In the past years, several local industries have generated demand for research and innovation in order to be competitive and access both local and international markets and to grow. Many and new entrepreneurs are demanding technology solutions and innovation to create new markets and businesses. Using Scientific Computing and Industrial modeling for design, simulation, analysis, prototyping and development of technology solutions creates very low cost alternative for Research, Development and innovation and serves as an innovation accelerator, shrinking time-to-insight and time-to-discovery. This provides the possibility for low resourced economies to create innovations that are competitive at very affordable costs.
The aim of the conference is to provide a forum to facilitate communication for mathematicians, Scientists and industry partners from varying disciplines to share cutting edge Mathematical concepts and state-of-the-art scientific computing resources to deliver cost-effective technology solutions.
Scientific Computing and Industrial modeling; a tool for accelerating innovation
Authors are invited to submit abstracts of manuscripts that present original research in all areas of Scientific Computing and Industrial modeling, including the development of experimental or commercial systems. Work focusing on emerging technologies is especially welcome. Topics of interest include, but are not limited to: High performance computing, Reservoir simulation, Imaging and Computer Vision, Data visualization, Biomathematical Modeling, Biogeochemical Modeling, Distributed and Grid Computing, Monte Carlo simulation, Computational Fluid Dynamics, Computational Finance, Nonlinear Optimization.
In the last four years, the National institute for Mathematical Sciences, the umbrella institute for Mathematical Sciences in Ghana has been focusing on setting up a Center of Excellence in Scientific computing and Industrial Modelling. Participants of the conference will also have an opportunity to witness the official launching of the centre and its activities.
Provost, College of Science
The discovery and commercialization of oil production has placed Ghana among High-Risk Zones which are characterised by high traffic density and the presence of navigational hazards. Even so, the size, location, and circumstances of any major oil spill remain unpredictable. Models have created a “coco” means of understanding our environment and a good way of dealing with any accident is to be prepared. Therefore, planning for an oil spill emergency helps minimise potential danger to human health and the environment by ensuring a timely and coordinated response. MOHID model was used to address scientific and environmental processes related to the oceanic dynamics in the EEZ of Ghana and the concept of nesting domains was used to reduce the cost and computer processing unit, CPU, time for computation. The hydrodynamics was modeled using three levels (the 2nd and 3rd levels were forced with atmospheric and oceanic parameters) and validated to have an average deviation of 14 % from field results. To locate possible shorelines /places that are susceptible to oil particles, random locations where oil exploitation takes place were used as discharge points for the simulations. In all, about 8 locations were included for the research and they are as follows: (-2.9 oE, 4.4 oN), (-1.7 oE, 4.6 oN), (-0.9 oE, 4.9 oN), (-0.9 oE, 4.6 oN), (-0.3 oE, 5.2 oN), (-0.3 oE, 4.6 oN), and (0.5 oE, 4.6 oN). In conclusion, it took the oil particles 2 days, 6 hours; 1 day, 17 hours; 3 days; 4 days; 1 day; 4 days, 1 hour; and 22 hours respectively to get beached at the shore. For particles located at this point (i.e. 0.5 oE, 5.5 oN), they were beached outside the EEZ of Ghana.
We consider a coupled system of PDEs modeling an erosion phenomenon in a heterogeneous porous media. It is an extremely difficult problem to study the properties of a micro non-homogeneous medium. A possible way is to attack the problem by applying asymptotic analysis which immediately leads to the concept of homogenization. We present here an application of the method of homogenization in a periodically perforated medium. At the end we hope to obtain a homogenized model with effective coefficients void of oscillations.
In this study, datasets obtained from ground magnetic and electrical geophysical surveys were integrated with existing geological datasets to map out structures in the Sefwi Belt of Southern Ghana. The magnetic data obtained were gridded and data enhancement filters were applied to the Total Magnetic Intensity (TMI) grid generated to enhance the magnetic anomalies to depth sources. The induced polarization-chargeability data were measured in the time domain. The positive anomalies on the IP-chargeability map coincided with the shears, margins, alterations and contact zones. Two Pole–dipole sections, carried out in the study area, were inverted and from the results obtained, an interpreted definitive geological map of the study area consisting of the geology, structures and hydrothermally altered zones was produced. The outcome of the electrical resistivity and IP inversions indicated that depths ranging from 50 to 200 m suggest conductive and chargeable bodies. The low-resistivity zones coincided with sheared and altered acidic meta-sediments. The geophysical signatures obtained from the enhanced magnetic data and the electrical data showed that the study area is structurally complex with a few of the structures corresponding to D1 deformation and most structures corresponding to D2 deformation. The study resulted in better illuminating geological structures and lithological boundaries, and thus has demonstrated the worth of geophysical data as an enhancement tool in mapping possible geological structures that host hydrothermal gold mineralization within the Sefwi Gold Belt of Ghana. Seven diamond drill holes were intuitively planned to test the hypothesized model and determine the depth of the resistive-chargeable anomalous units as well as litho-structural boundaries.
