This paper derives the equations that describe the interaction forcesbetween two identical cylinders
spinningaround their stationary and parallel axes in a fluid that isassumed to be in-viscous, steady, in-vortical,
and in-compressible. The paper starts by deriving the velocity field from Laplace equation,governing this
problem,and the system boundary conditions. It then determines the pressure field from the velocity field using
Bernoulli equation. Finally, the paper integrates the pressure around either cylinder-surface to find the force
acting on its axis.All equations and derivationsprovided in this paper are exact solutions. No numerical
analysesor approximations are used.The paper finds that such identical cylinders repel or attract each other in
inverse relation with separation between their axes, according to similar or opposite direction of rotation,
respectively.
The Interaction Forces between Two non-Identical Cylinders Spinning around th...iosrjce
This paper derives the equations that describe the interaction forcesbetween two non-identical
cylinders spinningaround their stationary and parallel axes in a fluid that isassumed to be in-viscous, steady, invortical,
and in-compressible. The paper starts by deriving the velocity field from Laplace equation,governing
this problem,and the system boundary conditions. It then determines the pressure field from the velocity field
using Bernoulli equation. Finally, the paper integrates the pressure around both cylinder-surface to find the
force acting on their axes.All equations and derivationsprovided in this paper are exact solutions. No numerical
analysesor approximations are used.The paper finds that such cylinders repel or attract each other in inverse
relation with separation between their axes, according to similar or opposite direction of rotation, respectively.
Heat Transfer in the flow of a Non-Newtonian second-order fluid over an enclo...IJMERJOURNAL
ABSTRACT : The problem of the heat transfer in the flow of an incompressible non-Newtonian second-order fluid over an enclosed torsionally oscillating discs in the presence of the magnetic field has been discussed. The obtained differential equations are highly non-linear and contain upto fifth order derivatives of the flow and energy functions. Hence exact or numerical solutions of the differential equations are not possible subject to the given natural boundary conditions; therefore the regular perturbation technique is applied. The flow functions 퐻, 퐺, 퐿 and 푀 are expanded in the powers of the amplitude (taken small) of the oscillations. The behaviour of the temperature distribution at different values of Reynolds number, phase difference, magnetic field and second-order parameters has been studied and shown graphically. The results obtained are compared with those for the infinite torsionally oscillating discs by taking the Reynolds number of out-flow 푅푚 and circulatory flow 푅퐿 equals to zero. Nusselt number at oscillating and stator disc has also been calculated and its behaviour is represented graphically.
Study of different flows over typical bodies by FluentRajibul Alam
This document summarizes numerical simulations of inviscid and viscous flows over wedges and flat plates. For inviscid flow over a wedge, the governing equations are presented and solved analytically and numerically for Mach numbers of 3 and 5. Numerical solutions match analytical results closely after mesh refinement. For viscous flow over a flat plate, the boundary layer equations are derived and Blasius' analytical solution is summarized, providing the velocity profile as a function of similarity variable.
Effect of an Inclined Magnetic Field on Peristaltic Flow of Williamson Fluid ...QUESTJOURNAL
ABSTRACT: This paper deals with the influence ofinclined magnetic field on peristaltic flow of an incompressible Williamson fluid in an inclined channel with heat and mass transfer. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
1) A hollow cylinder filled with fluid is accelerated along its axis. A ball with the same density as the fluid is initially at rest near the front lid.
2) The author sets up equations to model the small changes in fluid density under pressure and acceleration. This leads to a linear increase in density from front to rear.
3) Forces on the ball are calculated, showing it will oscillate with damped motion toward the rear lid due to added mass forces from surrounding fluid. However, viscous forces will prevent it from reaching the rear lid.
I am Titus B. I am a Mechanical Engineering Assignment Expert at mechanicalengineeringassignmenthelp.com. I hold a Ph.D. in Mechatronics Engineering Alcorn State University USA. I have been helping students with their assignments for the past 11 years. I solve assignments related to Mechanical Engineering.
Visit mechanicalengineeringassignmenthelp.com or email info@mechanicalengineeringassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Mechanical Engineering.
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...IOSRJM
This paper deals with the influence of magnetic field on peristaltic flow of an incompressible Williamson fluid in a symmetric channel with heat and mass transfer. Convective conditions of heat and mass transfer are employed. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
The Interaction Forces between Two non-Identical Cylinders Spinning around th...iosrjce
This paper derives the equations that describe the interaction forcesbetween two non-identical
cylinders spinningaround their stationary and parallel axes in a fluid that isassumed to be in-viscous, steady, invortical,
and in-compressible. The paper starts by deriving the velocity field from Laplace equation,governing
this problem,and the system boundary conditions. It then determines the pressure field from the velocity field
using Bernoulli equation. Finally, the paper integrates the pressure around both cylinder-surface to find the
force acting on their axes.All equations and derivationsprovided in this paper are exact solutions. No numerical
analysesor approximations are used.The paper finds that such cylinders repel or attract each other in inverse
relation with separation between their axes, according to similar or opposite direction of rotation, respectively.
Heat Transfer in the flow of a Non-Newtonian second-order fluid over an enclo...IJMERJOURNAL
ABSTRACT : The problem of the heat transfer in the flow of an incompressible non-Newtonian second-order fluid over an enclosed torsionally oscillating discs in the presence of the magnetic field has been discussed. The obtained differential equations are highly non-linear and contain upto fifth order derivatives of the flow and energy functions. Hence exact or numerical solutions of the differential equations are not possible subject to the given natural boundary conditions; therefore the regular perturbation technique is applied. The flow functions 퐻, 퐺, 퐿 and 푀 are expanded in the powers of the amplitude (taken small) of the oscillations. The behaviour of the temperature distribution at different values of Reynolds number, phase difference, magnetic field and second-order parameters has been studied and shown graphically. The results obtained are compared with those for the infinite torsionally oscillating discs by taking the Reynolds number of out-flow 푅푚 and circulatory flow 푅퐿 equals to zero. Nusselt number at oscillating and stator disc has also been calculated and its behaviour is represented graphically.
Study of different flows over typical bodies by FluentRajibul Alam
This document summarizes numerical simulations of inviscid and viscous flows over wedges and flat plates. For inviscid flow over a wedge, the governing equations are presented and solved analytically and numerically for Mach numbers of 3 and 5. Numerical solutions match analytical results closely after mesh refinement. For viscous flow over a flat plate, the boundary layer equations are derived and Blasius' analytical solution is summarized, providing the velocity profile as a function of similarity variable.
