Internal Friction Of Fluids Research Articles

Statistical investigation of heat transfer efficiency on a ternary nanoflow over a stretching surface

Ternary hybrid nanofluids offer significant advantages in heat transfer due to their enhanced thermal conductivity compared to traditional fluids. Investigating their flow behavior over expanding surfaces can contribute to the development of more efficient heat exchangers and cooling systems in the electronics, energy production, and transportation industries. This paper presents a detailed analysis of the flow and thermal transfer characteristics of a ternary hybrid nanofluid (SiO2–Cu–Al2O3/H2O) over a radially stretching surface, utilizing both numerical techniques (Keller Box method) and statistical techniques. Furthermore, the model includes factors like convective heat transport, radiation, internal fluid friction, and suction. Similarity transformations transform the governing equations for fluid flow and heat transfer into a system of nonlinear ordinary differential equations. Then the equations are solved numerically using the Keller Box method with the aid of MATLAB software. Additionally, we have employed statistical techniques such as correlation analysis, probable error estimation, and multivariate regression to validate and ensure the accuracy of the numerical outcomes. Parameter ranges for Biot number, radiation, unsteadiness, Eckert number, and energy generation parameter are 0.1 ≤ Mp ≤ 0.7, 0.1 ≤ Ks ≤ 1.5, 0.2 ≤ Ps ≤ 0.8, 0.1 ≤ EN ≤ 0.4, 0.5 ≤ RN ≤ 2, 1≤ BT ≤ 2.5, 0.1 ≤ Hsc ≤ 0.7 and found to significantly impact heat transfer behavior within these bounds. This study utilizes 3D surface plots to visualize the influence of key parameters on engineering quantities. R squared values show that the data strongly match the regression model. The findings demonstrate excellent concordance between the predicted values and the actual Nusselt number and skin friction measurements. Biot number has a very strong correlation with heat transfer, with minimal chance of error. And the same is evidenced by the correlation matrix and multiple regression analysis. Eckert number also exhibits a negative correlation with the Nusselt number. This translates to a situation where increasing Eckert number directly leads to a decrease in the rate of heat transfer, with no margin for error. Energy generation parameter demonstrates a negative correlation with heat transfer. However, there's a slight possibility of error, with an estimated error percentage of around 0.0017.

Computational framework for thermal transportation in radiative bio-convective micropolar nanomaterial flow over stretched sheet with entropy optimization

Entropy generation (EG) in bioconvective nanofluid flows is a phenomenon that occurs due to the presence of nanoparticles and microorganisms within fluid flows. EG leads to an increase in thermodynamic irreversibility within the system. Understanding and quantifying EG in nanofluid flows is essential to optimize heat transfer processes and design efficient systems. Current investigation aims to analyze the irreversibility of the magnetized bioconvective flow of non-Newtonian micropolar type nanofluid. Flow features are modeled considering flow by a Darcy Forchheimer permeable surface of a stretched sheet. Phenomenon of bioconvection in a micropolar fluid is considered to regulate the solid tiny particles. Magnetic field and permeability effects are accounted in momentum relation. Energy relation is constructed in the presence of Joule heating, internal fluid friction, and thermal radiation. Mass concentration equation is formulated by accounting Arrhenius kinetics and binary chemical reaction. Furthermore, the buoyancy effects of solid tiny particles are considered in thermal and concentration relations. Total entropy of the considered flow is modeled using the second thermodynamics law. System of dimensional equations representing the flow is obtained with the implementation of boundary layer approximations. The acquired system is altered into an ordinary one through transformations and then tackled by numerical scheme Runge-Kutta-Fehlberg method (RKF-45) in the Mathematica package. Behavior of velocity, thermal field, entropy production, mass concentration, motile density, and Bejan number versus sundry variables is investigated through plots. Skin friction, local heat, mass, and density rates are numerically examined. Thermal field upsurges and velocity field decays versus higher Hartmann number. Entropy generation improved versus higher Brinkman number, microorganisms diffusion variable, and temperature difference ratio variable.

