A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates

Mushtaq Ahmed

doi:doi:10.11648/j.fm.20190501.13

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A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates

Mushtaq Ahmed

Published in Fluid Mechanics (Volume 5, Issue 1)

Received: 13 March 2019 Accepted: 7 May 2019 Published: 4 June 2019

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Abstract

The objective of this article is to communicate a class of new exact solutions of the plane equation of momentum with body force, energy and continuity for moderate Peclet number in von-Mises coordinates. Viscosity of fluid is variable but its density and thermal conductivity are constant. The class characterizes the streamlines pattern through an equation relating two continuously differentiable functions and a function of stream function ψ. Applying the successive transformation technique, the basic equations are prepared for exact solutions. It finds exact solutions for class of flows for which the function of stream function varies linearly and exponentially. The linear case shows viscosity and temperature for moderate Peclet number for two variety of velocity profile. The first velocity profile fixes both the functions of characteristic equation whereas the second keeps one of them arbitrary. The exponential case finds that the temperature distribution, due to heat generation, remains constant for all Peclet numbers except at 4 where it follows a specific formula. There are streamlines, velocity components, viscosity and temperature distribution in presence of body force for a large number of the finite Peclet number.

Published in	Fluid Mechanics (Volume 5, Issue 1)
DOI	10.11648/j.fm.20190501.13
Page(s)	15-25
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Successive Transformation Technique, Variable Viscosity Fluids, Navier-Stokes Equations with Body Force, Martin’s Coordinates, Von-MisesCoordinates

References

[1]	Naeem, R. K.; Exact solutions of flow equations of an incompressible fluid of variable viscosity via one – parameter group: The Arabian Journal for Science and Engineering, 1994, 19 (1), 111-114.
[2]	Naeem, R. K.; Srfaraz, A. N.; Study of steady plane flows of an incompressible fluid of variable viscosity using Martin’s System: Journal of Applied Mechanics and Engineering, 1996, 1 (1), 397-433.
[3]	Naeem, R. K.; Steady plane flows of an incompressible fluid of variable viscosity via Hodograph transformation method: Karachi University Journal of Sciences, 2003, 3 (1), 73-89.
[4]	Naeem, R. K.; On plane flows of an incompressible fluid of variable viscosity: Quarterly Science Vision, 2007, 12 (1), 125-131.
[5]	Naeem, R. K.; Mushtaq A.; A class of exact solutions to the fundamental equations for plane steady incompressible and variable viscosity fluid in the absence of body force: International Journal of Basic and Applied Sciences, 2015, 4 (4), 429-465. www.sciencepubco.com/index.php/IJBAS, doi: 10.14419/ijbas.v4i4.5064.
[6]	Gerbeau, J. -F.; Le Bris, C., A basic Remark on Some Navier-Stokes Equations With Body Forces: Applied Mathematics Letters, 2000, 13 (1), 107-112.
[7]	Giga, Y.; Inui, K.; Mahalov; Matasui S.; Uniform local solvability for the Navier-Stokes equations with the Coriolis force: Method and application of Analysis, 2005, 12, 381-384.
[8]	Landau L. D. and Lifshitz E. M.; Fluid Mechanics, Pergmaon Press, vol 6.
[9]	Mushtaq A., On Some Thermally Conducting Fluids: Ph. D Thesis, Department of Mathematics, University of Karachi, Pakistan, 2016.
[10]	Mushtaq A.; Naeem R. K.; S. Anwer Ali; A class of new exact solutions of Navier-Stokes equations with body force for viscous incompressible fluid,: International Journal of Applied Mathematical Research, 2018, 7 (1), 22-26. www.sciencepubco.com/index.php/IJAMR, doi: 10.14419/ijamr.v7i1.8836.
[11]	Mushtaq Ahmed, Waseem Ahmed Khan,: A Class of New Exact Solutions of the System ofPDEfor the plane motion of viscous incompressible fluids in the presence of body force,: International Journal of Applied Mathematical Research, 2018, 7 (2), 42-48. www.sciencepubco.com/index.php/IJAMR, doi: 10.14419/ijamr.v7i2.9694.
[12]	Mushtaq Ahmed, Waseem Ahmed Khan, S. M. Shad Ahsen: A Class of Exact Solutions of Equations for Plane Steady Motion of Incompressible Fluids of Variable viscosity in presence of Body Force: International Journal of Applied Mathematical Research, 2018, 7 (3), 77-81. www.sciencepubco.com/index.php/IJAMR, doi: 10.14419/ijamr.v7i2.12326.
[13]	Mushtaq Ahmed, (2018), A Class of New Exact Solution of equations for Motion of Variable Viscosity Fluid In presence of Body Force with Moderate Peclet number, International Journal of Fluid Mechanics and Thermal Sciences, 4 (4)429- www.sciencepublishingdroup.com/j/ijfmts doi: 10.11648/j.ijfmts.20180401.12.
[14]	D. L. R. Oliver & K. J. De Witt, High Peclet number heat transfer from adroplet suspended in an electric field: Interior problem, Int. J. Heat Mass Transfer, vol. 36: 3153-3155, 1993.
[15]	Z. G. Feng, E. E. Michaelides, Unsteady heat transfer from a spherical particle atfinite Peclet numbers, J. Fluids Eng. 118: 96-102, 1996.
[16]	Fayerweather Carl, Heat Transfer From a Droplet at Moderate Peclet Numbers with heat Generation. PhD. Thesis, U of Toledo, May 2007.
[17]	Martin, M. H.; The flow of a viscous fluid I: Archive for Rational Mechanics and Analysis, 1971, 41 (4), 266-286.
[18]	Daniel Zwillinger; Handbook of differential equations; Academic Press, Inc. (1989).
[19]	Weatherburn C. E., Differential geometry of three Dimensions, Cambridge University Press, (1964).