Effect of an Inclined Magnetic Field on Peristaltic Flow of Williamson Fluid ...QUESTJOURNAL
ABSTRACT: This paper deals with the influence ofinclined magnetic field on peristaltic flow of an incompressible Williamson fluid in an inclined channel with heat and mass transfer. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
1) A hollow cylinder filled with fluid is accelerated along its axis. A ball with the same density as the fluid is initially at rest near the front lid.
2) The author sets up equations to model the small changes in fluid density under pressure and acceleration. This leads to a linear increase in density from front to rear.
3) Forces on the ball are calculated, showing it will oscillate with damped motion toward the rear lid due to added mass forces from surrounding fluid. However, viscous forces will prevent it from reaching the rear lid.
I am Titus B. I am a Mechanical Engineering Assignment Expert at mechanicalengineeringassignmenthelp.com. I hold a Ph.D. in Mechatronics Engineering Alcorn State University USA. I have been helping students with their assignments for the past 11 years. I solve assignments related to Mechanical Engineering.
Visit mechanicalengineeringassignmenthelp.com or email info@mechanicalengineeringassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Mechanical Engineering.
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...IOSRJM
This paper deals with the influence of magnetic field on peristaltic flow of an incompressible Williamson fluid in a symmetric channel with heat and mass transfer. Convective conditions of heat and mass transfer are employed. Viscous dissipation and Joule heating are taken into consideration.Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.
This document provides an introduction to fluid mechanics. It discusses how fluids are essential to life and have shaped history. It then provides brief biographies of some important figures in the history of fluid mechanics, such as Archimedes, Newton, Euler, Stokes, and Reynolds. It also discusses the significance of fluid mechanics across many fields including weather, vehicles, physiology, sports, and engineering. The document concludes by outlining the key components of analytical fluid dynamics, experimental fluid dynamics, and computational fluid dynamics.
ANALYSIS OF VORTEX INDUCED VIBRATION USING IFSIJCI JOURNAL
Interaction of fluid structure (IFS) is one of the upcoming field in calculation and simulation of multiphysics problems. IFS play an important role in calculating offshore structures deformations caused by the vortex induced loads. The complexity interaction nature of fluid around the solid geometries pose the difficulties in the analysis, but IFS analysis technique overshadow the challenges. In this paper, Analysis is done by considering a cylindrical member which is similar to the part of offshore platform. The IFS analysis is done by using the commercial package ANSYS 14.0. The Vortex induced loads simulation with IFS is purely a mesh dependent, for that we have to simulate many problems for getting optimum grid size. Computational Fluid Dynamics (CFD) analysis of a two dimensional model have been done and the obtained results were validated with the literature findings. CFD analysis is performed on the extruded version of the two dimensional mesh and the results were compared with the previously obtained two dimensional results. Preliminary IFS analysis is done by coupling the structural and fluid solvers together at smaller time steps and the dynamic response of the structural member to the periodically varying Vortex induced vibrations (VIV) loads were observed and studied.
1) Bernoulli's equation states that for steady flow in an inviscid fluid, the total mechanical energy per unit weight remains constant along a streamline. This includes potential energy, kinetic energy, and pressure energy.
2) It summarizes that an applied force equals the rate of change of mechanical energy, and derives the one-dimensional Euler and Bernoulli equations from conservation of energy and momentum principles.
3) The document provides examples of applying Bernoulli's equation to problems involving pipe flow and nozzle discharge to determine quantities like jet velocity and suction pressure.
This document provides an introduction to fluid mechanics fundamentals, including:
- The four primary dimensions in fluid mechanics are mass, length, time, and force. All other variables can be expressed in terms of these.
- Liquids can be treated as incompressible for most fluid mechanics problems since pressure changes are typically not large enough to cause changes in density.
- Viscosity describes a fluid's resistance to shear forces or layer sliding, and can result in either laminar (smooth) flow or turbulent (chaotic) flow.
The document presents a new approach for dynamic analysis of parallel manipulators based on the principle of virtual work. It illustrates the approach using a simple 4-bar linkage example, calculating the inertial forces and moments, virtual displacements, and input torque. It then generalizes the approach for dynamic analysis of a 6 degree-of-freedom parallel manipulator like a Gough-Stewart platform. The approach leads to faster computation than traditional Newton-Euler methods by not requiring calculation of constraint forces between links.
Vector calculus in Robotics EngineeringNaveensing87
Vector calculus concepts such as angular velocity vectors, linear and rotational velocities, and Jacobians are important applications in robotics engineering. Vector calculus is used to represent points and vectors in robotic mechanisms with three coordinates and their magnitudes and directions. It allows determining the velocity of parts in a robot by propagating velocities between links using rotational and angular velocities. Jacobians relate Cartesian velocities to joint velocities in robots and are used for forward and inverse kinematics analysis to calculate robot positions and orientations.
The document provides an overview of theory of machines and mechanisms. It discusses topics like kinematics, degrees of freedom, different types of linkages including four-bar linkages and slider crank mechanisms. It also describes inversions of slider crank mechanisms, quick return mechanisms, and analyzes the motion of components in mechanisms using concepts like transmission angle and quick return ratio. Examples of various mechanisms used in applications like steam engines and pumps are also discussed.
Leonhard Euler developed the theory of rigid body mechanics between 1738 and 1775 through several seminal works. In his 1752 work "Discovery of a new principle in mechanics", Euler introduced the concept of an instantaneous rotation axis and derived equations now known as Euler's equations that describe the rotation of a rigid body. Euler determined that a rigid body in motion can be described as translating plus rotating about an axis through its center of gravity. This established the foundation for describing the general motion of rigid bodies in mechanics.
This document presents explicit analytical solutions for pressure across oblique shock and expansion waves in supersonic flow. It begins by introducing the need for explicit pressure-deflection solutions in solving aerodynamic problems. It then presents:
1) Exact explicit solutions for pressure coefficient and ratio across weak and strong oblique shock waves as functions of deflection angle.
2) Third-order accurate explicit unitary solutions for pressure coefficient and ratio across oblique shocks and expansions as functions of deflection angle.
3) Numerical validation showing good agreement of the new explicit solutions with exact solutions for a range of Mach numbers and deflection angles.
The document discusses pressure drop in different types of flow reactors. It defines pressure drop as the difference in pressure between two points in a fluid network. Pressure drop occurs due to frictional forces and fouling. The general steps to calculate pressure drop are presented, along with the specific equations to calculate pressure drop in packed bed reactors, plug flow reactors, and continuous stirred tank reactors. An example calculation is shown to demonstrate how to determine the pressure drop across a packed bed reactor.