Flooding is a global phenomenon that causes casualties and property loss on every inhabited continent. In this study, we consider a hydrodynamic model that describes the phenomenon of flood inundation in one-dimensional (1D) spatial representation of the floodplain flow. We derive the number and type of well-posed characteristic boundary conditions that describe the flood waves in a river catchment area. With these establishments, we consider energy estimate analysis for the well-posedness of the flood inundation model. The equations are discretized using high-order accurate finite-difference methods that satisfy a Summation-by-Parts (SBP) property. Penalty terms known as the Simultaneous Approximation Term (SAT) technique are used to impose the characteristic boundary conditions. This lead to energy stable high-order finite difference scheme for the flood inundation model.
ABSRACT
There is a growing recognition of the pivotal role mathematics plays in the development of society.
This thesis reviews how mathematics influences human activities and its impact on the society, policy formation and research on how mathematics contributes to the national development. It includes possible initiatives/programs a nation can adopt for sustainable development.
The thesis is set in the contest of how mathematics contributes to development at the personal level and the societal level. It also outlines the key role mathematics plays in the development of the developing economies, specifically the South Korean economy.
The work further presents the contribution of mathematics to the developed economies, specifically the United Kingdom (UK) and the Netherlands economies, their implication on the working industry, the linkage and strength that emerges among mathematics works, the industry and development.
Consequently, we reports on initiatives/recommendations that a growing economy likes Ghana could consider as a major policy towards achieving socio-economic prosperity
Tight oil is petroleum that accumulates in relatively impermeable reservoir rocks,often shale or tight sandstones. Globally, tight oil resources provide signi?cant amount of petroleum for the world's energy needs. The flow behavior of tight oil in unconventional reservoirs are described by peculiar complexities that presents a challenging task in ?nding immediate solutions for reservoir engineers. It is there-fore critical to implement approaches that solve such problems without loosingvital information of the flow phenomenon. This study demonstrates a general concept to explain the behavior of tight oil in unconventional reservoirs. In this study, an investigation into the application of similarity transformations for the analysis of complex unconventional reservoirs exhibiting two phase phenomena during transient radial flow is done. The similarity transformation is carried out with the Boltzmann variable. The techniques adopted in the transformation process aids in converting highly nonlinear partial-differential equations (PDEs) governing the two phase flow phenomenon, to nonlinear ordinary di?erential equations (ODEs). The resulting ODEs, consequently simplify the determination of the reservoir performance and avoid the tedious calculation ingrained in solving the original PDEs. From a theoretical point of view, the successful conversionof the highly nonlinear PDEs to ODEs permits the derivation of saturation andpressure equations as unique functions of the Boltzmann variable, which in turn,guarantees the expression of saturation as a unique function of pressure. Further research is carried out to investigate the constant gas-oil ratio (GOR) that is typically observed in some hydraulically fractured tight oil reservoirs during constant pressure two-phase production. The similarity transformation approach sets up a foundation to develop an analytical solution to the model adopted in this study. The analytical solution yielding from this work is used to obtain similar forms to well-known equations (flow rate and cumulative production) for single phase flow, which enhance our understanding of multiphase flow behavior.
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\textbf{\huge Abstract}
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Elementary analysis dwells so much on the notion of limit of real-valued function or of a sequence in $ \mathbb{R}. $
In spite of the numerous applications of sequences, a sequence may not serve much purpose if its limiting value does not exist. This is because the limit of a sequence is the fundamental notion on which the whole of analysis ultimately rests. For instance, certain basic concepts in Mathematics such as \textbf{continuity, differentiability, integration e.t.c} are defined in terms of limit. Specifically of interest to us is class of sequences that is bounded. The question we pose is this: Is every bounded sequence in $ \mathbb{R} $ convergent? To answer this, we investigate the conditions under which a bounded sequence converges and recommend a criterion for the convergence of a bounded sequence.
For instance the sequence $ a_n = (-1)^n $ is bounded but does not converge. Again, the sequence $ a_n = (-1)^n+ \dfrac{1}{n} $ also fails to converge even though it is bounded. What then accounts for the convergence of a bounded sequence in $ \mathbb{{R}}? $.\\
We therefore conclude that a bounded sequence $a_n$, $n=1,2,3,\cdots $ in $\mathbb{R}$ will converge if any of the following conditions hold:
\begin{itemize}
\item The sequence $a_n$ is monotonic increasing or monotonic decreasing, that is $a_n$ is monotone.
\item $\lim\limits_{n\to \infty }\inf a_n\;\; =\;\; \lim\limits_{n\to \infty }\sup a_n$ or $ \inf A $ = $ \sup A $, where $A$ is the set of subsequential limits.
\end{itemize}
The sequences $ a_n = \dfrac{1}{2^n} $, $n=1,2,3,\cdots $ and $ a_n = \dfrac{1}{n^2} $, $n=1,2,3,\cdots $ are all bounded and convergent since they are all monotone and $\lim\limits_{n\to \infty }\inf a_n\;\; =\;\; \lim\limits_{n\to \infty }\sup a_n$ for all the sequence.