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Mushtaq Ahmed. (2019). A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates. Fluid Mechanics, 5(1), 15-25. https://doi.org/10.11648/j.fm.20190501.13

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ACS Style

Mushtaq Ahmed. A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates. Fluid Mech. 2019, 5(1), 15-25. doi: 10.11648/j.fm.20190501.13

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AMA Style

Mushtaq Ahmed. A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates. Fluid Mech. 2019;5(1):15-25. doi: 10.11648/j.fm.20190501.13

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@article{10.11648/j.fm.20190501.13,
  author = {Mushtaq Ahmed},
  title = {A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates},
  journal = {Fluid Mechanics},
  volume = {5},
  number = {1},
  pages = {15-25},
  doi = {10.11648/j.fm.20190501.13},
  url = {https://doi.org/10.11648/j.fm.20190501.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.fm.20190501.13},
  abstract = {The objective of this article is to communicate a class of new exact solutions of the plane equation of momentum with body force, energy and continuity for moderate Peclet number in von-Mises coordinates. Viscosity of fluid is variable but its density and thermal conductivity are constant. The class characterizes the streamlines pattern through an equation relating two continuously differentiable functions and a function of stream function ψ. Applying the successive transformation technique, the basic equations are prepared for exact solutions. It finds exact solutions for class of flows for which the function of stream function varies linearly and exponentially. The linear case shows viscosity and temperature for moderate Peclet number for two variety of velocity profile. The first velocity profile fixes both the functions of characteristic equation whereas the second keeps one of them arbitrary. The exponential case finds that the temperature distribution, due to heat generation, remains constant for all Peclet numbers except at 4 where it follows a specific formula. There are streamlines, velocity components, viscosity and temperature distribution in presence of body force for a large number of the finite Peclet number.},
 year = {2019}
}