Entropy optimization in bio-convective chemically reactive flow of micropolar nanomaterial with activation energy and gyrotactic microorganisms

The main objective of this research paper is to scrutinize the entropy generation(EG) in the bioconvective flow of micropolar fluid by porous surface of a stretched sheet. Mixed convection effects are considered. Thermal radiation, internal fluid friction, Joule heating, and heat generation aspects are accounted for in the energy relation. Mass concentration relation is developed because of the binary chemical reactions associated with Arrhenius kinetics. The governing equations for the micropolar fluid are developed based on Buongiorno's model with the help of boundary layer restrictions. By invoking the transformation procedure, dimensional model is made dimensionless. The well-known shooting technique RKF-45 in Mathematica software is implemented to solve the transformed equations. Performance of micropolar fluid velocities, thermal field, entropy generation, mass concentration, Bejan number, and motile density versus influential variables are highlighted. Furthermore, the intensity of skin friction, Nusselt quantity, motile density, and Sherwood number is computed and analyzed. It is observed that the rate of EG improves while the Bejan number diminishes versus higher Brinkman number, and diffusion parameter due to gyrotactic microorganisms. Magnitude of heat transfer rate boosts through higher Γ, Qs, Ec, Ha and Rd at an average percentage of 44, 82, 189, 156 respectively.

The Impact of Thermal Load and Turbulent Flow on Minimum Film Thickness of a Large Tilting-Pad Journal Bearing

Abstract Thermal loads induced by internal fluid friction significantly influence the operating behavior of high-speed journal bearings. These loads increase on average if critical Reynolds numbers are exceeded. This paper presents a detailed thermo-elasto-hydrodynamic (TEHD) analysis of a large five-pad tilting-pad journal bearing operating up to surface speeds of 94 m/s. The structure mechanical analysis is conducted by a FE-Code that is implemented as a submodule of the bearing code to improve computational efficiency. The nonlinear structure model includes the pads, their aligning ring, and the housing while the journal's thermal expansion is approximated analytically. The contact of pads and ring is modeled with a contact algorithm. Validation with test data from literature indicates the relevance of structure model's complexity. Besides the pad deflection, in particular, the consideration of the ring and housing thermal expansion is decisive in predicting the effective clearance properly. Furthermore, comparisons between measurement and simulation show that the experimentally identified nonlinear speed-dependent characteristic of minimum film thickness that features an increase due to turbulent flow is well predicted by the theoretical model. Generally, results provide an insight on the impact of the different effects dominating the behavior of the bearing in the respective speed ranges, and therefore, improve the understanding of complex bearing systems. Finally, the quality of results of approximation formulas that are easier to implement and more time efficient than the complex FE-code is investigated.

Controlling melt flow by nanoparticles to eliminate surface wave induced surface fluctuation

The high surface roughness is one of the major challenges encountered in laser metal additive manufacturing (AM) processes, which is closely related to the melt flow behavior. However, how to control the melt flow in laser metal AM processes to improve surface finish is not clear. Here we reveal the effects of nanoparticles on melt flow behavior at every location of melt pool during laser metal AM process for the first time using Al6061 + TiC nanoparticles system and achieve significant improvement of surface finish by using TiC nanoparticles to control the melt flow and damp the surface wave. Based on the in-situ x-ray imaging observation, the surface wave is fully damped after adding TiC nanoparticles, compared with only 56% damping without nanoparticles during LPBF of Al6061. Our in-depth in-situ x-ray imaging analysis and viscosity measurement enable us to identify that nanoparticle-induced increase of viscosity causes the fully damping of the surface wave by (1) increasing the internal fluid friction for more efficient wave amplitude reduction, (2) controlling the melt flow to increase the surface wave number, (3) controlling the melt flow to increase the wave damping time. Furthermore, we also quantified the relative contributions of increasing fluid friction, increasing wave number and increasing damping time to wave damping, which account for 61%, 25% and 14%, respectively. Our research provides the mechanisms and potential method to address the surface finish challenge in laser metal AM processes. • Experimentally revealed effects of nanoparticles on melt flow for the first time. • Quantitatively understood the mechanisms of nanoparticle-induced wave damping. • Uncovered the correlation among melt flow, surface wave, and surface finish. • Achieved surface fluctuation free keyhole-mode laser polishing.

