Acta Mechanica Sinica (2013) 29(2):166–178 DOI 10.1007/s10409-013-0016-3 RESEARCH PAPER Viscoelastic modeling of the diffusion of polymeric pollutants injected into a pipe flow T. Chinyoka · O.D. Makinde Received: 5 July 2012 / Revised: 30 November 2012 / Accepted: 4 December 2012 ©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2013 Abstract This study focuses on the transient analysis of nonlinear dispersion of a polymeric pollutant ejected by an external source into a laminar pipe flow of a Newtonian liq- uid under axi-symmetric conditions. The influence of den- sity variation with pollutant concentration is approximated according to the Boussinesq approximation and the nonlin- ear governing equations of momentum, pollutant concentra- tion are obtained together with and Oldroyd-B constitutive model for the polymer stress. The problem is solved numer- ically using a semi-implicit finite difference method. Solu- tions are presented in graphical form for various parameter values and given in terms of fluid velocity, pollutant concen- tration, polymer stress components, skin friction and wall mass transfer rate. The model can be a useful tool in un- derstanding the dynamics of industrial pollution situations arising from improper discharge of hydrocarbon pollutants into, say, water bodies. The model can also be quite useful for available necessary early warning methods for detecting or predicting the scale of pollution and hence help mitigate related damage downstream by earlier instituting relevant de- contamination measures. Keywords Axi-symmetric flow · Polymeric pollutant dis- persion · Oldroyd-B model Buoyancy forces · Semi-implicit finite difference method T. Chinyoka( ) Center for Research in Computational and Applied Mechanics University of Cape Town, Rondebosch 7701, South Africa e-mail: tchinyok@vt.edu O.D. Makinde Institute for Advanced Research in Mathematical Modeling and Computations, Cape Peninsula University of Technology, P.O. Box 1906, Bellville 7535, South Africa 1 Introduction Pollution of, say piped water sources for daily consumption, for example due to industrial waste discharge is a serious socio-ecological hazard. If the problem is not carefully con- trolled and monitored extensive communities can be exposed to severe health risks. Early detection of such pollution ac- cidents, both in terms of extent and impact, is thus of major primary importance and the subsequent requirement to take immediate corrective measures to redress the pollution prob- lem and mitigate against its impact equally so. The develop- ment of accurate methods to predict the spread of a pollutant once a discharge has been detected and hence also the devel- opment of equally reliable preventive/corrective techniques is thus of paramount importance [1]. Spread of pollutants in a fluid flow depends largely on concentration coefficients [2]. These can be determined empirically for each type of pollutant. Investigations such as Ref. [3] and also the current one can help identify the pollutant physical properties (and the related mathematical parameters) likely to cause the greatest harm in spreading the pollutant downstream. The importance of these kind of investigations as well as the complimentary experimental works, say in, large scale water treatment and redistribution networks thus makes them of great relevance [4–9]. It should however be remarked that to date, the litera- ture on the transient analysis of problem consisting of buoy- ancy effects and nonlinear pollutant injection is still quite sparse [10]. Moreover, investigations on polymeric pollu- tants is practically absent from the current literature and hence most pollution investigations have been carried out un- der Newtonian assumptions. Polymer blends often lead to products of superior quality [11], and hence there has been an ever increasing growth of the polymer industry. Most such products (from pharmaceuticals to beauty products) are liq- uid by nature and so are the waste products derived from their manufacture. Such waste products usually find their way into, say, water sources into which they are discharged