This document provides a table of contents for topics in fluid dynamics. It begins with mathematical notations for vectors, tensors, divergence, gradient, and other vector calculus topics. It then outlines the basic laws of fluid dynamics, such as conservation of mass and momentum. Later sections cover specific fluid dynamics concepts like boundary layers, compressible flow, shock waves, and potential flow. The document provides the framework for equations and analyses of fluid flows.
Lagrange's theorem describes fluid motion using a Lagrangian description that tracks individual fluid particles over time rather than describing the fluid properties at fixed spatial locations like the Eulerian description. The Lagrangian description follows Newton's laws of motion for individual particles, making it easier to apply concepts from solid mechanics. However, the Eulerian description is more commonly used in fluid mechanics problems because it is not practical to track every particle in complex flows. Lagrange's equations, derived using the calculus of variations, provide an alternative formulation of classical mechanics that has advantages over Newtonian mechanics such as applying to any coordinate system.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document discusses transport phenomena and provides definitions and examples of key concepts in vector and tensor analysis used to describe transport phenomena. It defines transport phenomena as dealing with the movement of physical quantities in chemical or mechanical processes. There are three main types of transport: momentum, energy, and mass transport. Vector and tensor quantities like velocity, stress, and strain gradient are used to describe transport phenomena. Tensors have a magnitude and direction(s) and transform under coordinate system rotations. The document provides examples of scalar, vector, and tensor notation and the Kronecker delta, alternating unit tensor, and mathematical operations on vectors like addition, dot product, and cross product.
Radiation Effects on MHD Free Convective Rotating Flow with Hall EffectsIJERA Editor
In this paper, we have studied the unsteady an incompressible MHD rotating free convection flow of Viscoelastic fluid through a porous medium with simultaneous heat and mass transfer near an infinite vertical oscillating porous plate under the influence of uniform transverse magnetic field. The governing equations of the flow field are solved by a regular perturbation method for small elastic parameter. The expressions for the velocity, temperature, concentration have been derived analytically and also its behaviour is computationally discussed with reference to different flow parameters with the help of graphs. The skin friction, the Nusselt number and the Sherwood number are also obtained and their behaviour discussed.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and TechnologyIJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
1) The document analyzes the interaction forces between two non-identical cylinders spinning around their stationary and parallel axes in an ideal fluid.
2) It derives the exact equations for the velocity field of the fluid using Laplace's equation and the boundary conditions.
3) It then determines the pressure field from the velocity field using Bernoulli's equation and integrates the pressure around the cylinder surfaces to find the forces acting on their axes.
1) The document analyzes the interaction forces between two non-identical cylinders spinning around their stationary and parallel axes in an ideal fluid.
2) It derives the exact equations for the velocity field of the fluid using Laplace's equation and the boundary conditions.
3) It then determines the pressure field from the velocity field using Bernoulli's equation and integrates the pressure around the cylinder surfaces to find the forces acting on their axes.
4) It finds that the cylinders will repel or attract each other in inverse relation to their separation distance, depending on whether their directions of rotation are similar or opposite.
Influence of MHD on Unsteady Helical Flows of Generalized Oldoyd-B Fluid betw...IJERA Editor
Considering a fractional derivative model for unsteady magetohydrodynamic (MHD)helical flows of an
Oldoyd-B fluid in concentric cylinders and circular cylinder are studied by using finite Hankel and Laplace
transforms .The solution of velocity fields and the shear stresses of unsteady magetohydrodynamic
(MHD)helical flows of an Oldoyd-B fluid in an annular pipe are obtained under series form in terms of
Mittag –leffler function,satisfy all imposed initial and boundary condition , Finally the influence of model
parameters on the velocity and shear stress are analyzed by graphical illustrations.
Methods to determine pressure drop in an evaporator or a condenserTony Yen
This articles aims to explain how one can relatively easily calculate the pressure drop within a condenser or an evaporator, where two-phase flow occurs and the Navier-Stokes equation becomes very tedious.
DERIVATION OF THE MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMIN...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
This document provides an introduction to fluid mechanics. It discusses how fluids are essential to life and have shaped history. It then provides brief biographies of some important figures in the history of fluid mechanics, such as Archimedes, Newton, Euler, Stokes, and Reynolds. It also discusses the significance of fluid mechanics across many fields including weather, vehicles, physiology, sports, and engineering. The document concludes by outlining the key components of analytical fluid dynamics, experimental fluid dynamics, and computational fluid dynamics.
ANALYSIS OF VORTEX INDUCED VIBRATION USING IFSIJCI JOURNAL
Interaction of fluid structure (IFS) is one of the upcoming field in calculation and simulation of multiphysics problems. IFS play an important role in calculating offshore structures deformations caused by the vortex induced loads. The complexity interaction nature of fluid around the solid geometries pose the difficulties in the analysis, but IFS analysis technique overshadow the challenges. In this paper, Analysis is done by considering a cylindrical member which is similar to the part of offshore platform. The IFS analysis is done by using the commercial package ANSYS 14.0. The Vortex induced loads simulation with IFS is purely a mesh dependent, for that we have to simulate many problems for getting optimum grid size. Computational Fluid Dynamics (CFD) analysis of a two dimensional model have been done and the obtained results were validated with the literature findings. CFD analysis is performed on the extruded version of the two dimensional mesh and the results were compared with the previously obtained two dimensional results. Preliminary IFS analysis is done by coupling the structural and fluid solvers together at smaller time steps and the dynamic response of the structural member to the periodically varying Vortex induced vibrations (VIV) loads were observed and studied.
1) Bernoulli's equation states that for steady flow in an inviscid fluid, the total mechanical energy per unit weight remains constant along a streamline. This includes potential energy, kinetic energy, and pressure energy.
2) It summarizes that an applied force equals the rate of change of mechanical energy, and derives the one-dimensional Euler and Bernoulli equations from conservation of energy and momentum principles.
3) The document provides examples of applying Bernoulli's equation to problems involving pipe flow and nozzle discharge to determine quantities like jet velocity and suction pressure.
This document provides an introduction to fluid mechanics fundamentals, including:
- The four primary dimensions in fluid mechanics are mass, length, time, and force. All other variables can be expressed in terms of these.
- Liquids can be treated as incompressible for most fluid mechanics problems since pressure changes are typically not large enough to cause changes in density.
- Viscosity describes a fluid's resistance to shear forces or layer sliding, and can result in either laminar (smooth) flow or turbulent (chaotic) flow.
The document presents a new approach for dynamic analysis of parallel manipulators based on the principle of virtual work. It illustrates the approach using a simple 4-bar linkage example, calculating the inertial forces and moments, virtual displacements, and input torque. It then generalizes the approach for dynamic analysis of a 6 degree-of-freedom parallel manipulator like a Gough-Stewart platform. The approach leads to faster computation than traditional Newton-Euler methods by not requiring calculation of constraint forces between links.
