DOCSLIB.ORG

Research Article Heat and Mass Transfer in Three-Dimensional Flow of an Oldroyd-B Nanofluid with Gyrotactic Micro-Organisms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Research Article Heat and Mass Transfer in Three-Dimensional Flow of an Oldroyd-B Nanofluid with Gyrotactic Micro-Organisms Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 6790420, 15 pages https://doi.org/10.1155/2018/6790420 Research Article Heat and Mass Transfer in Three-Dimensional Flow of an Oldroyd-B Nanofluid with Gyrotactic Micro-Organisms M. Sulaiman,1,2 Aamir Ali ,1 and S. Islam 2 1 Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock, 43600, Pakistan 2Department of Mathematics, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, 23200, Pakistan Correspondence should be addressed to Aamir Ali; aamir [email protected] Received 10 December 2017; Revised 28 February 2018; Accepted 29 April 2018; Published 6 September 2018 Academic Editor: Efstratios Tzirtzilakis Copyright © 2018 M. Sulaiman et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tis paper discusses the three-dimensional fow of the gyrotactic bioconvection of an Oldroyd-B nanofuid over a stretching surface. Teory of microorganisms is utilized to stabilize the suspended nanoparticles through bioconvection induced by the efects of buoyancy forces. Analytic solution for the governing nonlinear equations is obtained by using homotopy analysis method (HAM). Te efects of involved parameters on velocity, temperature, nanoparticles concentration, and density of motile microorganisms are discussed graphically. Te local Nusselt, Sherwood, and motile microorganisms numbers are also analyzed graphically. Several known results have been pointed out as the particular cases of the present analysis. It is found that the non-Newtonian fuid parameters, i.e., relaxation time parameter �1 and retardation time parameter �2, have opposite efects on the velocity profle. Te velocity of the fuid and boundary layer thickness decreases for increasing relaxation time while it decreases for increasing retardation time efects. 1. Introduction non-Newtonian fuid models [2–10]. However, in practical applications sometimes, the fow is three-dimensional. In In recent years the signifcance of the study of non- view of such motivation, researchers have studied the three- Newtonian fuids has remarkably increased due to its appli- dimensional fow for diferent fow geometries (see ref. [11– cations in engineering, industry, and biological sciences, e.g., 17]). in material processing, food, and cosmetic industries [1]. Te problem of heat and mass transfer analysis over a Tese applications generally involve complex geometries and stretched boundary layer is of great interest to researchers due require that the response of the fuid to given temperature to its engineering and industrial applications like glass fber, conditions is controlled. Te basic equations that governs manufacture and drawing of plastics and rubber sheets, paper the fow of non-Newtonian fuids are highly nonlinear. Te production, cooling of continuous stripes and an infnite mathematical description of a non-Newtonian fuid requires metallic sheet, the continuous casting, polymer extrusion a constitutive equation that determines the rheological prop- process, heat treated materials travelling on conveyer belts, erties of the fuid. Due to the complexity, the aforementioned food processing, and many others. Sakiadis [18] has done the fuids are generally divided into three types, namely, the pioneer work in the analysis of the boundary layer fow on diferential type, the rate type, and the integral type. Te a moving surface, which was later extended by Crane [19] simplest example of rate type fuid is Maxwell fuid model by considering the linearly stretching surface. Rashad and which narrates the properties of relaxation time. But it cannot Kabeir [20] studied the heat and mass transfer in transient predict the retardation time efects. Te Oldroyd-B fuid fow by mixed convection boundary layer over a stretching is one subclass of the rate type fuid which describes the sheet embedded in a porous medium with chemically reactive properties of both the relaxation time and the retardation species. Sandeep and Sulochana [21] studied the general- time. Tere are several investigations in this direction