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This article presents a new general purpose stochastic algorithm for large-scale optimization problems on mobile phones and embedded systems. Application of the algorithm to the Griewank function was possible in up to 100 million decision variables in double precision on a Samsung Note 4 mobile phone, and 10 million decision variables on a Raspberry Pi 3. Solving the Griewank function for 1,000 decision variables took less than 1 second on both the Samsung mobile and Raspberry Pi. In Africa where High-Performance Computing hardware is limited and expensive to setup, this algorithm opens up new possibilities to the advancement of computational sciences and research with already existing and cheap infrastructure.
The Curie-Weiss model is used to model decision making involving social interaction and generalises socio-economic discrete choice model. In this work we study how educational and employment choices are influenced by social attributes of gender and residence of individuals. The model has a social and a private incentive part with coefficients which are the measure of influence individuals have on each other and the external influence which is dependent on the vector of socio-economic attribute respectively. These coefficients are the parameters of the model. There is a satisfaction function of the model representing the utility of individuals as a result of their choices. It is reasonable to assume that individuals with same socio-economic attributes have the same behaviour. The work focuses on how the population of size N is divided into subgroups according to their socio-economic attributes with unequal subgroup sizes and with different attributes function contrary to an earlier work of Contucci et al. Each individual in the population of size N is assigned to two of socio-economic attributes of gender and residence. For gender, an individual could be male or female and for residence an individual could be living in the rural or urban. For this purpose, the population is divided into 4 subgroups, where the individuals in the same subgroup share the same gender and residence. Average decisions for the four groups are calculated from data and fitted into our Currie-Weiss model. Least squares method is used to estimate the parameters of the model. The estimates of the parameters indicate that place of residence has a significant effect on the educational level attained by an individual and the type of employment choices they make.
Modeling with reaction-diffusion systems involves constituents locally transformed into each other by chemical reactions and transported in space by diffusion. With this in mind, the attention to mathematical and disease epidemiology has increased, as disease epidemics have become a predominant worldwide health issue. The cases of V. Cholerae and Malaria are no different. Factors that affect the transmission of such diseases include mainly both human and environmental factors.
Proposing a Reaction-Diffusion SIR-B mathematical model for Cholera and an SEIR mathematical model of Malaria epidemiology with proliferate stability analysis on the epidemic and endemic equilibrium, that incorporates an environmental reservoir in both cases, is formulated to capture the movement of human hosts and host organisms in a heterogeneous environment.
Findings here are supported by the results of numerical experiments and based on these results, an evolutionary process that involves organism distribution and their interaction of spatially distributed population with local diffusion is presented.
Results show that the model dynamics exhibit a diffusion-controlled formation of different patterns which attribute diffusion has a great influence on the spread of the disease.
Mathematical models are used to capture the dynamics of diseases. This project seeks to investigate a SIR model for diarrhea infection with treatment as a compartment. Four nonlinear ordinary differential equations for the SITR model with one infective compartment is established. Then the reproductive number R 0 is determined. Mathematical analyses show that the model is locally and globally stable. A sensitivity analysis is performed for the disease transmission, to determine the right parameters responsible for its transmission and prevalence.
The model is fitted to the diarrhea incidence data from 2008 to 2015 by MOH-Ghana for validation. It was observed that the model fits well to the Ghanaian diarrhea incidence data; this shows that the model can also be used in other countries. The trend of diarrhea cases over a long period of time is predicted and the results forecast a decreasing trend. Optimal control analysis is performed to know the efforts needed to control the diarrhea disease.
The analytical results obtain on incorporating the three optimal controls; public health education u_1(t), the minimization of the bottlenecks of administering vaccination u_2(t) such as unavailability of vaccines, inaccessibility to the communities and the minimization of the bottlenecks of the fraction of individuals (infective class) that take oral rehydration therapy (ORT) at home u_3(t), such as: unavailability of ORS component, using untreated water to make up the solution into the model system were verified through simulations. The dynamics of the infected population for different scenarios were discussed. Finally, the simulations show that all the three controls stated above must be used simultaneously for effective controlling of the disease in Ghana.
Ponzi schemes are gradually getting into competition with regular investment institutions due to their outrageously high interest rates. People have embraced these schemes in spite of the much higher risk involved compared with regular investment companies. In order to understand the dynamics of the spread of Ponzi schemes in a given population, an epidemiological model is set up, precisely, the SIR model. The local stability analysis of both the disease free' and theendemic' equilibria are performed. Interest rate is incorporated into the model as an exponentially distributed random variable yielding the probability of not being `infected' and the related probability of recovering an investment. The study demonstrates that there exist a threshold value of the interest rate, above which the scheme is bound to collapse and below which it persists in the population. Numerical Simulations were also evaluated in order to investigate the effect of the interest rate variations on the sustainability of the scheme and did support the analytical results of the model.
The fundamental group of a topological space is explored with Seifert van-Kampen theorem and how they contribute to differentiating between some geometric surfaces. Some useful results and concepts of group theory together with classification of surfaces serve as a prerequisite to the study.
In this talk, we study a problem of optimal proportional, excess of loss
and investment strategy for an ambiguity averse insurer who is concern
with the misspecification errors and uncertainty around the fiscal model.
