Skip to main content

Dynamics of Weakly Nonlinear Waves Propagating in the Region with Mixed Nonlinearity

  • Chapter
  • First Online:
River, Sediment and Hydrological Extremes: Causes, Impacts and Management

Part of the book series: Disaster Resilience and Green Growth ((DRGG))

  • 186 Accesses

Abstract

Our research focuses on the behavior of weakly nonlinear waves in mixed nonlinear fluids. We use a multiple-scale approach together with the equation of state for a van der Waals fluid to obtain a transport equation from the Navier-Stokes equations. Using the resulting transport equation, we further investigate the effect of van der Waals variables on wave evolution.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 219.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 279.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Ballou DP (1970) Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions. Trans Am Math Soc 152(2):441–460

    Article  Google Scholar 

  • Colonna P, Guardone A (2006) Molecular interpretation of nonclassical gas dynamics of dense vapors under the van der Waals model. Phys Fluids 18(5):56101

    Article  Google Scholar 

  • Colonna P, Guardone A, Nannan NR (2007) Siloxanes: a new class of candidate Bethe-Zel’dovich-Thompson fluids. Phys Fluids 19(8):86102

    Article  Google Scholar 

  • Cramer MS (1989) Negative nonlinearity in selected fluorocarbons. Physics of Fluids A Fluid Dyn 1(11):1894–1897

    Article  Google Scholar 

  • Cramer MS (1991) Nonclassical dynamics of classical gases. In: Nonlinear waves in real fluids. Springer, Cham, pp 91–145

    Chapter  Google Scholar 

  • Cramer MS, Kluwick A (1984) On the propagation of waves exhibiting both positive and negative nonlinearity. J Fluid Mech 142:9–37

    Article  Google Scholar 

  • Cramer MS, Sen R (1992) A general scheme for the derivation of evolution equations describing mixed nonlinearity. Wave Motion 15(4):333–355

    Article  Google Scholar 

  • Cramer MS, Webb C (2007) A modified Zabolotskaya–Khokhlov equation for systems having small quadratic nonlinearity. Wave Motion 44(5):323–339

    Article  Google Scholar 

  • Cramer MS, Kluwick A, Watson LT, Pelz W (1986) Dissipative waves in fluids having both positive and negative nonlinearity. J Fluid Mech 169:323–336

    Article  Google Scholar 

  • Crighton DG (1986) The Taylor internal structure of weak shock waves. J Fluid Mech 173:625–642

    Article  Google Scholar 

  • Das J, Manikanta V, Nikhil Teja K, Umamahesh NV (2022) Two decades of ensemble flood forecasting: a state-of-the-art on past developments, present applications and future opportunities. Hydrol Sci J 67(3):477–493

    Article  Google Scholar 

  • Gupta LK, Pandey M, Raj PA, Shukla AK (2022) Fine sediment intrusion and its consequences for river ecosystems: a review fine sediment intrusion and its consequences for river ecosystems: a review. October. https://doi.org/10.1061/(ASCE)HZ.2153-5515.0000729

  • Jiang G-S, Shu C-W (1996) Efficient implementation of weighted ENO schemes. J Comput Phys 126(1):202–228

    Article  Google Scholar 

  • Kluwick A (1991) Nonlinear waves in real fluids. Springer, Cham

    Book  Google Scholar 

  • Kluwick A (2001) Rarefaction shocks. In: Handbook of shock waves. Amsterdam, Elsevier, pp 339–411

    Chapter  Google Scholar 

  • Kluwick A, Cox EA (1998) Nonlinear waves in materials with mixed nonlinearity. Wave Motion 27(1):23–41

    Article  Google Scholar 

  • Kluwick A, Meyer G (2010) Shock regularization in dense gases by viscous–inviscid interactions. J Fluid Mech 644:473–507

    Article  Google Scholar 

  • Oleinik OA (1959) Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Uspekhi Matematicheskikh Nauk 14(2):165–170

    Google Scholar 

  • Quartapelle L, Castelletti L, Guardone A, Quaranta G (2003) Solution of the Riemann problem of classical gas dynamics. J Comput Phys 190(1):118–140

    Article  Google Scholar 

  • Saikumar G, Pandey M, Dikshit PKS (2022) Natural river hazards: their impacts and mitigation techniques. In: River dynamics and flood hazards: studies on risk and mitigation. Cham, Springer, pp 3–16

    Google Scholar 

  • Taniguchi S, Mentrelli A, Ruggeri T, Sugiyama M, Zhao N (2010) Prediction and simulation of compressive shocks with lower perturbed density for increasing shock strength in real gases. Phys Rev E 82(3):36324

    Article  Google Scholar 

  • Thompson PA (1971) A fundamental derivative in gas dynamics. Phys Fluids 14(9):1843–1849

    Article  Google Scholar 

  • Thompson PA, Lambrakis KC (1973) Negative shock waves. J Fluid Mech 60(1):187–208

    Article  Google Scholar 

  • Zhang X, Shu C-W (2010) On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J Comput Phys 229(23):8918–8934

    Article  Google Scholar 

  • Zhao N, Mentrelli A, Ruggeri T, Sugiyama M (2011) Admissible shock waves and shock-induced phase transitions in a van der Waals fluid. Phys Fluids 23(8):86101

    Article  Google Scholar 

  • Zheng Y (2001) Systems of conservation laws: two-dimensional Riemannian problems. Springer, Cham

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Triveni P. Shukla .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Shukla, T.P. (2023). Dynamics of Weakly Nonlinear Waves Propagating in the Region with Mixed Nonlinearity. In: Pandey, M., Gupta, A.K., Oliveto, G. (eds) River, Sediment and Hydrological Extremes: Causes, Impacts and Management. Disaster Resilience and Green Growth. Springer, Singapore. https://doi.org/10.1007/978-981-99-4811-6_7

Download citation

Publish with us

Policies and ethics