for two- ized model for momentum and heat transfer behavior of dimensional fow confgurations with diferent features of the Jefrey, Maxwell, and Oldroyd-B nanofuids over a stretching 2 Mathematical Problems in Engineering →∞,→ ∞,→ ∞ →0, v → 0 Nanoparticles and micro-organisms = w = w = w v=v(y) =by = () = Figure 1: Geometry of the problem. surface with nonuniform heat source/sink. Afer that, several In present study, we have entered a new concept in the investigations have been done by considering diferent fow problem of boundary layer fow passing a stretching sheet. We confgurations [22–28]. have considered the three-dimensional bioconvection fow of With the advent of nanoscience, nanofuids have become an Oldroyd-B nanofuid which contains self-impelled motile a focus of attention in the study of fuid fow in the presence gyrotactic microorganisms, over a bidirectional stretching of nanoparticles. Nanofuids can be used to increase the surface.TisworkisanextensionoftheworkofW.A. thermal conductivity of the fuids and are stable fuids having Khan et al. [16] by considering the gyrotactic microorganisms much better writing, dispersion, and spreading properties in the three-dimensional fow of Oldroyd-B nanofuid. Te on solid surfaces. Choi and Eastman [29] initially presented problem is formulated in terms of governing equations of the idea of suspension of nanoparticles in base fuids. Later, motion with appropriate boundary conditions, which are Buongiorno [30] extended the concept by considering Brow- then transformed into a system of ordinary diferential nian motion and thermophoresis movement of nanoparticles equations using similarity transformations. Te analytical in view of application in hybrid power engine, thermal solution is found using Homotopy Analysis Methods (HAM). management, heat exchanger, domestic refrigerators, etc. Te efects of involved physical parameters on the fow felds Tey have enhanced heat conduction power as compared to are analyzed with the help of graphs. base fuids and are useful in cancer therapy and medicine. With the presentation of a reliable model of nanofuid 2. Problem Formulation given by Buongiorno [30], nanofuids have become a subject of great interest for researchers in the last few decades We have considered the steady, incompressible three- [31–34]. Recently, Nabwey et al. [35] studied the group dimensional fow of an Oldroyed-B nanofuid containing method analysis of mixed convection stagnation-point fow gyrotactic microorganisms past a stretching surface. Te sur- of non-Newtonian nanofuid over a vertical stretching sur- face is maintained at constant temperature ��, the nanopar- face. ticlevolumefraction��, and the number of motile microor- Bioconvection is a phenomenon that is used to describe ganisms ��; the ambient values of temperature, nanoparticle the unstructured pattern and instability produced by the volume fraction, and number of motile microorganisms are microorganisms, which are swimming to the upper part of denoted by �∞, �∞,and�∞, respectively, while the reference a fuid and has a lesser density. Due to swimming upwards, values are by �0, �0,and�0, respectively. Te microorganisms these involved microorganisms such as gyrotactic microor- are taken into the nanofuid to stabilize the nanoparticles due ganisms like algae tend to concentrate in the upper part of the to bioconvection. Te mechanical analog for the Oldroyd- fuid layer, causing unstable top heavy density stratifcation B model and the geometry of the present analysis is shown (see [36]). Nanofuid bioconvection phenomenon describes in Figure 1. Te boundary layer form of the continuity, the unstructured density stratifcation and formation of momentum, energy, nanoparticle concentration, and density patterns due to synchronized interaction of the nanoparticles, of microorganisms for three-dimensional Oldroyd-B fuid is thicker self-propelled microorganisms and buoyancy forces. as follows ([16, 37, 50]): Brownian motion and thermophoresis efect produce the motion of these particles, which is independent of the move- �� �V �� + + =0, (1) ment of motile microorganisms. Tat is why the combined �� �� �� interface of bioconvection and nanofuids becomes signif- 2 2 2 icant for microfuidic appliances. Uddin et al. studied the �� �� �� 2 � � 2 � � 2 � � � + V +� +�1 (� + V +� computational investigation of Stefan blowing and multiple- �� �� �� ��2 �� 2 ��2 slip efects on buoyancy-driven bioconvection nanofuid fow �2� �2� �2� �2� with microorganisms [37] and numerical solutions for gyro- +2�V +2V� +2�� )=] [ tactic bioconvection in nanofuid saturated porous media ���� ���� ���� ��2 with Stefan blowing and multiple slip efects [38]. Diferent �3� �3� �3� �� �2� types of microorganisms, theoretical biconvection models, +� (� + V +� − have been studied [39–51]. 