We assume that she insurer can invest in a risk-free and risky asset whose
price process follows a Heston stochastic volatility model. This problem
can be seen as a robust optimal control for the insurer. Using dynamic
programming, we obtain an explicit expression for optimal strategies when
the utility of exponential type. Numerical simulation are also performed
to study the sensitivity of the model parameters on the optimal control
and its economic implication.
ABSTRACT
Economic time series analysis has mostly been done in time domain and frequency domain independently. In an attempt to analyze a time series in both domains concurrently, wavelet analysis is used as an alternative approach to other classical approaches. The objective of the study seeks to analyze the time-frequency relationship between stock returns on three different sectors on the Ghana Stock Exchange (GSE). The study employs daily closing returns data of GCB Bank Ltd, Produce Buying Company (PBC) and Unilever Ghana Limited (UNIL). Correlations between stocks are determined by measures of wavelet technique such as wavelet power spectrum and wavelet coherency as alternative to Pearson's linear correlation. Results form the wavelet coherency showed pairs of returns correlation changing over each scale with respect to diverse time horizon.
Optimal allocation of shuttle terminals is a crucial problem encountered in setting up a university shuttle system. Determining the required number of terminals to set up a shuttle system is obviously a factor that can not be ignored for the system to run efficiently. The design to obtain this required terminals is a special type of facility location problem classified as NP-hard problems. On the other hand Genetic Algorithm (GA) is a meta-heuristic optimization technique which works based on evolutionary principle of natural selection. The Genetic Algorithm is helpful in finding solutions to such facility location problems.
In this paper, we first state this facility location problem as a mixed integer programming (MIP) problem. Furthermore, we encode all the possible terminal locations as a set of combinations of candidate terminals based on our constraints and specifically formulate the problem as a bus terminal location problem. Finally we use the genetic algorithm to determine the optimal terminal allocations for the University shuttle system.
The reaction mechanism for the formation hydrocarbons and gasoline for methanol and ethanol
was explored over zeolites FER (Ferrierite) and LTL (Perlialite) using density functional theory
calculations, so as to elucidate the specific elementary step leading to the formation of hydrocarbon
or ethylene by interaction with a hydrocarbon pool. The adsorption energies were studied with
varying Si/Al ratios. No stable protonated alcohol was observed in any of the adsorption sites but
rather only a physiosorbed alcohol. The cleaving of the C-O bond in dehydration of the alcohols
was observed when the starting geometry was of a protonated alcohol. The possible formation of
the dimethyl ether and ethanol from the adsorption of the methoxonium ion was observed with
energies -25kcal.mol and -46kcal/mol respectively. Methanol adsorption was strongest in the
zeolite FER. Adsorption of protonated ethanol in zeolite LTL led to the formation of carbonium
ion and surface ethoxy species with relative energies -9.9kcal/mol and -16kcal/mol respectively.
Both the surface ethoxy group and the carbonium ion formed is the precursors for the formation
of ethylene and long chain hydrocarbons. The results of this study offer great insight into the
optimization of zeolite catalysts to increase hydrocarbon yield in short-chain alcohol conversion.
A ROOT Reference Documentation has been implemented to generate all the lists of libraries needed for each ROOT class. Doxygen has no option to generate or add the lists of libraries for each ROOT class. Therefore shell scripting and a basic C++ program was employed to import the lists of libraries needed by each ROOT class. Finally, some plots showing the simulations of the discovery of Higgs particle at CERN were shown using ROOT.
Computational intelligent algorithms have emerged as new paradigm shift in geosciences computations and applications. The present study aims to perform coordinate transformation using a novel technique called the group method of data handling (GMDH). The data used for the coordinate transformation constitute the Ghana national triangulation network which is based on the two horizontal geodetic datums (Accra 1929 and Leigon 1977) utilised for geospatial applications in Ghana. The obtained GMDH model result was compared with other investigative methods such as backpropagation neural network (BPNN), radial basis function neural network (RBFNN), multivariate adaptive regression spline (MARS), 2D conformal, and 2D affine model. It was noticed that the proposed GMDH model is very efficient in transforming coordinates from the Leigon 1977 datum to the official mapping datum of Ghana, i.e. Accra 1929 datum. It was also found that GMDH could produce comparable results with the widely used BPNN, RBFNN and MARS. The classical transformation methods (2D affine and 2D conformal) performed poorly in comparison with the intelligent systems. The application of the GMDH will serve as a supplement to transformation procedures in the Ghana geodetic reference network. The proposed GMDH approach could also be used in developing countries yet to establish geocentric datum and have the same geodetic network infrastructure like Ghana.
The Legendre-Fenchel transport of a real-valued function is mathematical too used to represent a function in terms of its derivatives. In this work we studdied the introductory background of the Legendre-Fenchel transform and also how the Legendre transform was generalised by Fenchel to a non-differentiable and non-convex function called the Legendre-Frencel transform. We alo studied the concept of convex function, strictly convex function in relation to the transform. Again we focused on the central importance of the Legendre-Fenchel transform in physics especialy classical and statistical mechanics where it provides the connection between the Hamiltonian and the Legrangian(deriving Hamiltonian from a Lagrangian) and establish the relation between the internal energy and various thermodynamics potentials like Gibbs and Helmholtz free energy, ect. . A general motivation follow up with standard were provided to add flesh to the work.