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TY  - JOUR
T1  - A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates
AU  - Mushtaq Ahmed
Y1  - 2019/06/04
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N1  - https://doi.org/10.11648/j.fm.20190501.13
DO  - 10.11648/j.fm.20190501.13
T2  - Fluid Mechanics
JF  - Fluid Mechanics
JO  - Fluid Mechanics
SP  - 15
EP  - 25
PB  - Science Publishing Group
SN  - 2575-1816
UR  - https://doi.org/10.11648/j.fm.20190501.13
AB  - The objective of this article is to communicate a class of new exact solutions of the plane equation of momentum with body force, energy and continuity for moderate Peclet number in von-Mises coordinates. Viscosity of fluid is variable but its density and thermal conductivity are constant. The class characterizes the streamlines pattern through an equation relating two continuously differentiable functions and a function of stream function ψ. Applying the successive transformation technique, the basic equations are prepared for exact solutions. It finds exact solutions for class of flows for which the function of stream function varies linearly and exponentially. The linear case shows viscosity and temperature for moderate Peclet number for two variety of velocity profile. The first velocity profile fixes both the functions of characteristic equation whereas the second keeps one of them arbitrary. The exponential case finds that the temperature distribution, due to heat generation, remains constant for all Peclet numbers except at 4 where it follows a specific formula. There are streamlines, velocity components, viscosity and temperature distribution in presence of body force for a large number of the finite Peclet number.
VL  - 5
IS  - 1
ER  -

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Author Information

Mushtaq Ahmed

Department of Mathematics, University of Karachi, Karachi, Pakistan

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Plain Text BibTeX RIS

APA Style

Mushtaq Ahmed. (2019). A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates. Fluid Mechanics, 5(1), 15-25. https://doi.org/10.11648/j.fm.20190501.13

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ACS Style

Mushtaq Ahmed. A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates. Fluid Mech. 2019, 5(1), 15-25. doi: 10.11648/j.fm.20190501.13

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AMA Style

Mushtaq Ahmed. A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates. Fluid Mech. 2019;5(1):15-25. doi: 10.11648/j.fm.20190501.13

Copy | Download

@article{10.11648/j.fm.20190501.13,
  author = {Mushtaq Ahmed},
  title = {A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates},
  journal = {Fluid Mechanics},
  volume = {5},
  number = {1},
  pages = {15-25},
  doi = {10.11648/j.fm.20190501.13},
  url = {https://doi.org/10.11648/j.fm.20190501.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.fm.20190501.13},
  abstract = {The objective of this article is to communicate a class of new exact solutions of the plane equation of momentum with body force, energy and continuity for moderate Peclet number in von-Mises coordinates. Viscosity of fluid is variable but its density and thermal conductivity are constant. The class characterizes the streamlines pattern through an equation relating two continuously differentiable functions and a function of stream function ψ. Applying the successive transformation technique, the basic equations are prepared for exact solutions. It finds exact solutions for class of flows for which the function of stream function varies linearly and exponentially. The linear case shows viscosity and temperature for moderate Peclet number for two variety of velocity profile. The first velocity profile fixes both the functions of characteristic equation whereas the second keeps one of them arbitrary. The exponential case finds that the temperature distribution, due to heat generation, remains constant for all Peclet numbers except at 4 where it follows a specific formula. There are streamlines, velocity components, viscosity and temperature distribution in presence of body force for a large number of the finite Peclet number.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates
AU  - Mushtaq Ahmed
Y1  - 2019/06/04
PY  - 2019
N1  - https://doi.org/10.11648/j.fm.20190501.13
DO  - 10.11648/j.fm.20190501.13
T2  - Fluid Mechanics
JF  - Fluid Mechanics
JO  - Fluid Mechanics
SP  - 15
EP  - 25
PB  - Science Publishing Group
SN  - 2575-1816
UR  - https://doi.org/10.11648/j.fm.20190501.13
AB  - The objective of this article is to communicate a class of new exact solutions of the plane equation of momentum with body force, energy and continuity for moderate Peclet number in von-Mises coordinates. Viscosity of fluid is variable but its density and thermal conductivity are constant. The class characterizes the streamlines pattern through an equation relating two continuously differentiable functions and a function of stream function ψ. Applying the successive transformation technique, the basic equations are prepared for exact solutions. It finds exact solutions for class of flows for which the function of stream function varies linearly and exponentially. The linear case shows viscosity and temperature for moderate Peclet number for two variety of velocity profile. The first velocity profile fixes both the functions of characteristic equation whereas the second keeps one of them arbitrary. The exponential case finds that the temperature distribution, due to heat generation, remains constant for all Peclet numbers except at 4 where it follows a specific formula. There are streamlines, velocity components, viscosity and temperature distribution in presence of body force for a large number of the finite Peclet number.
VL  - 5
IS  - 1
ER  -

Copy | Download