A layer of yield-stress material on a flat plate that moves suddenly

The flow of an unbounded Newtonian fluid above a flat solid plate, set into motion suddenly with uniform velocity, is a canonical problem investigated by Stokes (On the effect of the internal friction of fluids on the motion of pendulums, Trans. Camb. Phil. Soc., vol. 9, 1851) and Rayleigh (Lond. Edinb. Dubl. Philos. Mag. J. Sci., vol. 21, issue 126, 1911, pp. 697–711). We tackle the same problem but with the fluid replaced by a yield-stress material of finite height with a free surface; the latter ensures both yielded (fluid-like) and rigid behaviour. The interface between the yielded and rigid regions – the so-called ‘yield surface’ – always emerges from the plate at a rate faster than that for momentum diffusion. The rigid region, adjacent to the free surface, is accelerated by the constant yield stress acting at this yield surface. As the velocity of this region increases following start-up, its acceleration climaxes and subsequently diminishes. In this latter period, the thickness of the rigid region increases owing to relaxation of the velocity gradients and the stress. The yield surface eventually collides with the plate in finite time, after which the entire material moves in concert with the plate as a rigid body. Analytical and numerical results are presented that can form the basis for practical applications. This includes the development of a ‘rheological microscope’ to directly detect and measure the yield stress. The analysis focuses on exploring the flow physics of a Bingham material. This is shown to be similar to that of a general Herschel–Bulkley material with shear-thinning or shear-thickening rheology.

Numerical analysis of lubricant viscosity variations on operating condition of helical gear system

This study presents a dynamic model to investigate the effects of lubricant viscosity due to oil degradations or oxidations on the vibration signal of helical gear system. The modelling has been extended to consider the effect of elastohydrodynamic lubrication (EHL) to include frictional effects between meshed gears. A ten degree-of-freedom (10-DOF) model has been developed to combine lateral, torsional and axial vibrations of helical gear transmissions with supporting bearings, powerful motor and applied load. The variation of viscosity takes into account through the time-varying of EHL friction coefficient. The results conclude that higher lubricant viscosity could increase internal fluid friction and vibrational excitation, where higher motor power is required to overcome the higher friction of thicker lubricants. The vibration signal can be a good indicator for lubrication condition and it is possible to be used for monitoring the lubrication condition and obtain accurate diagnostic result for tooth surface defects.

A power loss model for Archimedes screw generators

This paper presents a complete power loss model for an Archimedes screw used for power generation (ASG) including a non-dimensional model to predict power losses due to outlet submersion flooding. This model amends a prior idealized, frictionless ASG performance model to include power losses due to bearing friction, outlet exit effects, internal hydraulic friction and outlet submersion. This study presents data and a derived relationship for power losses due to outlet submergence and found that unmodified Manning’s coefficients can be used to model internal fluid friction losses. Laboratory experiments on a scale-model ASG were conducted to determine variable relationships and validate power loss models. The performance of a 7 kW grid-connected ASG was measured and used to validate model predictions. The proposed ASG power loss model improves the prior frictionless power model significantly and was generally capable of predicting the power output of a real-world ASG.

Semianalytical methods for heat and fluid flow between two parallel plates

The present study attempts to investigate the effects the viscous dissipation on the unsteady temperature distribution in the conduction limit for both hydrodynamically and thermally fully developed, laminar flow of Newtonian fluid between two asymmetrically heated infinitely long parallel plates. Utilizing the assumptions routinely employed in the literature, we devise here a semi-analytical formalism to investigate the temperature distribution for two different flow configurations, i.e., the poiseuille flow and the simple shear driven flow. In the analysis, we give focus to the viscous dissipative effect arises because of the two individual aspects in case of shear-driven flow: the shear heating produced by the movable upper plate along with fluid friction, while only due to the internal fluid friction in case of Poiseuille flow. Finally, we show the variation of velocity and the temperature distribution in the flow field for several nondimensional parameters as emerge in the present study, and highlight their individual role in delineating the temperature distribution in the flow field, which essentially alters transient thermal transport characteristics of heat in different cases of flow dynamics.

Entropy generation during fully-developed forced convection in parallel-plate micro-channels at high zeta-potentials

Entropy generation during laminar fully-developed flow in parallel-plate micro-channels with an imposed pressure gradient and a uniform wall heat flux has been investigated taking into consideration the effects of an EDL at the solid–fluid interfaces and viscous dissipation due to internal fluid friction. The rates of local volumetric and total entropy generation per unit channel volume due to heat transfer and viscous dissipation were calculated using analytical expressions for the velocity and temperature profiles. Comparing results from expressions using the local-temperature formulation to others using a reference-temperature formulation have shown that the latter is not accurate for low Brinkman number or small channel sizes. Effects of Brinkman number, Peclet number, channel size, and zeta-potential on the rate of entropy production are presented and discussed. Entropy production due to heat transfer reaches a maximum, while entropy production due to viscous dissipation and the ratio of pumping power to heat transfer reach a minimum within a specific range of zeta-potential. These trends were attributed to the effects of EDL on the velocity and temperature profiles.