The study of multi-phase flow of fluids in reservoirs is of particular interest in the field of petroleum recovery. This process is studied with estimated or experimentally determined parameter values and some assumptions. Capillary pressure is one of the effective parameters influencing fluid flow in hydrocarbon reservoirs. However, it is assumed negligible by most researchers despite its importance. In this work, a two-phase (oil-water) flow with the effect of capillary pressure was modelled and transformed using fractional flow formulation. The model equations obtained from the fractional formulation comprise an elliptic-pressure equation and a parabolic-saturation equation. The Finite element (FE) method was employed to discretize the elliptic-pressure equation and the corresponding parabolic-saturation equation discretized by the Finite volume (FV) method. Results from the numerical simulation of the model revealed that capillary pressure has effect on saturation profile which was observed in diffusive dominated problems but with less effect in advective-diffusive problems which was observed at points where saturation gradient was high.
Combined Effect of variable viscosity and Thermal Conductivity on dissipative flow of oil-based nanofluid over a permeable vertical plate
Christian John Etwire1,a*, Ibrahim Yakubu Seini2,b, Rabiu Musah3,c
1Faculty of Mathematical Sciences, University for Development Studies, P.O. Box 24, Navrongo, UER, Ghana
2School of Engineering, University for Development Studies, Nyankpala Campus, Tamale, NR, Ghana
3Faculty of Applied Sciences, University for Development Studies, P.O. Box 24, Navrongo, UER, Ghana
jecpapa@yahoo.coma, yakubuseini@yahoo.comb, mrabiu@uds.edu.ghc
Abstract: The combined effect of variable viscosity and thermal conductivity on dissipative flow of oil-based nanofluid over a permeable vertical plate with suction has been studied. The governing partial differential equations were transformed into a coupled third order ordinary differential equations by similarity techniques. The third order ordinary differential equations were then reduced into a system of first order ordinary differential equations and solved numerically using the fourth order Runge-Kutta algorithm with a shooting method. The results were presented in tabular and graphically forms for various controlling parameters. It was revealed that increasing the viscosity parameter of CuO oil-based nanofluid increases the skin friction coefficient but slows the rate of heat transfer at the surface of the plate whilst increasing the thermal conductivity of CuO oil-based nanofluid depreciates both the skin friction coefficient and rate of heat transfer at the surface of the plate. Also, increasing the viscosity and thermal conductivity of CuO oil-based nanofluid, Prandtl number, suction parameter and Biot number weaken the thermal boundary layer.
Keywords: Cohesive, Collision, Lubricant, Piston, Viscosity
1 Introduction
Lubricant plays significant role in the automobile industry. It cushions the engine’s bearings from the shocks of cylinder firing, neutralizes the corrosive elements created during combustion, seals the engine’s metal surfaces from rust and cools internal engine parts that cannot be directly cooled by the engine’s water-cooling system. Cooling is critical for sustaining the desired performance and reliability of vehicle engines but the water cooling system of a vehicle cannot dissipate heat from the piston head of the engine due to detonation and pre-ignition. The piston, as a moving wall of the combustion chamber converts the heat generated as a result of the combustion of the fuel into mechanical work and drives the crankshaft through the connecting rod. Continuous heating of the piston without efficient coolant can lead to high fuel and oil consumption, harmful exhaust emissions, reduction in engine power output or permanent engine damage. To ensure durability, reliability and prolong lifespan of the engine, there is the need for oil with enhanced heat transfer characteristics. Nano-oil is the panacea and it exhibits improvement in thermophysical properties such as thermal conductivity, thermal diffusivity, viscosity, and convective heat transfer coefficients (Choi and Eastman, 1995). Pirhayati et al. (2012) examined the pressure drop of CuO-base oil nanofluid flow inside an inclined tube and found a reduction in pressure with the inclination of the tube at constant volume fraction concentrations and for Reynolds numbers less than 170. Pirhayati et al. (2014) further extended their work to include convective heat transfer characteristics. Their results showed that the heat transfer coefficient of nanofluid increased with increasing Reynolds number inside horizontal and inclined tubes. Bhaumik and Pathak (2015) analyzed the anti-wear properties of CuO nanoparticles in Mineral Oil using Pin-On-Disk Tribometer. Convective heat transfer and stability of Oil–based Nanofluid was investigated by Karamallah and Hussein (2016). Recently, Haq et al. (2017) discussed MHD pulsatile flow of engine oil based carbon nanotubes between two concentric cylinders.