Vector calculus in Robotics EngineeringNaveensing87
Vector calculus concepts such as angular velocity vectors, linear and rotational velocities, and Jacobians are important applications in robotics engineering. Vector calculus is used to represent points and vectors in robotic mechanisms with three coordinates and their magnitudes and directions. It allows determining the velocity of parts in a robot by propagating velocities between links using rotational and angular velocities. Jacobians relate Cartesian velocities to joint velocities in robots and are used for forward and inverse kinematics analysis to calculate robot positions and orientations.
The document provides an overview of theory of machines and mechanisms. It discusses topics like kinematics, degrees of freedom, different types of linkages including four-bar linkages and slider crank mechanisms. It also describes inversions of slider crank mechanisms, quick return mechanisms, and analyzes the motion of components in mechanisms using concepts like transmission angle and quick return ratio. Examples of various mechanisms used in applications like steam engines and pumps are also discussed.
Leonhard Euler developed the theory of rigid body mechanics between 1738 and 1775 through several seminal works. In his 1752 work "Discovery of a new principle in mechanics", Euler introduced the concept of an instantaneous rotation axis and derived equations now known as Euler's equations that describe the rotation of a rigid body. Euler determined that a rigid body in motion can be described as translating plus rotating about an axis through its center of gravity. This established the foundation for describing the general motion of rigid bodies in mechanics.
This document presents explicit analytical solutions for pressure across oblique shock and expansion waves in supersonic flow. It begins by introducing the need for explicit pressure-deflection solutions in solving aerodynamic problems. It then presents:
1) Exact explicit solutions for pressure coefficient and ratio across weak and strong oblique shock waves as functions of deflection angle.
2) Third-order accurate explicit unitary solutions for pressure coefficient and ratio across oblique shocks and expansions as functions of deflection angle.
3) Numerical validation showing good agreement of the new explicit solutions with exact solutions for a range of Mach numbers and deflection angles.
The document discusses pressure drop in different types of flow reactors. It defines pressure drop as the difference in pressure between two points in a fluid network. Pressure drop occurs due to frictional forces and fouling. The general steps to calculate pressure drop are presented, along with the specific equations to calculate pressure drop in packed bed reactors, plug flow reactors, and continuous stirred tank reactors. An example calculation is shown to demonstrate how to determine the pressure drop across a packed bed reactor.
This document provides a table of contents for topics in fluid dynamics. It begins with mathematical notations for vectors, tensors, divergence, gradient, and other vector calculus topics. It then outlines the basic laws of fluid dynamics, such as conservation of mass and momentum. Later sections cover specific fluid dynamics concepts like boundary layers, compressible flow, shock waves, and potential flow. The document provides the framework for equations and analyses of fluid flows.
Lagrange's theorem describes fluid motion using a Lagrangian description that tracks individual fluid particles over time rather than describing the fluid properties at fixed spatial locations like the Eulerian description. The Lagrangian description follows Newton's laws of motion for individual particles, making it easier to apply concepts from solid mechanics. However, the Eulerian description is more commonly used in fluid mechanics problems because it is not practical to track every particle in complex flows. Lagrange's equations, derived using the calculus of variations, provide an alternative formulation of classical mechanics that has advantages over Newtonian mechanics such as applying to any coordinate system.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document discusses transport phenomena and provides definitions and examples of key concepts in vector and tensor analysis used to describe transport phenomena. It defines transport phenomena as dealing with the movement of physical quantities in chemical or mechanical processes. There are three main types of transport: momentum, energy, and mass transport. Vector and tensor quantities like velocity, stress, and strain gradient are used to describe transport phenomena. Tensors have a magnitude and direction(s) and transform under coordinate system rotations. The document provides examples of scalar, vector, and tensor notation and the Kronecker delta, alternating unit tensor, and mathematical operations on vectors like addition, dot product, and cross product.
Radiation Effects on MHD Free Convective Rotating Flow with Hall EffectsIJERA Editor
In this paper, we have studied the unsteady an incompressible MHD rotating free convection flow of Viscoelastic fluid through a porous medium with simultaneous heat and mass transfer near an infinite vertical oscillating porous plate under the influence of uniform transverse magnetic field. The governing equations of the flow field are solved by a regular perturbation method for small elastic parameter. The expressions for the velocity, temperature, concentration have been derived analytically and also its behaviour is computationally discussed with reference to different flow parameters with the help of graphs. The skin friction, the Nusselt number and the Sherwood number are also obtained and their behaviour discussed.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and TechnologyIJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
1) The document analyzes the interaction forces between two non-identical cylinders spinning around their stationary and parallel axes in an ideal fluid.
2) It derives the exact equations for the velocity field of the fluid using Laplace's equation and the boundary conditions.
3) It then determines the pressure field from the velocity field using Bernoulli's equation and integrates the pressure around the cylinder surfaces to find the forces acting on their axes.
1) The document analyzes the interaction forces between two non-identical cylinders spinning around their stationary and parallel axes in an ideal fluid.
2) It derives the exact equations for the velocity field of the fluid using Laplace's equation and the boundary conditions.
3) It then determines the pressure field from the velocity field using Bernoulli's equation and integrates the pressure around the cylinder surfaces to find the forces acting on their axes.
4) It finds that the cylinders will repel or attract each other in inverse relation to their separation distance, depending on whether their directions of rotation are similar or opposite.
Influence of MHD on Unsteady Helical Flows of Generalized Oldoyd-B Fluid betw...IJERA Editor
Considering a fractional derivative model for unsteady magetohydrodynamic (MHD)helical flows of an
Oldoyd-B fluid in concentric cylinders and circular cylinder are studied by using finite Hankel and Laplace
transforms .The solution of velocity fields and the shear stresses of unsteady magetohydrodynamic
(MHD)helical flows of an Oldoyd-B fluid in an annular pipe are obtained under series form in terms of
Mittag –leffler function,satisfy all imposed initial and boundary condition , Finally the influence of model
parameters on the velocity and shear stress are analyzed by graphical illustrations.
Methods to determine pressure drop in an evaporator or a condenserTony Yen
This articles aims to explain how one can relatively easily calculate the pressure drop within a condenser or an evaporator, where two-phase flow occurs and the Navier-Stokes equation becomes very tedious.
DERIVATION OF THE MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMIN...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
The document discusses simulating the dynamics of a two-dimensional aerofoil with a trailing edge flap. It aims to investigate factors governing the aerofoil's response and relate these to analytical techniques. It derives potential and kinetic energy equations for the aerofoil system using a Lagrangian approach. This allows obtaining equations of motion in matrix form representing the aerofoil's vibrating system dynamics under aerodynamic and external disturbance forces.