An Analytical Approach to the Effect of Viscous Dissipation on Shear-driven Flow between two Parallel Plates with Constant Heat Flux Boundary Conditions

An investigation has been made to analyze the effects of viscous dissipation on the heat transfer characteristics for both hydro-dynamically and thermally fully developed, laminar shear driven flow between two infinitely long parallel plates, where the upper plate is moving in an axial direction at a constant speed. On the basis of some routine assumptions made in the literature, a close form analytical expression of Nusselt numbers for the flow of Newtonian fluid with constant properties has been developed for three different cases of constant heat flux boundary conditions. In the analysis, focus is on the viscous dissipative effect due to the shear produced by the moving upper plate apart from the viscous heating due to internal fluid friction for the flow of a Newtonian fluid.

Viscous dissipation effects on the limiting value of Nusselt numbers for a shear-driven flow through an asymmetrically heated annulus

In this study, an analytical investigation for analyzing the effects of viscous dissipation on the limiting Nusselt number for a hydro-dynamically fully developed laminar shear-driven flow through an asymmetrically heated annulus of two infinitely long concentric cylinders has been made, where the inner cylindrical rod is moving in an axial direction at a constant speed. On the basis of some common assumptions, an analytical framework has been devised to explore the effects of viscous dissipation on the heat transfer characteristics for the flow of Newtonian fluid, and, consequently, closed-form expressions for the limiting Nusselt numbers are evaluated. In the analysis, focus has been given on the viscous dissipative effect due to the shear produced by the movable inner cylindrical rod apart from the viscous dissipation due to internal fluid friction for the flow of a Newtonian fluid. The interactive effects of the Brinkman number and degree of asymmetry on the limiting Nusselt number are analytically investigated. It is observed from this study that the limiting Nusselt number becomes independent of Brinkman number when both the walls of the annulus are kept at an equal temperature. Moreover, the temperature profile in the conduction limit obtained with the consideration of viscous dissipation effect provides a boundary condition required for solving energy equation including the axial conduction in the fluid.

History force on coated microbubbles propelled by ultrasound

In this paper the unsteady translation of coated microbubbles propelled by acoustic radiation force is studied experimentally. A system of two pulsating microbubbles of the type used as contrast agent in ultrasound medical imaging is considered, which attract each other as a result of the secondary Bjerknes force. Optical tweezers are used to isolate the bubble pair from neighboring boundaries so that it can be regarded as if in an unbounded fluid and the hydrodynamic forces acting on the system can be identified unambiguously. The radial and translational dynamics, excited by a 2.25 MHz ultrasound wave, is recorded with an ultrahigh speed camera at 15×106 frames/s. The time-resolved measurements reveal a quasisteady component of the translational velocity, at an average translational Reynolds number ⟨Ret⟩≈0.5, and an oscillatory component at the same frequency as the radial pulsations, as predicted by existing models. Since the coating enforces a no-slip boundary condition, an increased viscous dissipation is expected due to the oscillatory component, similar to the case of an oscillating rigid sphere that was first described by Stokes [“On the effect of the internal friction of fluids on the motion of pendulums,” Trans. Cambridge Philos. Soc. 9, 8 (1851)]. A history force term is therefore included in the force balance, in the form originally proposed by Basset and extended to the case of time-dependent radius by Takemura and Magnaudet [“The history force on a rapidly shrinking bubble rising at finite Reynolds number,” Phys. Fluids 16, 3247 (2004)]. The instantaneous values of the hydrodynamic forces extracted from the experimental data confirm that the history force accounts for the largest part of the viscous force. The trajectories of the bubbles predicted by numerically solving the equations of motion are in very good agreement with the experiment.

