Nonlinear Mathematical Models Involving Non-Fourier Heat Flux

Abstract

Heat transfer analysis in non-Newtonian fluids is a continuous focus of the present researchers.Instead of diffusion this mechanism is analyzed as a wave owing to its several applications in skin burns and nanomaterial. No doubt this phenomenon arises due to variations in temperature within the whole system. Fourier firstly proposed the well-known law of heat flux. This law has limitation for development of parabolic energy expression. The energy expression clearly illustrates that the initial disturbance is immediately observed by the medium. This argument is named as "Paradox of heat conduction". Fourier's law of heat conduction is modified in various ways and conditions to avoid this feature. Advanced form of Fourier's law of heat conduction by adding the thermal relaxation time contribution is provided by Cattaneo. He noticed that the hyperbolic energy expression appears due to presence of thermal relaxation time. The modification in Cattaneo theory is provided by Christov. He employed Oldroyd type derivative instead of material derivation in Cattaneo law. Further the stagnation flows may be characterized as inviscid or viscous, steady or unsteady, two-dimensional or three-dimensional, symmetric or asymmetric, normal or oblique, homogenous or two immiscible fluids and forward or reverse flows. The behavior of stagnation point flow is very important in engineering and industrial phenomena. The flow over the tips of submarines, rockets, oilships and aircrafts are few examples of stagnation point flows. Another interesting example is the blood flow at a junction within an artery. Recently the researchers and scientists are still interested to explore the characteristics of fluid flow subject to simultaneous effect of heat and mass transfer. This is due to rapid advancements and developments in the technological and industrial processes. In fact the investigators are interested to enhance the efficiency of various machines by increasing the rate of heat transfer and quality of final products with desired characteristics through rate of heating/cooling. The combined effects of heat and mass transfer are further significant in many natural, biological, industrial and geophysical processes. Such phenomena include designing of many chemical processing equipment, formation and dispersion of fog, distribution of temperature and moisture over agricultural fields, damaging of crops due to freezing, grooves of fluid trees, environmental pollution, drying of porous solids, packed bed catalytic reactors, geothermal reserves, enhanced oil recovery, thermal insulation and underground energy transport. Motivated by all such facts, the structure of the present thesis is as follows: Literature review about boundary layer flows, description of solution procedure and laws of conservation of mass, linear momentum and energy are given in chapter one. Chapter two addresses the stagnation point flow towards a stretching boundary moving in a nonlinear manner. Modeling is based upon Eyring-Powell liquid. Boundary has variable thickness. Fluid thermal conductivity alters with temperature. Convergent series solutions for nonlinear systems are constructed. The solutions have been explored for the contributions of involved sundry variables. Attention is mainly given to the results of velocity, temperature and skin friction coefficient. Physical insight is outlined by graphical results. The outcomes of this chapter are published in “European Physical Journal Plus 131 (2016) 355”. Chapter three extends the analyses of chapter two for chemical reaction and stratification (through heat and mass) effects. Here Sherwood number is also studied. This material is published in “Results in Physics 7 (2017) 99–106”. Contribution of chapter three for second grade fluid model is modified in chapter four. Such consideration of second grade can predict the normal stress effect. Here chemical reaction is absent. The observations of this chapter have been published in “Chinese journal of Physics 55 (2017) 230-241”. Chapter five extends the contents of chapter four in direction of chemical reaction. Here first order chemical reaction is discussed. The obtained results are published in “Journal of Molecular Liquids 234 (2017) 444-451”. The two chapters namely six and seven have been prepared for the flows analyses modeled by constitutive relations of Walter B-liquid. Here analyses involve stagnation point flow, nonlinear stretching sheet of variable thickness, stratification, temperature dependent conductivity and heat flux by Cattaneo-Christov theory. Transformation procedure leads to development of nonlinear ordinary different systems. Homotopic convergent solutions with appropriate domains are derived. Salient parameters effects on physical quantities of interest are pointed out. Materials of these two chapters have been published in Neural Computing & Applications DOI 10.1007/s00521-017-3013-9 and Journal of Molecular Liquids 238 (2017) 229-235. Prime focus in chapter eight is to introduce nonlinear convection in two-dimensional stagnation point of micropolar fluid. Cattaneo-Christov theory and temperature dependent of thermal conductivity are entertained. Sheet of varying thickness is stretched with nonlinear velocity. Free stream velocity is also nonlinear. Homotopic process is implemented for the convergent series solutions. Skin friction coefficient in present case of micropolar fluid is sketched and analyzed. The material of this chapter is accepted for publication to “Journal of the Brazilian Society of Mechanical Sciences and Engineering”.