This document discusses two-dimensional ideal fluid flow. It begins by defining an ideal fluid as having no viscosity, compressibility, or surface tension. The continuity equation is then derived, stating that the net flow out of a control volume must equal the change in mass within the volume. Euler's equations are also derived, forming a set of partial differential equations that can be solved to determine pressure and velocity fields. Bernoulli's equation is obtained by integrating the Euler equations, relating total pressure, velocity, and elevation. The concepts of rotational and irrotational flow are introduced, with irrotational flow defined as having zero rotation of any fluid element.
The document describes a project to design, simulate, and test a water rocket. A control volume analysis and flight model from previous research was used to simulate the rocket's performance in MATLAB and predict a maximum height of 55±2 meters. Two test launches were conducted, with heights of 22±5 meters and 43±6 meters, which did not match the predicted height due to non-vertical flight and mass changes not accounted for in the model. Adjusting the model to account for these factors resulted in a predicted height of 51±3 meters, closely matching the higher test result and demonstrating the model's usefulness when assumptions are validated with test conditions.
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푢(푋, 푡)
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The Interaction Forces between Two Identical Cylinders Spinning around their Stationary and ParallelAxes in an IdealFluid
1. IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE)
e-ISSN: 2278-1684,p-ISSN: 2320-334X, Volume 12, Issue 6 Ver. II (Nov. - Dec. 2015), PP 01-07
www.iosrjournals.org
DOI: 10.9790/1684-12620107 www.iosrjournals.org 1 | Page
The Interaction Forces between Two Identical Cylinders Spinning
around their Stationary and ParallelAxes in an IdealFluid
Abdullah Baz1,
Salman M. Al-Kasimi2
Assistant Professor College of Computer and Information System
UQU, Saudi Arabia
Associate Professor the Research Institute UQU, Saudi Arabia
Abstract: This paper derives the equations that describe the interaction forcesbetween two identical cylinders
spinningaround their stationary and parallel axes in a fluid that isassumed to be in-viscous, steady, in-vortical,
and in-compressible. The paper starts by deriving the velocity field from Laplace equation,governing this
problem,and the system boundary conditions. It then determines the pressure field from the velocity field using
Bernoulli equation. Finally, the paper integrates the pressure around either cylinder-surface to find the force
acting on its axis.All equations and derivationsprovided in this paper are exact solutions. No numerical
analysesor approximations are used.The paper finds that such identical cylinders repel or attract each other in
inverse relation with separation between their axes, according to similar or opposite direction of rotation,
respectively.
Keywords: Rotating identical cylinders, ideal fluid, Laplace equation,velocity field,Bernoulli equation,
pressure, force, inverse law, repulsion, attraction.
I. Introduction
Determining the force acting on an objectdue to its existence in a fluid is an important topic, and has
several important applications. One of these applications is evaluating the lift force acting on an aeroplane wing
due to the flow of the air. The solution to such a problem might be analytical or numerical, depending upon the
complexity of the system and the required level of accuracy of the solution. Cylindrical Objects in fluid-flows
constitute one category of such problems and have vast applications.
According to the literature reviewed, several such systems have already been studied both numerically
and analytically,while other systems have attracted no attention. An example ofsuch studiedsystems[1]is the lift
force acting on a cylinder rollingin aflow. Another example[2] is the interaction forces between two concentric
cylinders with fluid internal and/or external to them. A third example[3]is the interaction forces between two
cylinders rotating around two parallel floating axes. No study to the knowledge of both authors has been done
on the interaction forceswhen the two parallel axes are fixed.
This paper is dedicated to find the interaction forcesbetween two identical cylindersspinning in an ideal
fluid (in-viscous, steady, in-vortical, andin-compressible);when their two parallel axes of rotation are made
fixed.These forces act on both axes. The system has never been studied before. For simplicity, the cylinders are
assumed infinitely long, so asto have a two-dimensional problem in 𝑥𝑦-plane, where rotational axes are parallel
to the ignored 𝑧-axis.
II. Problem Statement
Fig.1 shows the top view of the system targeted by this paper. It depictstwo identical circles (for the
two identical cylinders)of radius 𝑅. The distance between the two centres (for the two axes) is:2𝑎, where:
𝑎 > 𝑅.
Fig.1: Top view of the system targeted by this paper
2. The Interaction Forces between Two Identical Cylinders Spinning around their Stationary...
DOI: 10.9790/1684-12620107 www.iosrjournals.org 2 | Page
Each circle (cylinder) is rotating around its centre (axis) with a fixed angular velocity,𝜔, in a fluid that
is in-viscous, steady, in-vortical, and in-compressible. The aim of the paper is to derive the exact equations
describing the forcesboth cylindersexert on their fixed axes after the entire system reached steady state.
The next section uses the Laplace equation[4]governing such problems, to find the velocity field of the
fluid, satisfying its boundary conditions, which are:
1. the velocity of the fluid at the circle (i.e. the surface of the cylinder)is tangential to it, with a magnitude
of: 𝜔 ∙ 𝑅; and:
2. the velocity of the fluid at infinity is zero.
III. Fluid Velocity
As the governing Laplace equation is linear, super-position can be applied to simplify the solution.
Considering Cylinder-A alone,the steady-state fluid-velocity vector:
𝑉𝐴 𝑥, 𝑦 = 𝑉𝐴 𝑥 𝑥, 𝑦 , 𝑉𝐴 𝑦 𝑥, 𝑦 ,(1)
isknown [5] to be as shown in Fig.2, where its two components are givenby:
𝑉𝐴 𝑥 𝑥, 𝑦 = −𝜔 ∙ 𝑅2
∙
𝑦
𝑥2+𝑦2
, and:(2)
𝑉𝐴 𝑦 𝑥, 𝑦 = 𝜔 ∙ 𝑅2
∙
𝑥
𝑥2+𝑦2
, provided:(3)
𝑥2
+ 𝑦2
> 𝑅2
.
These velocity equations satisfyboth boundary conditions mentioned above. Furthermore, the velocity
of the fluid due to the spinning of Cylinder-Ais seen to be directly proportional to 𝜔, and inversely proportional
tothe distance from the cylinder axis;i.e. the further from cylinder axis, the slower the fluid is.