2 ����2 ���� 2 ��3 �� ��2 Mathematical Problems in Engineering 3 �� �2V �� �2� 1 � � � − − )] + [� �(1−� )(� 1 and 2 are the relaxation time and retardation time, � is 2 2 � ∞ �� �� �� �� �� the density of nanofuid, �� is the density of nanoparticles, �� is the density of microorganisms particles, � is the gravity, −�∞)−(�� −��)(�−�∞)−�(�� −��)(� �=(��)�/(��)� is the heat capacity ratio, �� and �� are the stretching velocities, �, � > 0 are the stretching rate, and � ,� ,� −�∞)] g, 1 1 1 are the
Load more

Recommended publications
  • General@ Electric Space Sciences Laboratory Theoretical Fluid Physics Section
    4 3 i d . GPO PRICE $ CFSTI PRICE(S) $ R65SD50 Microfiche (MF) , 7.3, We53 July85 THE STRUCTURE OF THE VISCOUS HYPERSONIC SHOCK LAYER SPACE SCIENCES LABORATORY . MISSILE AND SPACE DIVISION GENERAL@ ELECTRIC SPACE SCIENCES LABORATORY THEORETICAL FLUID PHYSICS SECTION THE STRUCTURE OF THE VISCOUS HYPERSONIC SHOCK LAYER BY L. Goldberg . Work performed for the Space Nuclear Propulsion Office, NASA, under Contract No. SNPC-29. f This report first appeared as part of Contract Report DIN: 214-228F (CRD), October 1, 1965. Permission for release of this publication for general distribution was received from SNPO on December 9, 1965. R65SD50 December, 1965 MISSILE AND SPACE DIVISION GENERAL ELECTRIC CONTENTS PAGE 1 4 I List of Figures ii ... Abstract 111 I Symbols iv I. INTRODUCTION 1 11. DISCUSSION OF THE HYPER NIC CO TTINUI 3 FLOW FIELD 111. BASIC RELATIONS 11 IV . BOUNDARY CONDITIONS 14 V. NORMALIZED SYSTEM OF EQUATIONS AND BOUNDARY 19 CONDITIONS VI. DISCUSSION OF RESULTS 24 VII. CONCLUSIONS 34 VIII. REFERENCES 35 Acknowledgements 39 Figures 40 1 LIST OF FIGURES 'I PAGE 1 1 1. Hypersonic Flight Regimes 40 'I 2. Coordinate System 41 3. Profiles Re = 15, 000 42 S 3 4. Profiles Re = 10 43 S 3 5. Profiles Re = 10 , f = -0.4 44 S W 2 6. Profiles Re = 10 45 S 2 7. Profiles Re = 10 , f = do. 4 46 S W Profiles Re = 10 47 8. s 9. Normalized Boundary Layer Correlations 48 10. Reduction in Skin Friction and Heat Transfer with Mass 49 Transfer 11. Normalized Heat Transfer 50 12. Normalized Skin Friction 51 13.
    [Show full text]
  • Inertia-Less Convectively-Driven Dynamo Models in the Limit
    Inertia-less convectively-driven dynamo models in the limit of low Rossby number and large Prandtl number Michael A. Calkins a,1,∗, Keith Julienb,2, Steven M. Tobiasc,3 aDepartment of Physics, University of Colorado, Boulder, CO 80309 USA bDepartment of Applied Mathematics, University of Colorado, Boulder, CO 80309 USA cDepartment of Applied Mathematics, University of Leeds, Leeds, UK LS2 9JT Abstract Compositional convection is thought to be an important energy source for magnetic field generation within planetary interiors. The Prandtl number, Pr, characterizing compositional convection is significantly larger than unity, suggesting that the inertial force may not be important on the small scales of convection as long as the buoyancy force is not too strong. We develop asymptotic dynamo models for the case of small Rossby number and large Prandtl number in which inertia is absent on the convective scale. The rele- vant diffusivity parameter for this limit is the compositional Roberts number, q = D/η, which is the ratio of compositional and magnetic diffusivities. Dy- namo models are developed for both order one q and the more geophysically relevant low q limit. For both cases the ratio of magnetic to kinetic energy densities, M, is asymptotically large and reflects the fact that Alfv´en waves have been filtered from the dynamics. Along with previous investigations of asymptotic dynamo models for Pr = O(1), our results show that the ratio M is not a useful indicator of dominant force balances in the momentum equa- tion since many different asymptotic limits of M can be obtained without changing the leading order geostrophic balance.
