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IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 17, Issue 3 Ser. I (May – June 2021), PP 42-48 www.iosrjournals.org Hydromagnetic Free Convection Turbulent Unsteady Fluid Flow over A Vertical Infinite Heat Absorbing Plate 1 Evaline Chepkemoi1, Wilys O. Mukuna1* Department of Mathematics and Computer Science, University of Kabianga, P. O. Box 2030 – 20200, KERICHO, KENYA Abstract Analysis of hydromagnetic free convection turbulent fluid flow over an infinite vertical plate was carried out. The fluid flow was modeled using conservation equations of energy and momentum. The governing equations were then non-dimensionalised which gave rise to non-dimensional parameters.The approximate numerical solution for the non-linear partial differential equations were determined by use of the finite difference method and solved using MATLAB computer software. The various non-dimensional parameters were thereafter examined of their effects on the velocities and temperature profiles. Thereafter the solutions presented in graphs and an analysis of the results given. The study of magnetohydrodynamics is important as it has applications in areas like meteorology and astrophysics, biological, environmental, aerospace and aeronautical engineering among others. It is evident from the results that the primary velocity increases with decreasing magnetic parameter (M), increases with increase in Hall parameter and also increases with increase in Grashoff number. Also, the secondary velocity increases with decreasing magnetic parameter (M) and decreases with increasing Hall parameter. It is also found that the temperature profile decreases with decreasing magnetic parameter (M), decreases with increasing Hall parameter and increases with decrease in Prandtl number. Keywords: Unsteady, Hydromagnetic, Turbulent, Free convection, Vertical plate, Finite difference --------------------------------------------------------------------------------------------------------------------------------------Date of Submission: 28-04-2021 Date of Acceptance: 12-05-2021 --------------------------------------------------------------------------------------------------------------------------------------Nomenclature u,v,w x,y,z t p T  k ρ m E Gr g Pr M Ho ɸ α Cp u’, v’ w’ a Q W Cartesian velocity components (ms-1) Cartesian coordinate variables Time (s) Pressure (Nm-2) Temperature (K) Dynamic viscosity (Kgm-2s) Kinematic viscosity (m2s-1) Thermal conductivity (wm-1k-1) Gradient operator, Density (Kg/m3) Mass of the fluid particle (Kg) Electric field (Vm-1), internal energy (J) Thermal Grashoff number Acceleration due to gravity (ms-2) Prandtl number Turbulent Prandtl number Magnetic parameter External applied transverse magnetic field intensity(wbm-2) Viscous dissipative rate Thermal diffusivity Thermal expansion coefficient (K-1) Specific heat at constant pressure (JKg-1K-1) Fluctuating components of velocity Mean velocities Acceleration (m/s2) Heat (J) Work (J) DOI: 10.9790/5728-1703014248 www.iosrjournals.org 42 | Page Hydromagnetic Free Convection Turbulent Unsteady Fluid Flow Over A Vertical .. Material derivative given by L B H J Partial derivative with respect to time Electrical conductivity (Ω-1m-1) Characteristic length (m) Electron permeability (H/m) Magnetic flux density (wbm-2) Magnetic field intensity (wbm-2) Current density vector I. Introduction The subject of fluid flow is of great importance to scientists and engineers of different fields like geophysists, meteorologists and industrialists among others. Fluid can flow in a channel, in a pipe or over a plate. In this study our interest is on the flow of fluid over a plate. Generally, the motion of a fluid within a boundary layer can either be laminar or turbulent. However, in real life situations fluid flows are turbulent and not laminar. Therefore, in this study we are interested in turbulent flow. Laminar flow is a fluid flow where the particles of a fluid move parallel to each other and there is no mixing between the adjacent layers of the fluid. On the other hand, turbulent flow is a fluid flow in which the fluid particles do not move parallel to each other and the adjacent layers of the fluid cross each other. A boundary layer of fluid refers to the immediate vicinity of the boundary surface where the effects of viscosity are significant. The region in which flow adjusts from zero velocity at the wall to a maximum in the main stream of the flow is called boundary layer. When there is fluid flow over a surface a thermal boundary layer must develop if there is bulk temperature difference. The