On the parameter β=Re/KC= D2/ νT

This brief communication is a summary of the facts regarding the universalization in 1976 of a parameter β to fully nonlinear unsteady separated flows about bluff bodies nearly 125 years after its first appearance in a linearized analysis of unseparated viscous flow with very slow oscillations about a cylindrical rod and sphere by Sir George Gabriel Stokes [1851. On the effect of the internal friction of fluids on the motion of pendulums. Transactions of the Cambridge Philosophical Society 9(II), 8–106]. The primary purpose of Stokes was to show that “ the index of friction” (the kinematic viscosity), in the equations of motion may be deduced from experiments for the vindication of the heuristic reasoning that went into the derivation by Navier [1827. Mémoire sur les lois du movement des fluides. Mémoires de l’Académie de Sciences 6, 389–416.], Poisson [1831. Nouvelle théorie de l’action capillaire Bachelier, Paris.], de Saint-Venant [1843. Note á joindre un memoire sur la dynamique des fluides. Comptes Rendus 17, 1240–1244.] and Stokes [1845. On the theories of internal friction of fluids in motion. Transactins of the Cambridge Philosophical Society 8, 287–305.] of what are now called the Navier–Stokes equations.

Analysis of slug tests in bypassed wells

In Berlin many wells are constructed with an additional small diameter bypass tube flanged on to the large diameter casing and running parallel to the later upward to the surface. The response to a slug test conducted in such a well is usually characterised by significant water level oscillations developing inside the branched pipe system. In the present paper, a general non-linear model is developed for the analysis of slug tests performed in bypassed wells. The model includes skin effects, non-linear head losses due to internal well bore fluid friction, minor losses originating at radius changes along the flow path inside the well, and inertial effects of the water columns contained within the primary casing and the bypass. Pipe flow inside the branched tubing system is described by the one-dimensional conservation law of the mechanical energy of the well bore fluid. Aquifer flow is quantified by a convolution-type approach. The coupled well-aquifer system is governed by an integro-differential equation representing the mean mechanical energy of the total well bore fluid and, coupled to it, by a non-linear ordinary differential equation which accounts for mechanical energy fluctuations between the water columns inside the main casing and the bypass. Both equations are solved by finite difference approximations coupled to point iterative numerical techniques. An application of the model to field data suggests that non-linear hydraulic head losses originating inside the well bore may cause the measured head responses to depend on the initial displacement. A packer used to initiate a slug test can induce these non-linear distortions through the combined effect of minor losses originating at the inlet and the outlet of a small diameter packer flow-through tube and internal fluid friction potentially developing inside that packer passage. As classical linear theories do not allow for the analysis of such phenomena, a model accounting for these non-linear processes should be used if field data show significant dependencies on the initial slug height.

Viscous and rotational splitting of the translational oscillations of Earth's solid inner core

The small oscillations of the inner core about its central position present an interesting problem in fluid dynamics with roots in the classical literature. It is a generalization of the problem Stokes [Stokes, G.G., 1851. On the effect of the internal friction of fluids on the motion of pendulums, Trans. Cambridge Philos. Soc. 9, 8.] solved of a sphere oscillating in an unbounded, non-rotating, viscous fluid to the case of a sphere oscillating in the contained, rotating fluid outer core. In the inviscid case, the influence of the Coriolis acceleration splits the oscillation into three distinct motions with differing periods, one along the axis of rotation, one prograde in the equatorial plane, and one retrograde in the equatorial plane. In this paper, we consider the effect of the presumed high viscosity at the inner core boundary on the rotational splitting. We obtain novel exterior analytic solutions of the Poincaré equation for the flow outside the boundary and solve the Ekman boundary-layer problem to match the no-slip condition at the inner core boundary. Continuity of normal displacement at the inner core boundary, and in far-field approximation at the core–mantle boundary, determine the free constants in the two independent Poincaré solutions. Conservation of linear momentum between the inner core, outer core and shell establishes a centre of mass reference frame. Analytic expressions for the pressure and viscous drag are obtained for both the axial and equatorial modes of oscillation. These are then used in the equation of motion for the inner core to deduce viscous and rotational splitting laws. Viscosity is found to reduce the effect of rotation on the periods of all three modes; most noticeably, it reduces the large rotational splitting of the two equatorial modes. We expect the geophysically interesting value of the viscosity just outside the inner core might be recovered from the observed splitting, particularly, that of the two equatorial modes.