    [Show full text]
  • Summary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer 1. Nusselt Number Average Nusselt Number: Nul = Convective
    Jingwei Zhu http://jingweizhu.weebly.com/course-note.html Summary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer 1. Nusselt number Average Nusselt number: convective heat transfer ℎ퐿 Nu = = L conductive heat transfer 푘 where L is the characteristic length, k is the thermal conductivity of the fluid, h is the convective heat transfer coefficient of the fluid. Selection of the characteristic length should be in the direction of growth (or thickness) of the boundary layer; some examples of characteristic length are: the outer diameter of a cylinder in (external) cross flow (perpendicular to the cylinder axis), the length of a vertical plate undergoing natural convection, or the diameter of a sphere. For complex shapes, the length may be defined as the volume of the fluid body divided by the surface area. The thermal conductivity of the fluid is typically (but not always) evaluated at the film temperature, which for engineering purposes may be calculated as the mean-average of the bulk fluid temperature T∞ and wall surface temperature Tw. Local Nusselt number: hxx Nu = x k The length x is defined to be the distance from the surface boundary to the local point of interest. 2. Prandtl number The Prandtl number Pr is a dimensionless number, named after the German physicist Ludwig Prandtl, defined as the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. That is, the Prandtl number is given as: viscous diffusion rate ν Cpμ Pr = = = thermal diffusion rate α k where: ν: kinematic viscosity, ν = μ/ρ, (SI units : m²/s) k α: thermal diffusivity, α = , (SI units : m²/s) ρCp μ: dynamic viscosity, (SI units : Pa ∗ s = N ∗ s/m²) W k: thermal conductivity, (SI units : ) m∗K J C : specific heat, (SI units : ) p kg∗K ρ: density, (SI units : kg/m³).
    [Show full text]
  • The Analogy Between Heat and Mass Transfer in Low Temperature Crossflow Evaporation
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by AUT Scholarly Commons The analogy between heat and mass transfer in low temperature crossflow evaporation Reza Enayatollahi*, Roy Jonathan Nates, Timothy Anderson Department of Mechanical Engineering, Auckland University of Technology, Auckland, New Zealand. Corresponding Author: Reza Enayatollahi Email Address: [email protected] Postal Address: WD308, 19 St Paul Street, Auckland CBD, Auckland, New Zealand Phone Number: +64 9 921 9999 x8109 Abstract This study experimentally determines the relationship between the heat and mass transfer, in a crossflow configuration in which a ducted airflow passes through a planar water jet. An initial exploration using the Chilton-Colburn analogy resulted in a coefficient of determination of 0.72. On this basis, a re-examination of the heat and mass transfer processes by Buckingham’s-π theorem and a least square analysis led to the proposal of a new dimensionless number referred to as the Lewis Number of Evaporation. A modified version of the Chilton-Colburn analogy incorporating the Lewis Number of Evaporation was developed leading to a coefficient of determination of 0.96. 1. Introduction Heat and mass transfer devices involving a liquid interacting with a gas flow have a wide range of applications including distillation plants, cooling towers and aeration processes and desiccant drying [1-5]. Many studies have gone through characterising the heat and mass transfer in such configurations [6-9]. The mechanisms of heat and mass transfer are similar and analogical. Therefore, in some special cases where, either the heat or mass transfer data are not reliable or may not be available, the heat and mass transfer analogy can be used to determine the missing or unreliable set of data.
