## THERMAL SCIENCE

International Scientific Journal

### MULTIPLE INTEGRAL-BALANCE METHOD: BASIC IDEA AND AN EXAMPLE WITH MULLIN’S MODEL OF THERMAL GROOVING

**ABSTRACT**

A multiple integration technique of the integral-balance method allowing solving high-order diffusion equations is conceived in this note. The new method termed multiple-integral balance method is based on multiple integration procedures with respect to the space co-ordinate and is generalization of the widely applied heat-balance integral method of Goodman and the double integration method of Volkov. The method is demonstrated by a solution of the linear diffusion models of Mullins for thermal grooving.

**KEYWORDS**

PAPER SUBMITTED: 1970-01-01

PAPER REVISED: 2017-05-02

PAPER ACCEPTED: 2017-05-03

PUBLISHED ONLINE: 2017-05-06

**THERMAL SCIENCE** YEAR

**2017**, VOLUME

**21**, ISSUE

**Issue 3**, PAGES [1555 - 1560]

- Goodman, T.R, The heat balance integral and its application to problems involving a change of phase, Transactions of ASME, 80 (1958), 1-2, pp.335-342.
- Langford , D., The heat balance integral method, Int. J. Heat Mass Transfer, 16 (1973), 12, pp.2424-2428.
- Myers, J.G. Optimizing the exponent in the heat balance and refined integral methods. Int Comm Heat Mass Transfer, 36 (2009), 2, pp. 143-147.
- Mitchell, S. L., Myers, T. G. Application of standard and refined heat balance integral methods to onedimensional Stefan problems. SIAM Review 52 (2010) ,1, pp. 57-86.
- Hristov J (2009) The heat-balance integral method by a parabolic profile with unspecified exponent: Analysis and Benchmark Exercises, Thermal Science 13 (2009) , 2, pp.27-48
- Hristov J., Integral solutions to transient nonlinear heat (mass) diffusion with a power-law diffusivity: a semi-infinite medium with fixed boundary conditions, Heat Mass Transfer, in press, DOI: 10.1007/s00231-015-1579-2
- Hristov J., Double Integral-Balance Method to the Fractional Subdiffusion Equation: Approximate solutions, optimization problems to be resolved and numerical simulations, J. Vibration and Control, DOI: 10.1177/1077546315622773
- Volkov, V. N., Li-Orlov, V. K., A Refinement of the Integral Method in Solving the Heat Conduction Equation, Heat Transfer Sov. Res., 2 (1970), 2, pp. 41-47.
- J. A. Esfahani, S. M. M. Khazeni , S. Mhammadi , Accuracy analysis of predicted velocity profiles of laminar duct flow with entropy generation method , J.Appl. Math. Mech., 34, (2013),8, pp.971-985
- Kot, V.A. Method of boundary characteristics, J. Eng. Phys. ThermoPhys., 88(2015),6, pp. 1390-1408.
- Mullins, W.W. Theory of thermal grooving, J. Appl. Phys, 28 91957), 3, pp.333-339
- P. Broadbridge, Exact solvability of the Mullins nonlinear diffusion model of groove development, J. Math. Phys, 30 (1989), 7,pp.1648-1651.
- P. Broadbridge, Exact solution of a degenerate fully nonlinear diffusion equation, Z.angw.Math.Phys, 55 92004), pp.534-538. doi:10.1007/s00033-004-3015-1
- Kitada, A. On properties of a classical solution of nonlinear mass transport equation ut=uxx/(1+ux^2), J. Math. Phys, 27 (1986), 7,pp.1391-1392.
- Robertson, Grain-boundary growing by surface diffusion for finite slopes, J. Appl. Phys, 42 (1971), 1, pp.463-467.
- M. Abu Hamed, A.A. Nepomnyashchy, Physica D: Nonlinear Phenomena, 298-299 (2015),1,pp. 42-47