol{\mathcal{J}}}\cdot{\boldsymbol{\mathcal{J}'}} \right)\)

1439

ohmic_heat_mm

\(\mathrm{f}_1\mathrm{f}_4v_r\Theta\)

1440

rhot_vr_s

\(\mathrm{f}_1\mathrm{f}_4v_r'\Theta'\)

1441

rhot_vrp_sp

\(\mathrm{f}_1\mathrm{f}_4v_r'\overline{\Theta}\)

1442

rhot_vrp_sm

\(\mathrm{f}_1\mathrm{f}_4\,\overline{v_r}\Theta'\)

1443

rhot_vrm_sp

\(\mathrm{f}_1\mathrm{f}_4\,\overline{v_r}\,\overline{\Theta}\)

1444

rhot_vrm_sm

\(\mathrm{f}_1\mathrm{f}_4v_\theta \Theta\)

1445

rhot_vt_s

\(\mathrm{f}_1\mathrm{f}_4v_\theta'\Theta'\)

1446

rhot_vtp_sp

\(\mathrm{f}_1\mathrm{f}_4v_\theta'\overline{\Theta}\)

1447

rhot_vtp_sm

\(\mathrm{f}_1\mathrm{f}_4\,\overline{v_\theta}\Theta'\)

1448

rhot_vtm_sp

\(\mathrm{f}_1\mathrm{f}_4\,\overline{v_\theta}\,\overline{\Theta}\)

1449

rhot_vtm_sm

\(\mathrm{f}_1\mathrm{f}_4v_\phi \Theta\)

1450

rhot_vp_s

\(\mathrm{f}_1\mathrm{f}_4v_\phi'\Theta'\)

1451

rhot_vpp_sp

\(\mathrm{f}_1\mathrm{f}_4v_\phi'\overline{\Theta}\)

1452

rhot_vpp_sm

\(\mathrm{f}_1\mathrm{f}_4\,\overline{v_\phi}\Theta'\)

1453

rhot_vpm_sp

\(\mathrm{f}_1\mathrm{f}_4\,\overline{v_\phi}\,\overline{\Theta}\)

1454

rhot_vpm_sm

\(c_P \mathrm{f}_1 v_r T\)

1455

enth_flux_r

\(c_P \mathrm{f}_1 v_\theta T\)

1456

enth_flux_theta

\(c_P \mathrm{f}_1 v_\phi T\)

1457

enth_flux_phi

\(c_P \mathrm{f}_1 v_r' T'\)

1458

enth_flux_rpp

\(c_P \mathrm{f}_1 v_\theta' T'\)

1459

enth_flux_thetapp

\(c_P \mathrm{f}_1 v_\phi' T'\)

1460

enth_flux_phipp

\(c_P \mathrm{f}_1 v_r'\, \overline{T}\)

1461

enth_flux_rpm

\(c_P \mathrm{f}_1 v_\theta'\, \overline{T}\)

1462

enth_flux_thetapm

\(c_P \mathrm{f}_1 v_\phi'\, \overline{T}\)

1463

enth_flux_phipm

\(c_P \mathrm{f}_1\, \overline{v_r}\, T'\)

1464

enth_flux_rmp

\(c_P \mathrm{f}_1\, \overline{v_\theta}\, T'\)

1465

enth_flux_thetamp

\(c_P \mathrm{f}_1\, \overline{v_\phi}\, T'\)

1466

enth_flux_phimp

\(c_P \mathrm{f}_1\, \overline{v_r}\, \overline{T}\)

1467

enth_flux_rmm

\(c_P \mathrm{f}_1\, \overline{v_\theta}\, \overline{T}\)

1468

enth_flux_thetamm

\(c_P \mathrm{f}_1\, \overline{v_\phi}\, \overline{T}\)

1469

enth_flux_phimm

\(-c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\frac{\partial \Theta}{\partial r}\)

1470

cond_flux_r

\(-c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\frac{1}{r}\frac{\partial \Theta}{\partial \theta}\)

1471

cond_flux_theta

\(-c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\frac{1}{r \mathrm{sin}\theta}\frac{\partial \Theta}{\partial \phi}\)

1472

cond_flux_phi

\(-c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\frac{\partial \Theta'}{\partial r}\)

1473

cond_fluxp_r

\(-c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\frac{1}{r}\frac{\partial \Theta'}{\partial \theta}\)

1474

cond_fluxp_theta

\(-c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\frac{1}{r \mathrm{sin}\theta}\frac{\partial \Theta'}{\partial \phi}\)

1475

cond_fluxp_phi

\(-c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\frac{\partial \overline{\Theta}}{\partial r}\)

1476

cond_fluxm_r

\(-c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\frac{1}{r}\frac{\partial \overline{\Theta}}{\partial \theta}\)

1477

cond_fluxm_theta

\(-c_6\mathrm{f}_1\mathrm{f}_4\mathrm{f}_5\frac{1}{r \mathrm{sin}\theta}\frac{\partial \overline{\Theta}}{\partial \phi}\)

1478

cond_fluxm_phi