    [Show full text]
  • High-Lewis Number Premixed Flame Instabilities
    High-Lewis Number Premixed Flame Instabilities A Thesis Submitted to the Faculty of Drexel University by Yi Ma in partial fulfillment of the requirements for the degree of Doctor of Philosophy September 2009 ii Acknowledgements It has been a great blessing being a graduate student under the supervision of Dr. Howard Pearlman. My appreciation of his generosity, tolerance, support and guidance through my long journey of Ph.D. study is beyond words. He is not only a very knowledgeable scholar but also a very kind and inspiring person. What I have learned from him will benefit me for my whole life. In addition, I would like to thank my committee members, Dr. Nicholas P. Cernansky, Dr. David L. Miller, Dr. Alan Lau and Dr. Smitesh Bakrania for their time and energy to review this dissertation. I must acknowledge a few of my colleagues in Hess Lab, Mike Foster, Rodney Johnson, Ashutosh Gupta, and Seul H. Park, among others, and my best friend Meline Zeng for their friendship and assistance for all these years. I would also like to thank two former undergraduate students, Christopher Ricciuti and Brian Fellon, for their contribution to the nano-size catalytic platinum particle project. Finally, the staff at the Machine Shop, namely Mark Shirber and Richard Miller have been very helpful modifying my experimental facility during various stages of this research. I would like to give special thanks to my husband David B. Lenhert for his endless love, tremendous help and encouragement during my Ph.D. study. Finally, my thanks go to my parents Zhong Hejiao and Ma Qingyun for their unconditional love, care and support over the years.
    [Show full text]
  • The Effect of Induced Magnetic Field and Convective Boundary Condition
    Propulsion and Power Research 2016;5(2):164–175 HOSTED BY http://ppr.buaa.edu.cn/ Propulsion and Power Research www.sciencedirect.com ORIGINAL ARTICLE The effect of induced magnetic field and convective boundary condition on MHD stagnation point flow and heat transfer of upper-convected Maxwell fluid in the presence of nanoparticle past a stretching sheet Wubshet Ibrahimn Department of Mathematics, Ambo University, P.O. Box 19, Ambo, Ethiopia Received 5 February 2015; accepted 17 July 2015 Available online 30 May 2016 KEYWORDS Abstract The present study examines the effect of induced magnetic field and convective fl fl boundary condition on magnetohydrodynamic (MHD) stagnation point ow and heat transfer due Nano uid; fl Stagnation point flow; to upper-convected Maxwell uid over a stretching sheet in the presence of nanoparticles. fi Heat transfer; Boundary layer theory is used to simplify the equation of motion, induced magnetic eld, energy Convective boundary and concentration which results in four coupled non-linear ordinary differential equations. The condition; study takes into account the effect of Brownian motion and thermophoresis parameters. The Induced magnetic governing equations and their associated boundary conditions are initially cast into dimensionless field; form by similarity variables. The resulting system of equations is then solved numerically using Upper-convected Max- fourth order Runge-Kutta-Fehlberg method along with shooting technique. The solution for the well fluid governing equations depends on parameters such as, magnetic, velocity ratio parameter B,Biot number Bi, Prandtl number Pr, Lewis number Le, Brownian motion Nb, reciprocal of magnetic Prandtl number A, the thermophoresis parameter Nt, and Maxwell parameter β.
    [Show full text]
  • On Dimensionless Numbers
    chemical engineering research and design 8 6 (2008) 835–868 Contents lists available at ScienceDirect Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd Review On dimensionless numbers M.C. Ruzicka ∗ Department of Multiphase Reactors, Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Rozvojova 135, 16502 Prague, Czech Republic This contribution is dedicated to Kamil Admiral´ Wichterle, a professor of chemical engineering, who admitted to feel a bit lost in the jungle of the dimensionless numbers, in our seminar at “Za Plıhalovic´ ohradou” abstract The goal is to provide a little review on dimensionless numbers, commonly encountered in chemical engineering. Both their sources are considered: dimensional analysis and scaling of governing equations with boundary con- ditions. The numbers produced by scaling of equation are presented for transport of momentum, heat and mass. Momentum transport is considered in both single-phase and multi-phase flows. The numbers obtained are assigned the physical meaning, and their mutual relations are highlighted. Certain drawbacks of building correlations based on dimensionless numbers are pointed out. © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Dimensionless numbers; Dimensional analysis; Scaling of equations; Scaling of boundary conditions; Single-phase flow; Multi-phase flow; Correlations Contents 1. Introduction .................................................................................................................