Viscosity of oil is eminent in deciding its suitability for lubrication whether in the hydrodynamic or elasto-hydrodynamic regime. Oil’s viscosity is the measure of its thickness or resistance to flow. It is directly linked with how well oil lubricates and protects surfaces that move relative to each other. Viscous oil provides stronger oil film and the thicker the oil film, the more resistant it will be rubbed from lubricated surfaces. However, very thick oil offers excessive resistance to flow at low temperatures and as a result may not flow quickly enough to those parts requiring lubrication. It is therefore imperative that oil exhibits the right viscosity at both the highest and the lowest temperatures which is a requirement for proper functioning of the engine. Nanotechnology is paramount in developing oil with improved viscosity index. Viscosity index research has received remarkable admiration from scientists and engineers due to its industrial and engineering applications. Researchers such as; Makinde and Aziz (2010), Tshehla and Makinde (2011), Chinyoka and Makinde (2011), Moorthy and Senthilvadivu (2012), EL-Kabeir et al., (2013), Moorthy et al., (2013) , Eegunjobi and Makinde (2014) and Makinde et al., (2016) studied the effect of variable viscosity on conventional fluid.
However, Kuppalapalle et al., (2013) investigated the effect of variable viscosity on the flow and heat transfer of viscous Ag- water and Cu-water nanofluids. Shivakumara and Dhananjaya (2014) discussed the onset of convection in a Nanofluid saturated porous layer with temperature dependent viscosity. Uddin et al., (2014) analyzed g-Jitter mixed convective slip flow of Nanofluid past a permeable stretching sheet embedded in a Darcian porous media with variable viscosity. The effects of variable viscosity and thermal conductivity on natural-convection of Nanofluids past a vertical plate in porous media was reported by Noghrehabadi et al., (2014). James et al., (2015) discussed the effects of variable viscosity of nanofluid flow over a permeable wedge embedded in saturated porous medium with chemical reaction and thermal radiation. Nasrin and Alim (2015) investigated entropy generation by nanofluid with variable thermal conductivity and viscosity in a flat plate solar collector. Alvi et al., (2016) analyzed Peristalsis of non-constant viscosity Jeffrey fluid with nanoparticles. Ram et al., (2016) studied Variable Viscosity effects on time dependent magnetic Nanofluid flow past a stretchable rotating plate.
Most recently, Shahzadi et al., (2017) looked at the simultaneous effects of single wall carbon nanotube and effective variable viscosity on peristaltic flow through annulus having permeable walls. Chandra et al., (2017) studied effects of variable viscosity and thermal conductivity on MHD boundary layer flow of nanofluid with thermal radiation. Huda et al., (2017) analyzed the dynamics of variable-viscosity nanofluid flow with heat transfer in a flexible vertical tube under propagating waves. The effects of oil-based nanofluid on a stretching surface with variable suction and thermal conductivity was discussed by Etwire et al., (2017).
From the survey of literature, not much research work has been done on oil based nanofluid with variable viscosity which occurs in automobile industry since the engine of a vehicle operates at varying temperature and viscosity of oil depletes with temperature. The knowledge of the combined impact of viscosity and other thermophysical parameters on oil would help formulate lubricant which can maintain lubricating oil film at all operating temperature and ensures high temperature viscosity retention. Thus, this study sought to investigate the effects of variable viscosity and thermal conductivity on dissipative flow of oil-based nanofluid over a permeable vertical plate.
2 Mathematical Model
Consider a two-dimensional steady incompressible flow of dielectric and viscous CuO oil-based Nanofluid over a porous vertical plate with variable viscosity and thermal conductivity and suction. The x-axis is taken along the direction of the plate whilst the y-axis is taken normal to it as depicted in Figure 1. The left surface of the plate is heated by convection from a hot fluid at temperature , which provides a heat transfer coefficient while a stream of CuO oil-based Nanofluid at the free stream temperature moves over the right surface of the plate with a uniform free stream velocity .
Figure 1: Schematic diagram of flow problem
It is assumed that both the oil and CuO are in thermal equilibrium with no slip between them. The variation of density in the CuO oil-based nanofluid is taken into account using the Boussinesq approximation. The continuity, momentum and energy equations modeling the flow problem can be expressed as
(1) (2)
(3)
where and are and components of velocities respectively, is the temperature dependent dynamic viscosity of CuO oil-based nanofluid, is the temperature dependent thermal conductivity of CuO oil-based nanofluid, is the density of CuO oil-based nanofluid, is the thermal expansion coefficient of CuO oil-based nanofluid, is the permeability of the porous media, is the temperature of CuO oil-based nanofluid and is the heat capacitance of CuO oil-based nanofluid.
The boundary conditions on the right surface of the plate are;
at (4)
The boundary conditions of the CuO oil-based nanofluid at the far right surface of the plate are;
, as (5)
The properties of the nanofluid (Oztop and Abu-Nada, 2008) are defined as;
(6)
Where and are the densities of the oil and CuO respectively, and are the thermal expansion coefficients of the oil and CuO respectively, is the solid volume fraction of the CuO, and are the thermal conductivities of the oil and CuO nanoparticles respectively and is the dynamic viscosity of oil. The temperature dependent dynamic viscosity , and thermal conductivity , of CuO oil-based nanofluid are defined as;
(7)
(8)
Where is the dynamic viscosity of CuO oil-based nanofluid at ambient temperature , m is the viscosity parameter, is the thermal conductivity of CuO oil-based nanofluid at ambient temperature and q is the thermal conductivity parameter.
