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Computational and asymptotic methods for three-dimensional boundary-layer flow and heat transfer over a wedge

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Abstract

This paper is devoted to mathematical analysis of three-dimensional boundary-layer flow and heat transfer over a wedge. The boundary layer is formed due to the flow of a viscous and incompressible fluid and grows downstream along the walls of the wedge. This is appropriately approximated by a power of distances along both lateral directions. This analysis introduces a shear-to-strain-rate parameter \(\gamma\) in the study which plays an essential role in three-dimensional boundary-layer study. A set of boundary-layer equations along with the conservation of energy has been transformed into a coupled nonlinear ordinary differential equations. Two methods are employed. The fluid–wedge interaction problem is solved in the circumstances where the equations are linearized and obtained the solutions in terms of confluent hypergeometric functions, and otherwise, the full-nonlinear problem numerically using the Keller-box method. In the former case, the results of eigenfunction analysis that is based on asymptotic study of the problem qualitatively support the latter numerical results. In either methods, the existence of solutions and range of parameter domain depending on various physical quantities is successfully analyzed. The two different approaches give good agreement with each other in predicting the velocity and temperature profiles in three-dimensional boundary layer. However, both approaches show that a solution to the problem exist only \(-\,1< \gamma < \infty\). Since the Reynolds number is asymptotically large, the flow clearly divides into two regions showing the effects of viscosity and thermal conductivity. The velocity and thermal profiles are found to be benign. Flow always attached to the wedge surfaces and, thus, related thicknesses are becoming thinner. Extensive comparisons of the various results are made and a possible explanation on the results is discussed in some detail.

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Abbreviations

A :

Constant

\(C_\mathrm{p}\) :

Specific heat at constant pressure

\(\text{ erf }\) :

Error function

\(\text{ erfc }\) :

Complementary error function

f, g, \(\theta\) :

Dimensionless velocities and temperature

k :

Thermal conductivity of the fluid

m :

Pressure gradient

N :

Exponent wall temperature

p :

Pressure

Pr :

Prandtl number

\(q_\mathrm{w}\) :

Heat flux

T :

Temperature of the fluid

\(T_\mathrm{w}\), \(T_\infty\) :

Wall temperature and temperature of the mainstream flow

u, v, w :

Velocity components in (x, y, z) directions

U, V :

Mainstream flows in x- and y-directions

\(U_\infty\), \(V_\infty\) :

Strain and shear rates of mainstream flows

\(U_\mathrm{w}\), \(V_\mathrm{w}\) :

Wedge surface velocities

\(U_0\), \(V_0\) :

Strain and shear rates of the wedge velocities

\(\mathscr {F}(\eta)\), \(\mathscr {G}(\eta )\), \(\mathscr {L}(\eta)\) :

Asymptotic velocities and temperature

\(\gamma\) :

Shear-to strain-rate parameter or three-dimensionality parameter

\(\beta\) :

Pressure gradient

\(\rho\), \(\mu\) :

Density and dynamic viscosity

\(\nu\) :

Kinematic viscosity

\(\alpha = \frac{k}{\rho C_\mathrm{p}}\) :

Thermal diffusivity

\(\delta\), \(\delta _\mathrm{t}\) :

Thickness of momentum and thermal boundary layer

\(\psi _1\), \(\psi _2\) :

Stream functions

\(\eta\) :

Similarity variable

\(\lambda _1\), \(\lambda _2\) :

wedge velocities in x and y directions

\(\tau\) :

Wall shear stress

\(\varGamma\) :

Gamma function

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Acknowledgements

The authors would like to thank the anonymous reviewers for the educative comments that led to present form.

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Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Central College Campus, Bangalore, 560 001, India

    Ramesh B. Kudenatti & B. Jyothi

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  1. Ramesh B. Kudenatti
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Correspondence to Ramesh B. Kudenatti.

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Appendix: An algorithm of the Keller-box method

Appendix: An algorithm of the Keller-box method

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Kudenatti, R.B., Jyothi, B. Computational and asymptotic methods for three-dimensional boundary-layer flow and heat transfer over a wedge. Engineering with Computers 36, 1467–1483 (2020). https://doi.org/10.1007/s00366-019-00776-3

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