    [Show full text]
  • The Numerical Simulation of Double-Diffusive Laminar Mixed Convection Flow in a Lid-Driven Porous Cavity
    WSEAS TRANSACTIONS on HEAT and MASS TRANSFER Shashak Misra, A. Satheesh, C. G. Mohan, P. Padmanathan The Numerical Simulation of Double-Diffusive Laminar Mixed Convection Flow in a Lid-Driven Porous Cavity SHASHAK MISRA, A. SATHEESH*, C.G. MOHAN AND P. PADMANATHAN School of Mechanical and Building Sciences VIT University, Vellore – 632014 Tamil Nadu, INDIA Corresponding author: Tel.: +91-416-224-2164; mob: +91-959-787-2825; [email protected] Abstract: - This paper presents the numerical investigation of double-diffusive mixed convective flow in an impermeable enclosed cavity. The uniform temperature and concentration is imposed along the vertical walls and the horizontal walls are considered as insulation. The flow behavior is analyzed when the top wall moves left side at a constant velocity (Uo) and the other walls kept remains stationary. Transport equations are solved using finite volume technique. The pressure and velocity terms are coupled by SIMPLE algorithm. The third order deferred Quadratic Upwind Interpolation for Convection Kinematics (QUICK) scheme and second order central difference scheme are applied at the inner and boarder nodal points, respectively. The present numerical simulation is compared with the reported literature and found that are in good agreement. The heat and mass transfer results are presented in the form of iso-temperature and concentration. The Lewis number (Le) and aspect ratio are varied over a wide range to analyse non-dimensional horizontal (U) and vertical velocities (V), stream line contours, temperature and concentration gradients. The present analysis is carried out at constant Prandtl (Pr=0.7), Richardson (Ri=1.0), Darcy (Da=1.0) and Reynolds (Re=100) numbers.
    [Show full text]
  • Dimensional Analysis and Modeling
    cen72367_ch07.qxd 10/29/04 2:27 PM Page 269 CHAPTER DIMENSIONAL ANALYSIS 7 AND MODELING n this chapter, we first review the concepts of dimensions and units. We then review the fundamental principle of dimensional homogeneity, and OBJECTIVES Ishow how it is applied to equations in order to nondimensionalize them When you finish reading this chapter, you and to identify dimensionless groups. We discuss the concept of similarity should be able to between a model and a prototype. We also describe a powerful tool for engi- ■ Develop a better understanding neers and scientists called dimensional analysis, in which the combination of dimensions, units, and of dimensional variables, nondimensional variables, and dimensional con- dimensional homogeneity of equations stants into nondimensional parameters reduces the number of necessary ■ Understand the numerous independent parameters in a problem. We present a step-by-step method for benefits of dimensional analysis obtaining these nondimensional parameters, called the method of repeating ■ Know how to use the method of variables, which is based solely on the dimensions of the variables and con- repeating variables to identify stants. Finally, we apply this technique to several practical problems to illus- nondimensional parameters trate both its utility and its limitations. ■ Understand the concept of dynamic similarity and how to apply it to experimental modeling 269 cen72367_ch07.qxd 10/29/04 2:27 PM Page 270 270 FLUID MECHANICS Length 7–1 ■ DIMENSIONS AND UNITS 3.2 cm A dimension is a measure of a physical quantity (without numerical val- ues), while a unit is a way to assign a number to that dimension.
    [Show full text]
  • Numerical Solutions of an Unsteady 2-D Incompressible Flow with Heat
    Numerical solutions of an unsteady 2-d incompressible flow with heat and mass transfer at low, moderate, and high Reynolds numbers V. Ambethkara,∗, Durgesh Kushawahaa aDepartment of Mathematics, Faculty of Mathematical Sciences, University of Delhi Delhi, 110007, India Abstract In this paper, we have proposed a modified Marker-And-Cell (MAC) method to investigate the problem of an unsteady 2-D incompressible flow with heat and mass transfer at low, moderate, and high Reynolds numbers with no-slip and slip boundary conditions. We have used this method to solve the governing equations along with the boundary conditions and thereby to compute the flow variables, viz. u-velocity, v-velocity, P, T, and C. We have used the staggered grid approach of this method to discretize the governing equations of the problem. A modified MAC algorithm was proposed and used to compute the numerical solutions of the flow variables for Reynolds numbers Re = 10, 500, and 50, 000 in consonance with low, moderate, and high Reynolds numbers. We have also used appropriate Prandtl (Pr) and Schmidt (Sc) numbers in consistence with relevancy of the physical problem considered. We have executed this modified MAC algorithm with the aid of a computer program developed and run in C compiler. We have also computed numerical solutions of local Nusselt (Nu) and Sherwood (Sh) numbers along the horizontal line through the geometric center at low, moderate, and high Reynolds numbers for fixed Pr = 6.62 and Sc = 340 for two grid systems at time t = 0.0001s. Our numerical solutions for u and v velocities along the vertical and horizontal line through the geometric center of the square cavity for Re = 100 has been compared with benchmark solutions available in the literature and it has been found that they are in good agreement.