3 Similarity Procedure
Equation (1) is satisfied automatically by defining the stream function , in the usual way as:
and (9)
A similarity solution of equations (1) – (5) is achieved by defining an independent variable , a stream function in terms of a dependent variable and a dimensionless temperature , as;
(10)
Substituting equations (6) – (10) into equations (1) – (5), yield the desire coupled ordinary differential equations as;
(11)
(12)
Subject to the boundary conditions
at (13)
as (14)
where the prime symbol denotes differentiation with respect to , is the variable viscosity parameter of CuO oil-based nanofluid, is the thermal Grashof number, is the suction parameter, is the permeability parameter, is the variable thermal conductivity parameter of CuO oil-based nanofluid, is the Prandtl number, is the Biot number and is the Brinkman number. Parameters of engineering applications considered in this study are the skin-friction coefficient (Cf) and the Nusselt number (Nu) which are defined as;
(15)
where is the wall shear stress and is the wall heat flux which are given by
(16)
Substituting equation (16) into equation (15) yield
(19)
3 Numerical Procedure
The coupled third order nonlinear ordinary differential equations (11) and (12) are reduced into a system of first order ordinary differential equation by letting;
, (20)
Substituting equation (20) into equations (11) and (12) yield the required system of first order differential equations as;
(21)
Subject to the boundary conditions;
(22)
The Shooting technique is employed to guess the unknown b and c until the boundary conditions ( and ) is satisfied. The resulting differential equations are solved using the fourth order Runge Kutta integration scheme. Numerical computations are done using MAPLE 16 software package.
4 Results and Discussions
The thermophysical parameters of industrial and engineering applications considered in this study include; solid volume fraction of CuO ( ), Biot number (Bi), Brinkman number (Br), Prandtl number (Pr), Permeability parameter (K*), the viscosity (ᾳ) and the thermal conductivity (σ) of CuO oil-based nanofluid and Thermal Grashof number (GT). The effects of these parameters on the velocity profile, temperature profile, skin friction coefficient (Cf) and Nusselt number (Nu) were explored. The solid volume fraction of CuO was varied within the range . The permeability parameter and Grashof number are maintained at a constant value of 0.01. The thermophysical properties of oil and CuO are given in Table 1.
Table 1: Thermophysical properties of base fluid and nanoparticle
Physical property Cp (J/kgK) ρ (Kg/m3) k (W/mK) β x 10-5(K-1)
Oil 1670 920 0.138 64
CuO 540 6510 18 0.85
4.1 Numerical results
The numerical results of the present work for the plate surface temperature ( ) and the local Nusselt number represented by the rate of heat transfer ( ) were compared with the work of Aziz (2009) for varying values of the Biot number (Bi) and there was excellent agreement. The comparison is presented in Table 2.
Table 2: Computations showing comparison with Aziz (2009) for and
Aziz (2009) Present Work
Bi
0.05 0.1447 0.0428 0.1447 0.0428
0.10 0.2528 0.0747 0.2528 0.0747
0.20 0.4035 0.1193 0.4035 0.1193
0.40 0.5750 0.1700 0.5750 0.1700
0.60 0.6699 0.1981 0.6699 0.1981
0.80 0.7302 0.2159 0.7302 0.2159
1.00 0.7718 0.2282 0.7718 0.2282
The impact of the various thermo-physical parameters on the skin friction coefficient and the rate of heat transfer at the surface of the plate are presented in Table 3.
Table 3: Computation showing and for different parameter values
Br ⱷ Pr σ ᾳ S Bi
100 0.10 100 0.1 0.1 0.01 1 0.386727 1.902234
150 0.10 100 0.1 0.1 0.01 1 0.404480 2.881563
200 0.10 100 0.1 0.1 0.01 1 0.422451 3.745695
100 0.13 100 0.1 0.1 0.01 1 0.393888 2.176423
100 0.16 100 0.1 0.1 0.01 1 0.398046 2.463991
100 0.20 100 0.1 0.1 0.01 1 0.399411 2.873316
100 0.10 200 0.1 0.1 0.01 1 0.370614 1.127871
100 0.10 300 0.1 0.1 0.01 1 0.364551 0.727374
100 0.10 400 0.1 0.1 0.01 1 0.361333 0.469120
100 0.10 100 1.0 0.1 0.01 1 0.380246 1.092332
100 0.10 100 2.0 0.1 0.01 1 0.377653 0.806703
100 0.10 100 3.0 0.1 0.01 1 0.376132 0.650537
100 0.10 100 0.1 1.0 0.01 1 0.442927 0.553412
100 0.10 100 0.1 2.0 0.01 1 0.504076 0.154223
100 0.10 100 0.1 2.5 0.01 1 0.535795 0.042832
100 0.10 100 0.1 0.1 0.10 1 0.417849 1.177565
100 0.10 100 0.1 0.1 0.20 1 0.459698 0.721107
100 0.10 100 0.1 0.1 0.30 1 0.504852 0.464480
100 0.10 100 0.1 0.1 0.01 2 0.384661 2.976997
100 0.10 100 0.1 0.1 0.01 3 0.383394 3.672943
100 0.10 100 0.1 0.1 0.01 4 0.382537 4.161911