transfer of energy in a fluid takes place by way of convection. Convection is a process in which energy is transferred through a fluid when there is motion of bulk fluid [2]. The theoretical investigation of fluid flow is referred to as computational fluid dynamics. The use of computational software, applied mathematics and physics to picture out in what manner fluid flows and how this fluid affects objects as it flows is known as computational fluid dynamics. A field of study in which magnetofluids is carried out is called magnetohydrodynamics (MHD). The study of magnetohydrodynamics is significant in areas like meteorology and astrophysics, biological, environmental, aerospace and aeronautical engineering among others [11]. II. Literature review Researchers have done some work on MHD stokes problem for a dissipative fluid which is heat generating with ion-slip and hall current, mass diffusion and radiation absorption [5]. The influence on the rate of mass transfer, concentration, velocity skin friction, the rate of heat transfer and temperature for the various parameters were analyzed. Also investigation on unsteady MHD fluid flow, mass and heat transmission features in an incompressible, viscous, Newtonian and electrically conducting fluid across a porous vertical plate has been done [4]. He considered reaction of chemical, thermal radiation and induced magnetic field. He then gives the solution of the governing equations using scheme of finite difference which is of Crank-Nicholson type. Also the effects of various non-dimensional parameters was carried out. It was found that the velocity decreases with increasing magnetic parameter (M). Also there was a decrease in concentration with increasing Schmidt number as well as chemical reaction. Studies on an hydromagnetic turbulent boundary layer fluid flow past an infinite vertical cylinder with Hall current was carried out by a number of scholars [11]. They used prandtl mixing length hypothesis to resolve Reynolds stresses which arose due to turbulence in conservation equations. They solved these equations using finite difference method. Also, the influence on temperature profiles and velocity were investigated for the flow parameters. It is found that primary velocity increases with increase in Hall parameter. A mathematical model of an MHD turbulent boundary layer fluid flow was analyzed [9]. The problem of the flow when the plate is impulsively started and the turbulence in the mass conservation equations gave rise to Reynold stresses which were then resolved using prandtl mixing length hypothesis. Finite difference method was used to find solutions to the governing equations of the unsteady free convection turbulent boundary layer fluid flow. He then presented the results obtained in form of graphs and the effects of time, Hall current and Eckert numbers on temperature and velocity profiles were discussed. He found that an increase in Eckert leads to an increase in the velocity profiles and temperature profiles in heating of the plate and does not affect the temperature profiles in the case of cooling the plate. DOI: 10.9790/5728-1703014248 www.iosrjournals.org 43 | Page Hydromagnetic Free Convection Turbulent Unsteady Fluid Flow Over A Vertical .. III. Mathematical Model We are considering a two-dimensional flow for this study. The infinite vertical plate is taken to be along the x-axis and the horizontal is the y-axis while the z+-axis is taken normal to the plate. The fluid is assumed to be incompressible and viscous. A strong magnetic field of uniform strength is applied normal to the direction of the flow. The induced magnetic field is considered negligible hence H = (0, 0, ), as shown in the figure below. The temperature of the plate and the fluid are assumed to be the same initially. At time t*>0 the plate is stationary and the fluid starts moving impulsively in its plane with velocity and at the same time the temperature of the plate is instantaneously raised to which is maintained constant later on. The above flow is governed by the following equations: 4.1 4.2 4.3 The boundary and initial conditions are: at Boundary and Initial Conditions , , , , everywhere , , , , at , ,, , as DOI: 10.9790/5728-1703014248 4.53 4.54 4.55 www.iosrjournals.org 44 | Page Hydromagnetic Free Convection Turbulent Unsteady Fluid Flow Over A Vertical .. Final Set of Governing Equations The equations whose solutions we seek and the boundary conditions are [8,13]: 4.56 4.57 Given that the turbulent prandtl number is given by Where then: , substituting this in equation 4.42 we obtain 4.58 , , , , , ,, , , , everywhere , at 4.59 4.60 4.61 , as IV. Finite Difference Scheme Considering that the systems of partial differential equations 3.55, 3.56 and 3.57 are highly non-linear their analytic solutions are not possible. We thus approximate their solutions by finite difference method. The mesh is shown in figure 2 and the equivalent finite difference scheme for equations 3.55, 3.56 and 3.57 are respectively: 4.62 4.63 4.64 Where i and j refer to z and t respectively. The values for k have been substituted as 0.4 and z have been substituted with . The boundary and initial conditions 4.45 now take the form: everywhere for 4.65 for 4.66 for 4.67 Using the boundary and initial conditions 4.49 we compute values for consecutive grid points for primary and secondary velocities and temperature that is: and , + 2 2 , , + , 1+ 2 , 1+ 2 4.68 4.69 4.70 DOI: 10.9790/5728-1703014248 www.iosrjournals.org 45 | Page Hydromagnetic Free Convection Turbulent Unsteady Fluid Flow Over A Vertical .. 4.7 Discussion of Results The numerical results obtained from the computer program are presented in this chapter. The trends of various fluid flow parameters are discussed and explained as they were observed upon varying. DOI: 10.9790/5728-1703014248 www.iosrjournals.org 46 | Page Hydromagnetic Free Convection Turbulent Unsteady Fluid Flow Over A Vertical .. 4.1 Primary velocity From figure 4.4 it is observed that: i) When there is decrease in Magnetic parameter results in increase in the primary velocity profile. The presence of magnetic field in an electrically conducting fluid introduces a force which acts against the flow if the magnetic field is applied hence the effect in the primary velocity. ii) An increase in Hall parameter leads to increase in the primary velocity iii) Finally an increase in Grashoff parameter decreases the primary velocity. The Grashoff number shows the relative effect of the buoyancy force to the viscous force in the boundary layer. 4.2 Secondary velocity From figure 4.5 it is observed that: i) For a decrease in Magnetic parameter there is a resulting increase in the secondary velocity. The presence of magnetic field in an electrically conducting fluid introduces a force which acts against the flow if the magnetic field is applied hence the change in the secondary velocity. ii) When the Hall parameter is increased, there is a decrease in the secondary velocity. iii) An increase in the Grashoff number doesnot affect the secondary velocity. 4.3 Temperature From figure 4.6 it is observed that: i) Increase Hall parameter decreases the temperature profiles. ii) Decrease in Prandtl number increases the temperature profiles. Physically, decrease in Prandtl number leads to an increase in thermal boundary layer and rise in the average temperature within boundary layer iii) For a decrease in Magnetic parameter there is a resulting decrease in the temperature profiles. V. Conclusion And Recommendations 5.1 Introduction In this chapter conclusions and recommendations of the research carried out were outlined as follows: 5.2 Conclusion 1) It is found that the primary velocity increases with decreasing magnetic parameter (M), increases with increase in Hall parameter and also increases with increase in Grashoff number. 2) It is found that the secondary velocity increases with decreasing magnetic parameter (M) and decreases with increasing Hall parameter. 3) It was also found that the temperature profile decreases with decreasing magnetic parameter (M), decreases with increasing Hall parameter and increases with decrease in Prandtl number. 5.3 Recommendations There is still a lot that has to be done from this research so as to be able to be much closer to the effects of the situations in real life. Therefore, the following recommendations will be of much help in further research related to this topic area; • Flows involving non-Newtonian fluids. • Compressible fluid flow. • Magnetic field inclined at an angle • Considering a rotational plate. References [1]. [2]. [3]. [4]. [5]. [6]. [7]. [8]. [9]. [10]. Alfven H. (1942). on existence of electromagnetic-hydrodynamic waves. Arkiv for material Astronomi och fysik. Bd., 29b(2), 140143. Bergman Ted and Adrienne Lavine (2010). Fundamentals of Heat and Mass Transfer, sixth edition.Wiley. Cowling T. G. (1957). Magnetohydrodynamics. Interscience, vol 3(5). Kiprop K. (2017). Unpublished Msc thesis. Hydrodynamic radiating fluid flow past an infinite vertical porous plate in presence of chemicals reaction and induced magnetic field. Jomo Kenyatta University of Agriculture and Technology. Kwanza J.K., Kinyajui M. and Uppal SM. (2005). 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