    [Show full text]
  • Multiphase Flow Handbook in the Liquid Static Pressure
    12 Multiphase Interactions 12.1 Cavitation ..........................................................................12-1 Cavitation Nuclei and Bubble Growth ● Collapse of Cavitation Bubbles and Erosion ● Acoustic Cavitation ● Hydrodynamic Bubble Cavitation ● Sheet Cavitation ● Vortex Cavitation ● ● G.F. Oweis Cavitation in Hydraulic Machinery Cavitation in Biomedical Applications ● Other Occurrences and Applications of S.L. Ceccio Cavitation University of Michigan 12.2 Bubble Physics ................................................................12-17 Y. Matsumoto Single Spherical Bubble Dynamics ● Bubble Deformation and University of Tokyo Fragmentation Due to the Collapse ● Bubbles Cloud Dynamics 12.3 Droplet Breakup, Coalescence, and Wall Impact ..........12-35 C. Tropea Droplet Breakup Mechanisms ● Binary Drop Collisions ● I.V. Roisman Drop-Wall Impact Technical University of Darmstadt 12.4 Particle Interactions ........................................................12-51 Particle–Particle Collision ● Particle–Wall Interaction ● Erosion Y. Tsuji 12.5 Particle and Droplet Dispersion in Turbulent Flows ....12-81 Osaka University Introduction ● Time Scales ● Particle Dispersion in Turbulent R. Lyczkowski Shear Flows ● Body Force Effects on Particle Dispersion ● Two- Way Coupling Effects on Particle Dispersion Argonne National Laboratory 12.6 Turbulence Modulation by Particles ..............................12-86 T.R. Troutt Introduction ● Mechanisms of Turbulence Modification ● Washington State University Analytical Approaches ● Methods
    [Show full text]
  • Gyrotactic Micro-Organism Flow of Maxwell Nanofluid Between Two
    www.nature.com/scientificreports OPEN Gyrotactic micro‑organism fow of Maxwell nanofuid between two parallel plates Yun‑Jie Xu1, Muhammad Bilal2, Qasem Al‑Mdallal3*, Muhammad Altaf Khan4 & Taseer Muhammad5 The present study explores incompressible, steady power law nanoliquid comprising gyrotactic microorganisms fow across parallel plates with energy transfer. In which only one plate is moving concerning another at a time. Nonlinear partial diferential equations have been used to model the problem. Using Liao’s transformation, the framework of PDEs is simplifed to a system of Ordinary Diferential Equations (ODEs). The problem is numerically solved using the parametric continuation method (PCM). The obtained results are compared to the boundary value solver (bvp4c) method for validity reasons. It has been observed that both the results are in best settlement with each other. The temperature, velocity, concentration and microorganism profle trend versus several physical constraints are presented graphically and briefy discussed. The velocity profle shows positive response versus the rising values of buoyancy convection parameters. While the velocity reduces with the increasing efect of magnetic feld, because magnetic impact generates Lorentz force, which reduces the fuid velocity. Symbols Uw Stretching velocity g Acceleration T∞ Free stream temperature ⇀ ⇀ j × B Lorenz force σe Electrical conductivity B0 Magnetic feld cp Specifc heat capacity Dn Difusivity of microorganism K Termal difusion ratio µ2 Magnetic permeability Pe Peclet number 1 Buoyancy convection coefcient due to temperature 3 Buoyancy convection coefcient due to microorganism Cf Skin friction Shx Sherwood number qw Heat fux qn Motile microorganism fux A Matrix coefcient ∗ h Gap between two plates T Fluid temperature β Heat source/sink parameter DT Termophoresis coefcient DB Brownian motion 1School of Engineering, Huzhou University, Huzhou 313000, People’s Republic of China.
    [Show full text]