Table 3 reveals that increasing the Brinkman number increases the magnitude of both the skin friction coefficient and the rate of heat transfer at the surface of the plate. This is as a result of enhanced viscous dissipation over thermal conduction. Similar trend is observed with the solid volume fraction of CuO due to ballistic heat transfer occurring in the CuO. Conversely, increasing the intensity of the Prandtl number and thermal conductivity of CuO oil-based nanofluid decrease the magnitude of both the skin friction coefficient and the rate of heat transfer at the surface of the plate. However, increasing both the viscosity parameter of CuO oil-based nanofluid and suction parameter increase the intensity of the skin friction coefficient but slow the rate of heat transfer at the surface of the plate since the enhancement in the viscosity parameter strengthens the cohesive forces between the molecules of CuO oil-based nanofluid which enhances the viscous shear stresses in the nanofluid whilst the suction parameter delays the onset of the boundary layer. Also increasing the Biot number depreciates the skin friction coefficient but enhances the rate at which heat is transferred at the surface of the plate
4.2 Graphical Results
4.2.1 Effects of Parameter Variation on the Velocity Profiles
Figures 2 and 3, illustrate the velocity profiles for varying values of the thermophysical parameters. It is noted in Figure 2 that increasing the magnitude of the viscosity parameter of CuO oil-based nanofluid increases the velocity profile of the nanofluid. An increase in the viscosity parameter slows down the motion of the molecules of CuO oil-based nanofluid and increases the collision rate of the molecules which enhances the shearing stresses of the nanofluid. This thickens the momentum boundary layer of CuO oil-based nanofluid. Similar trend is observed in Figure 3 as the suction parameter is increased.
Figure 2: Velocity Profile for varying values of Viscosity of CuO oil-based Nanofluid for , and
Figure 3: Velocity Profile for varying values of Suction parameter for , and
4.2.2 Effects of Parameter Variation on Temperature Profiles
Figures 4–10, present the temperature profiles for varying values of the thermophysical parameters. It is evident in Figures 4 and 5 that increasing the solid volume fraction of CuO and Brinkman number enhance the temperature profile. The combined enhancement in solid volume fraction of CuO and Brinkman number intensify the shearing stresses of the nanofluid due to viscous diffusion. This increases the temperature of CuO oil-based nanofluid within the vicinity of the plate which intend thickens the thermal boundary layer. However, in Figure 6 increasing the viscosity of CuO oil-based nanofluid depreciates the temperature profile. As viscosity is increased, the cohesive intermolecular forces between the molecules of the nanofluid strengthen due to the drop in the energy level of the molecules. This deteriorates the thermal boundary layer thickness. Similar trend was observed in Figures 7-10 as the magnitudes of thermal conductivity of CuO oil-based nanofluid, Prandtl number, suction parameter and Biot number were increased due to enhanced viscous diffusion rate and the delay in the development of the boundary layer.
Figure 4: Temperature Profile for varying values of Solid Volume Fraction of CuO for , and
Figure 5: Temperature Profile for varying values of Brinkman number for , and
Figure 6: Temperature Profile for varying values of Viscosity of CuO oil-based Nanofluid for , and
Figure 7: Temperature Profile for varying values of Thermal Conductivity of CuO oil-based Nanofluid for , and
Figure 8: Temperature Profile for varying values of Prandtl number for , and
Figure 9: Temperature Profile for varying values of Suction parameter for , and
Figure 10: Temperature Profile for varying values of Biot number for , and
5 Conclusions
The combined effect of variable viscosity and thermal conductivity on dissipative flow of oil-based nanofluid over a permeable vertical plate with suction has been examined. The partial differential equations governing the flow were transformed into ordinary differential equations by similarity transformation and then solved numerically using the fourth order Runge-Kutta algorithm with a shooting method. Numerical results were presented whilst the velocity and temperature profiles illustrated graphically and analyzed. The study revealed that increasing the magnitudes of both the viscosity and thermal conductivity parameters of CuO oil-based nanofluid depreciate the thermal boundary layer.
Nomenclature
Cartesian coordinates
Velocity components
Ambient temperature
Free-stream temperature
Temperature of the sheet
Temperature of CuO oil-based nanofluid
Thermal conductivity of oil
Thermal conductivity of CuO
Specific heat at constant pressure
Permeability of the porous media
Permeability parameter
= Heat transfer coefficient
Prandtl number
Cf = skin-friction coefficient
Thermal Grashof number
Re = Reynolds number
Nu = Nusselt number
Wall heat flux
Biot number
Brinkman number
Greek Symbols
Wall shear stress
Dynamic viscosity of CuO oil-based nanofluid
Kinematic viscosity of oil
Density of CuO oil-based nanofluid
Thermal diffusivity of oil
Density of oil
Density of CuO
Solid volume fraction of CuO
Heat capacitance of CuO oil-based nanofluid
Stream function
Variable thermal conductivity parameter of CuO oil-based nanofluid
Variable viscosity parameter of CuO oil-based nanofluid,
= Thermal expansion coefficient of CuO oil-based nanofluid
= Thermal expansion coefficient of oil
= Thermal expansion coefficient of CuO
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