Considering Cylinder-B alone, the steady-state fluid-velocity vector:
𝑉𝐵 𝑥, 𝑦 = 𝑉𝐵𝑥 𝑥, 𝑦 , 𝑉𝐵𝑦 𝑥, 𝑦 ,(4)
can be obtained from Eqs.2&3 (with a positive shift of: 2𝑎, along the 𝑥-axis) as:
𝑉𝐵𝑥 𝑥, 𝑦 = −𝜔 ∙ 𝑅2
∙
𝑦
𝑥−2𝑎 2+𝑦2
, and:(5)
𝑉𝐵𝑦 𝑥, 𝑦 = 𝜔 ∙ 𝑅2
∙
𝑥−2𝑎
𝑥−2𝑎 2+𝑦2
, provided:(6)
(𝑥 − 2𝑎)2
+ 𝑦2
> 𝑅2
.
Fig.2: The velocity field of the ideal fluid due to the spinning of Cylinder-A
3. The Interaction Forces between Two Identical Cylinders Spinning around their Stationary...
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Hence, applying super-position;the fluid velocity for the system of two cylinders shown in Fig.1, can
be found using Eqs.1-6 as:
𝑉 𝑥, 𝑦 = 𝑉𝐴 𝑥, 𝑦 + 𝑉𝐵 𝑥, 𝑦 = 𝑉𝑥 𝑥, 𝑦 , 𝑉𝑦 𝑥, 𝑦 ,where:(7)
𝑉𝑥 𝑥, 𝑦 = 𝑉𝐴 𝑥 𝑥, 𝑦 + 𝑉𝐵𝑥 𝑥, 𝑦 = −𝜔 ∙ 𝑅2
∙
𝑦
𝑥2+𝑦2
+
𝑦
𝑥−2𝑎 2+𝑦2
, and:(8)
𝑉𝑦 𝑥, 𝑦 = 𝑉𝐴 𝑦 𝑥, 𝑦 + 𝑉𝐵𝑦 𝑥, 𝑦 = 𝜔 ∙ 𝑅2
∙
𝑥
𝑥2+𝑦2
+
𝑥−2𝑎
𝑥−2𝑎 2+𝑦2
,provided:(9)
𝑥2
+ 𝑦2
> 𝑅2
, and: (𝑥 − 2𝑎)2
+ 𝑦2
> 𝑅2
; i.e. where fluid exists outside both cylinders.
The above fluid-velocity is plotted as shown in Fig.3 below.The next section uses the fluid velocity and
Bernoulli equation to obtain the pressure field.
Fig.3: The velocity field of the ideal fluid due to the spinning of Cylinder-A and Cylinder-B in the same
direction around their stationary and parallel axes
IV. Fluid Pressure
In this section, the pressure at the boundary of either cylinder is derived, in readiness to find the force
exerted on its axes. Ignoring the effect of the gravitational force in the fluid, Bernoulli equation relates the
pressure magnitude, 𝑃 𝑥, 𝑦 , to the velocity field, 𝑉 𝑥, 𝑦 , as:
𝑃(𝑥, 𝑦) +
1
2
𝜌 ∙ 𝑉(𝑥, 𝑦) 2
= 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡, where: 𝜌 is the density of the fluid. (10)
The above equation can be read as:the summation of both static and dynamic pressures is constant
everywhere in the fluid. In this respect, it is the square of the magnitude of the fluid velocity, |𝑉 𝑥, 𝑦 |², is what
really matters for the fluid pressure.
Applying Eq.10 at Cylinder-A border & infinity (where velocity diminishes), then:
𝑃𝐴(𝑥, 𝑦) +
𝜌
2
∙ 𝑉(𝑥, 𝑦) @ 𝐶𝑦𝑙𝑖𝑛𝑑𝑒𝑟 −𝐴 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦
2
= 𝑃∞, where:
𝑃𝐴(𝑥, 𝑦): is the pressure at Cylinder-A boundary, and:
𝑃∞: is the fluid pressure at ∞. Hence:
𝑃𝐴 𝑥, 𝑦 = 𝑃∞ −
𝜌
2
∙ 𝑉(𝑥, 𝑦) @ 𝐶𝑦𝑙𝑖𝑛𝑑𝑒𝑟 −𝐴 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦
2
.Using Eq.7, then:
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𝑃𝐴 𝑥, 𝑦 = 𝑃∞ −
𝜌
2
∙ 𝑉2
𝑥 𝑥, 𝑦 + 𝑉2
𝑦 𝑥, 𝑦
@ 𝐶𝑦𝑙𝑖𝑛𝑑𝑒𝑟 −𝐴 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦
.
This is simplified using Eqs.8&9with: 𝑥2
+ 𝑦2
= 𝑅2
to:
𝑃𝐴 𝑥, 𝑦 = 𝑃∞ −
𝜌∙𝜔2∙𝑅4
2
∙
𝑦
𝑅2
+
𝑦
𝑅2−4𝑎∙𝑥+4𝑎²
2
+
𝑥
𝑅2
+
𝑥−2𝑎
𝑅2−4𝑎∙𝑥+4𝑎2
2
.
This is reduced with: 𝑥2
+ 𝑦2
= 𝑅2
to:
𝑃𝐴 𝑥, 𝑦 = 𝑃∞ −
𝜌∙𝜔2∙𝑅2
2
∙ 1 +
3𝑅2−4𝑎∙𝑥
𝑅2−4𝑎∙𝑥+4𝑎²
,where: 𝑥 ∈ Circle-𝐴.
Converting to polar coordinates, with: 𝑥 = 𝑅 ∙ cos 𝜃, then:
𝑃𝐴 𝑥, 𝑦 = 𝑃𝐴 𝜃 = 𝑃∞ −
𝜌∙𝜔2∙𝑅2
2
∙ 1 +
3𝑅2−4𝑎∙𝑅∙cos 𝜃
𝑅2−4𝑎∙𝑅∙cos 𝜃+4𝑎²
, 𝜃 ∈ 0,2𝜋 .(11)
𝑃𝐴 𝜃 is seen to be symmetrical about the𝑥-axis.
V. Force Acting on the Rotational Axis of Cylinder-A
The fluid pressure,𝑃𝐴 𝜃 ,expressed by Eq.11 is acting perpendicular to the surface of Cylinder-A as
shown in Fig.4, and can be seen to cause infinitesimal repelling force, 𝑑𝐹𝐴 𝜃 , in the same direction, given by:
𝑑𝐹𝐴 𝜃 = 𝑃𝐴 𝜃 ∙ 𝐿 ∙ 𝑅 ∙ 𝑑𝜃, where:
𝐿: is the Length of either cylinder, which is assumed to be infinitely long.
