Physics-informed neural network approach to analyze the onset of oscillatory and stationary convections in chemically triggered Navier-Stokes-Voigt fluid layer heated and salted from below

doi:10.1007/s10483-025-3316-7

Abstract

Abstract:

The present work analyzes the linear and weakly nonlinear stability of double-diffusive convection (DDC) in a Navier-Stokes-Voigt (NSV) fluid, considering a chemical reaction and an internal heat source. The lower fluid layer is salted and heated. The quiescent state and dimensionless variables yield dimensionless parameters for the governing partial differential equations (PDEs). A two-dimensional scenario is investigated using the stream function. Stationary and oscillatory convection can be analyzed using the linear approach. The nonlinear equations are numerically solved using the Runge-Kutta Fehlberg (RKF-45) technique. Additionally, the physics-informed neural network (PINN) validates the mathematical outcomes. The Kelvin-Voigt parameter and the Prandtl number do not affect stationary convection. Thhe neutral stability diagrams show that the ratios of diffusivity, solute Rayleigh, and Kelvin-Voigt parameters stabilize oscillatory convection. However, internal heat and chemical reactions cause instability. The Kelvin-Voigt, internal heat, and chemical reaction parameters increase mass and heat transfer (MHT), while the solute Rayleigh number and the ratio of diffusivity decrease MHT.

Key words: stability analysis, Navier-Stokes-Voigt (NSV) fluid, internal heat generation, chemical reaction, double-diffusive convection (DDC)

2010 MSC Number:

O357

B. S. SANJU, R. NAVEEN KUMAR, R. S. VARUN KUMAR, A. ABDULRAHMAN. Physics-informed neural network approach to analyze the onset of oscillatory and stationary convections in chemically triggered Navier-Stokes-Voigt fluid layer heated and salted from below. Applied Mathematics and Mechanics (English Edition), 2025, 46(11): 2199-2220.

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Figures/Tables 23

Fig.?1

Table?1

A comparison of transition values from stationary to oscillatory convection RaTcS=RaTco for different values of RaC, τ−1, Λ, and Pr when Qh=KR=0, with results from Straughan[26] and Kumar et al.[36]"

Parameter				Straughan^[26]		Kumar et al.^[36]		Present study
$P r$	$τ − 1$	$R a C$	$Λ$	$α c 2$	$R a Tc S = R a Tc o$	$α c 2$	$R a Tc S = R a Tc o$	$α c 2$	$R a Tc S = R a Tc o$
50	21.67	1.495 154	0	4.935	689.96	4.935	689.955	4.935	689.955
50	21.67	1.928 934	1	4.903	699.36	4.903	699.364	4.903	699.364
50	21.67	2.362 713	2	4.872	708.75	4.872	708.753	4.872	708.753
6.99	70.44	0.153 605	0	4.935	668.33	4.935	668.335	4.935	668.335
6.99	70.44	0.437 677	1	4.865	688.34	4.865	688.343	4.865	688.341
6.99	70.44	0.720 471	2	4.800	708.26	4.800	708.280	4.800	708.259
6.99	111.2	0.061 330	0	4.935	664.33	4.935	664.331	4.935	664.331
6.99	111.2	0.174 820	1	4.889	676.95	4.889	676.950	4.889	676.949
6.99	111.2	0.287 949	2	4.847	689.53	4.847	689.536	4.847	689.529

Table?1

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Table?2

Comparison of obtained PINN values with RKF-45 values"

Time	$Q h$	$K R$	$N u$	$N u$ (PINN)	$V AE$	$S h$	$S h$ (PINN)	$V AE$
0	0.5	0.5	1	0.999 975 479	2.452 1 $× 10 − 5$	1	0.999 733 628	0.000 266 372
0.5			1.002 820 4	1.002 693 781	0.000 126 66	1.004 347 9	1.004 159 538	0.000 188 372
1			1.005 390 9	1.005 311 241	7.965 3 $× 10 − 5$	1.008 710 4	1.008 521 573	0.000 188 817
0	0.7		1	0.999 597 804	0.000 402 2	1	0.999 782 193	0.000 217 807
0.5			1.002 822 4	1.002 426 452	0.000 395 91	1.003 224 8	1.003 159 459	6.535 96 $× 10 − 5$
1			1.004 518 3	1.004 511 918	6.413 4 $× 10 − 6$	1.003 952 4	1.003 756 113	0.000 196 298
0	1.1		1	0.999 608 767	0.000 391 23	1	0.999 805 393	0.000 194 607
0.5			1.001 160 1	1.000 715 430	0.000 444 72	0.999 595 1	0.999 465 912	0.000 129 235
1			1.005 390 9	0.996 893 827	0.008 497 07	1.008 710 4	0.995 932 906	0.012 777 484
0	0.5	1	1	0.999 882 402	0.000 117 6	1	0.999 758 826	0.000 241 174
0.5			1.001 780 3	1.001 398 222	0.000 382 11	1.000 531 8	1.000 456 709	7.506 58 $× 10 − 5$
1			0.998 304 3	0.997 701 441	0.000 602 83	0.995 439 6	0.995 184 948	0.000 254 693
0		1.5	1	0.999 411 647	0.000 588 35	1	0.999 936 605	6.339 5 $× 10 − 5$
0.5			1.004 437 2	1.003 855 805	0.000 581 42	1.006 250 2	1.005 951 479	0.000 298 693
1			1.009 640 3	1.009 577 905	6.234 9 $× 10 − 5$	1.010 318 3	1.010 158 196	0.000 160 118

Table?2

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