An Antoine Type Equation for Liquid Viscosity Dependency to Temperature

The viscosities of fluids are required in most engineering calculations where fluid flow or mixing is an important factor, such as in the design of reactors, heat exchangers, fractionating towers, cooling towers, and distillation units. For Newtonian fluids, viscosity is defined as the measure of the internal fluid friction, which is the constant of proportionality between the shear stress per unit area at any point and the velocity gradient. Liquid viscosity is characterized by strong temperature dependency, especially close to the boiling and freezing points. An Antoine-type equation, containing 3 adjustable parameters, is presented which successfully correlates liquid viscosity to temperature over the full range of boiling to freezing point. (14 refs.)

XII. On the reflexion and refraction of solitary plane waves at a plane interface between two isotropic elastic mediums—fluid, solid, or ether

§ 1. “Elastic solid” includes fluid and ether; except conceivable dynamics of the mutual action across the interface of the two mediums. Maxwell's electro-magnetic equations for a homogeneous non-conductor of electricity are identical with the equations of motion of an incompressible elastic solid, or with the equations expressing the rotational components of the motion of an elastic solid compressible or incompressible; but not so their application to a heterogeneous non-conductor or to the interface between two homogeneous non-conductors. § 2. The equations of equilibrium of a homogeneous elastic solid, under the influence of forces X, Y, Z, per unit volume, acting at any point ( x, y, z ) of the substance are given in Stokes' classical paper “On the Theories of the Internal Friction of Fluids in Motion, and of the Equilibrium and Motion of Elastic Solids,” p. 115, vol. i. of his Mathematical Papers ; also in Thomson and Tait's Natural Philosophy [§ 698 (5) (6)].

On the Reflexion and Refraction of Solitary Plane Waves at a Plane Interface between two Isotropic Elastic Mediums—Fluid, Solid, or Ether

§ 1. “Elastic solid” includes fluid and ether; except conceivable dynamics of the mutual action across the interface of the two mediums. Maxwell's electro-magnetic equations for a homogeneous non-conductor of electricity are identical with the equations of motion of an incompressible elastic solid, or with the equations expressing the rotational components of the motion of an elastic solid compressible or incompressible; but not so their application to a heterogeneous non-conductor or to the interface between two homogeneous non-conductors.§ 2. The equations of equilibrium of a homogeneous elastic solid, under the influence of forces X, Y, Z, per unit volume, acting at any point (x, y, z) of the substance are given in Stokes' classical paper “On the Theories of the Internal Friction of Fluids in Motion, and of the Equilibrium and Motion of Elastic Solids,” p. 115, vol. i. of his Mathematical Papers; also in Thomson and Tait's Natural Philosophy [§ 698 (5) (6)].

III. On the motion of a sphere in a viscous liquid

1. The first problem relating to the motion of a solid body in a viscous liquid which was successfully attacked was that of a sphere, the solution of which was given by Professor Stokes in 1850, in his memoir “On the Effect of the Internal Friction of Fluids on Pendulums,” ‘Cambridge Phil. Soc. Trans.,’ vol. 9, in the following cases: (i.) when the sphere is performing small oscillations along a straight line; (ii.) when the sphere is constrained to move with uniform velocity in a straight line; (iii.) when the sphere is surrounded by an infinite liquid and constrained to rotate with uniform angular velocity about a fixed diameter: it being supposed, in the last two cases, that sufficient time has elapsed for the motion to have become steady. In the same memoir he also discusses the motion of a cylinder and a disc. The same class of problems has also been considered by Meyer and Oberbeck, the latter of whom has obtained the solution in the case of the steady motion of an ellipsoid, which moves parallel to any one of its principal axes with uniform velocity. The torsional oscillations about a fixed diameter, of a sphere which is either filled with liquid or is surrounded by an infinite liquid when slipping takes place at the surface of the sphere, forms the subject of a joint memoir by Helmholtz and Piotrowski. Very little appears to have been effected with regard to the solution of problems in which a viscous liquid is set in motion in any given manner and then left to itself. The solution, when the liquid is bounded by a plane which moves parallel to itself, is given by Professor Stokes at the end of his memoir referred to above; and the solutions of certain problems of two-dimensional motion have been given by Stearn. In the present paper I propose to obtain the solution for a sphere moving in a viscous liquid in the following cases:—(i.) when the sphere is moving in a straight line under the action of a constant force, such as gravity; (ii.) when the sphere is surrounded by viscous liquid and is set in rotation about a fixed diameter and then left to itself.