Fig.4: Top view of Cylinder-A showing fluid pressure
Decomposing: 𝑑𝐹𝐴 𝜃 into two components, and ignoring its y-component due to the symmetry of
𝑃𝐴 𝜃 about the 𝑥-axis; then:
𝑑𝐹𝐴 𝑥 𝜃 = −𝑃𝐴 𝜃 ∙ 𝐿 ∙ 𝑅 ∙ cos 𝜃 ∙ 𝑑𝜃.(12)
Integrating 𝑑𝐹𝐴 𝑥 𝜃 around Circle-A (the surface of Cylinder-A) yields the force, 𝐹, exerted on the
axis of rotation of Cylinder-A. Hence:
𝐹 = 𝑑𝐹𝐴 𝑥 𝜃
2𝜋
0
. This is expressed using Eqs.11&12, as:
𝐹 = −𝐿 ∙ 𝑅 𝑃∞ −
𝜌∙𝜔2∙𝑅2
2
∙ 1 +
3𝑅2−4𝑎∙𝑅∙cos 𝜃
𝑅2−4𝑎∙𝑅∙cos 𝜃+4𝑎²
∙ cos 𝜃 ∙ 𝑑𝜃
2𝜋
0
.
The first two of the above three integrandswill integrate to zero, leaving 𝐹as:
𝐹 =
1
2
∙ 𝜌 ∙ 𝐿 ∙ 𝜔2
∙ 𝑅4
∙ 𝐼, where:(13)
𝐼 =
3𝑅−4𝑎∙cos 𝜃
𝑅2−4𝑎∙𝑅∙cos 𝜃+4𝑎²
∙ cos 𝜃 ∙ 𝑑𝜃
2𝜋
0
.
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VI. Solving the Integral
The integral, 𝐼, can be solved [6] by usingthe complex transformation:
𝑧 = 𝑒 𝑖𝜃
= cos 𝜃 + 𝑖 ∙ sin 𝜃, where: 𝑖 = −1, 𝑖2
= −1, 𝑑𝜃 =
𝑑𝑧
𝑖𝑧
,and:
cos 𝜃 =
𝑧+
1
𝑧
2
=
𝑧2+1
2𝑧
;
whereby it converts to an integral over the closed contour, 𝐶, of the unit circle: 𝑧 = 1, in the complex 𝑧-plane.
Hence, removing𝜃:
𝐼 =
3𝑅−4𝑎∙cos 𝜃
𝑅2+4𝑎²−4𝑎∙𝑅∙cos 𝜃
∙ cos 𝜃 ∙ 𝑑𝜃
2𝜋
0
=
3𝑅−4𝑎∙
𝑧2+1
2𝑧
𝑅2+4𝑎2−4𝑎∙𝑅∙
𝑧2+1
2𝑧
∙
𝑧2+1
2𝑧
∙
𝑑𝑧
𝑖𝑧𝐶
.
This is simplified to:
𝐼 =
−𝑖
2
2𝑎∙𝑧4−3𝑅∙𝑧3+4𝑎∙𝑧2−3𝑅∙𝑧+2𝑎
𝑧2 2𝑎∙𝑅∙𝑧2− 𝑅2+4𝑎2 ∙𝑧+2𝑎∙𝑅
∙ 𝑑𝑧𝐶
,and can be expressed as:
𝐼 =
−𝑖
4𝑎∙𝑅
2𝑎∙𝑧4−3𝑅∙𝑧3+4𝑎∙𝑧2−3𝑅∙𝑧+2𝑎
𝑧2∙ 𝑧−𝛼 ∙ 𝑧−𝛽
∙ 𝑑𝑧𝐶
,where: 𝛼 and: 𝛽 are the roots of:
2𝑎 ∙ 𝑅 ∙ 𝑧2
− 𝑅2
+ 4𝑎2
∙ 𝑧 + 2𝑎 ∙ 𝑅 = 0, given by:
𝛼 =
𝑅
2𝑎
,and:
(14)
𝛽 =
2𝑎
𝑅
=
1
𝛼
.
(15)
Noting that:𝑎 > 𝑅, then: 𝛼 < 1, and is inside 𝐶, while: 𝛽 > 1 , and is outside 𝐶.Hence, the
integrand in 𝐼 has three poles within 𝐶, two at the origin and one at 𝛼.
Using Cauchy Theorem in complex integrals[6], 𝐼 is found as:
𝐼 =
−𝑖
4𝑎∙𝑅
∙ 2𝜋 ∙ 𝑖 ∙
2𝑎∙𝑧4−3𝑅∙𝑧3+4𝑎∙𝑧2−3𝑅∙𝑧+2𝑎
𝑧−𝛼 ∙ 𝑧−𝛽
′
@𝑧=0
+
2𝑎∙𝑧4−3𝑅∙𝑧3+4𝑎∙𝑧2−3𝑅∙𝑧+2𝑎
𝑧2∙ 𝑧−𝛽 @𝑧=𝛼
.This gives:
𝐼 =
𝜋
2𝑎∙𝑅
∙
−3𝑅∙𝛼∙𝛽+2𝑎∙ 𝛼+𝛽
𝛼2∙𝛽2
+
2𝑎∙𝛼4−3𝑅∙𝛼3+4𝑎∙𝛼2−3𝑅∙𝛼+2𝑎
𝛼2∙ 𝛼−𝛽
.Removing𝛽using Eq.15, this becomes:
𝐼 =
𝜋
2𝑎∙𝑅
∙ −3𝑅 + 2𝑎 ∙
𝛼2+1
𝛼
+
2𝑎∙𝛼4−3𝑅∙𝛼3+4𝑎∙𝛼2−3𝑅∙𝛼+2𝑎
𝛼∙ 𝛼2−1
.
Substituting for:𝛼using Eq.14, and simplifying gives: 𝐼 = −
2𝜋
𝑎
∙
2𝑎2−𝑅2
4𝑎2−𝑅2
.
Hence, putting this, in Eq.13, and simplifying gives:
𝐹 = −
𝜋∙𝜌∙𝐿∙𝜔2∙𝑅4
𝑎
∙
2𝑎2−𝑅2
4𝑎2−𝑅2
, remembering that: 𝑎 > 𝑅. (16)
The negative sign means that the fluid actually repels Cylinder-A away from Cylinder-B.
VII. InterpretingThe Repulsion Force
Considering: a, to be given as: 𝑎 = 𝑛 ∙ 𝑅,then: 𝑎 > 𝑅, implies that: 𝑛 > 1;and Eq.16 for
the force, 𝐹, can be simplified to:
𝐹 = −
𝑅
2𝑎
∙
𝑛2−0.5
𝑛2−0.25
∙ 𝑃 =
−1
2𝑛
∙
𝑛2−0.5
𝑛2−0.25
∙ 𝑃, where: 𝑛 > 1; and:
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𝑃 = 𝜋 ∙ 𝜌 ∙ 𝐿 ∙ 𝜔2
∙ 𝑅3
.
(17)
When the two identical cylinders are made very near to each other, then: 𝑛 = 1+
,and the
repulsion force is:𝐹1, given as:
𝐹1 = −
𝑃
3
= −0.33333𝑃.
The maximum repulsion, 𝐹𝑝, can be found to occur at:𝑛 =
5+ 17
8
= 1.06789, with value of:
𝐹𝑝 = −0.33675𝑃.
As the gap between the two identical cylinders ismade: 2𝑅, then:𝑛 = 2,and the repulsion force
is reduced to:𝐹2, given as:
𝐹2 = −
7𝑃
30
= −
7
15
∙
𝑅∙𝑃
𝑎
= −0.23333𝑃.
When gap is made: 4𝑅, then:𝑛 = 3,and the repulsion force is reduced to: 𝐹3, given as:
𝐹3 = −
17𝑃
105
= −
17
35
∙
𝑅∙𝑃
𝑎
= −0.16190𝑃.
When gap is made:6𝑅, then:𝑛 = 4,and the repulsion force is reduced to: 𝐹4, given as:
𝐹4 = −
31𝑃
252
= −
31
63
∙
𝑅∙𝑃
𝑎
= −0.12302𝑃.
As the two identical cylinders are spaced far enough (𝑛 > 4) of each other, then the repulsion force
goes asymptotically to:
𝐹𝑛 = −
𝑃
2𝑛
= −
𝑅∙𝑃
2𝑎
.
This is an inverse relationship with the separation between axes of rotation, i.e. the identical cylinders rotating in
the same directiontend to repel each other in inverse law with respect to their axial separation.
VIII. Effect of Opposing Direction
When Cylinder-B spins differentlyto Cylinder-A, namely in the negative sense, the fluid-velocity can
be plotted as shown in Fig.5 below.
Fig.5: The velocity field of the ideal fluid due to the spinning of Cylinder-A and Cylinder-B in different
directionsaround their stationary and parallel axes
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A similar treatment as above, gives the fluid force acting on the axis of Cylinder-A, as:
𝐹 =
𝜋∙𝜌∙𝐿∙𝜔2∙𝑅4
𝑎
∙
2𝑎2
4𝑎2−𝑅2
, remembering: 𝑎 > 𝑅. Using Eq.17, this can be simplified to:
𝐹 =
𝑅
2𝑎
∙
𝑎2
𝑎2−0.25𝑅2
∙ 𝑃 =
𝑅
2𝑎
∙
𝑛2
𝑛2−0.25
∙ 𝑃 =
1
2𝑛
∙
𝑛2
𝑛2−0.25
∙ 𝑃 , where: 𝑛 > 1.
This is an attraction force, reduced asymptotically for far enough spacing (𝑛 > 4), to:
𝐹𝑛 =
𝑃
2𝑛
=
𝑅∙𝑃
2𝑎
.
This is also an inverse relationship with the separation between axes of rotation. Since both cylinders are
identical, thenthe results obtained are valid for both of them.
IX. Conclusion
This paper derived the force acting on twoidentical cylinders spinning atconstant angular velocity
around their stationary and parallel axes in anin-viscid, steady,in-vortical, and in-compressible fluid.The
obtained equations showed that each cylinder axis,in such a system,experiences a repelling or attracting force
according to similar or opposite sense of rotation respectively.The magnitude of that force is inversely
proportional to the separation between the axes. It is also proportional to the density of the fluid, the cylinder
volume, its radius, and the product of the two angular velocities of the cylinders.
Nomenclature:
This section summarizes the symbols used in the paper in alphabetical order as follows:
𝜌: Density of the fluid
𝜔: Angular speed of spinning of either cylinder
𝑎: Half the distance between axes of cylinders
𝑑𝐹𝐴 𝜃 :Infinitesimal force acting on Cylinder-A
𝑑𝐹𝐴 𝑥 𝜃 : Component of:𝑑𝐹𝐴 𝜃 along the 𝑥-axis
𝐹: Interaction force acting on the axle of Cylinder-A
𝐹1: Value of: F when both cylinders are about to touch each other
𝐹2: Value of: F when both cylinders are spaced2𝑅 apart
𝐹3: Value of: F when both cylinders are spaced4𝑅 apart
𝐹4: Value of: F when both cylinders are spaced6𝑅 apart
𝐹𝑝: Peak value of: F
𝐿: Length of either cylinder
𝑃(𝑥, 𝑦): Pressure magnitude of the fluid
𝑃∞: Pressure magnitude of the fluidat ∞
𝑃𝐴 𝑥, 𝑦 : Pressuremagnitude of the fluidat Cylinder-A boundary in 𝑥y-coordinates
𝑃𝐴(𝜃): Pressure magnitude of the fluidat Cylinder-A boundary in 𝑟𝜃-coordinates
R: Radius of either cylinder
𝑉 𝑥, 𝑦 : Velocity vector of the fluid due to the spinning of both cylinders
𝑉𝑥 𝑥, 𝑦 :Component of: 𝑉 𝑥, 𝑦 along the 𝑥-axis
𝑉𝑦 𝑥, 𝑦 :Component of: 𝑉 𝑥, 𝑦 along the 𝑦-axis
𝑉𝐴 𝑥, 𝑦 : Velocity vector of the fluid due to the spinning of Cylinder-A
𝑉𝐴 𝑥 𝑥, 𝑦 : Component of: 𝑉𝐴 𝑥, 𝑦 along the 𝑥-axis
𝑉𝐴 𝑦 𝑥, 𝑦 : Component of: 𝑉𝐴 𝑥, 𝑦 along the 𝑦-axis
𝑉𝐵 𝑥, 𝑦 : Velocity vector of the fluid due to the spinning of Cylinder-B
𝑉𝐵𝑥 𝑥, 𝑦 : Component of: 𝑉𝐵 𝑥, 𝑦 along the 𝑥-axis
𝑉𝐵𝑦 𝑥, 𝑦 : Component of: 𝑉𝐵 𝑥, 𝑦 along the 𝑦-axis
References
[1]. Fluid Mechanics: An Introduction, E. Rathakrishnan, 3rd
edition. 2012.
[2]. Fluid Mechanics for Engineers: A Graduate Textbook, M. T. Schobeiri, 2010.
[3]. Hydrodynamics, Sir H. Lamb, 1945.
[4]. Solutions of Laplace’s Equation, D. R. Bland, 1961.
[5]. Vortex Dynamics, P. G. Saffman, 1995.
[6]. Complex Integration and Cauchy's Theorem, G. N. Watson, 2012.