scholarly journals A MATHEMATICAL MODEL OF NONSTATIONARY MOTION OF A VISCOELASTIC FLUID IN ROLLER BEARINGS

2019 ◽  
Vol 81 (4) ◽  
pp. 501-512
Author(s):  
I.A. Zhurba Eremeeva ◽  
D. Scerrato ◽  
C. Cardillo ◽  
A. Tran
Keyword(s):  
Mathematical Model ◽  
Viscoelastic Fluid ◽  
Viscoelastic Properties ◽  
Fluid Model ◽  
Maxwell Fluid ◽  
Initial Boundary ◽  
Deborah Number ◽  
Nonstationary Motion ◽  
Friction Forces ◽  
Roller Bearings

Nowadays, the emergence of new lubricants requires an enhancement of the rheological models and methods used for solution of corresponding initial boundary-value problems. In particular, models that take into account viscoelastic properties are of great interest. In the present paper we consider the mathematical model of nonstationary motion of a viscoelastic fluid in roller bearings. We used the Maxwell fluid model for the modeling of fluid properties. The viscoelastic properties are exhibited by many lubricants that use polymer additives. In addition, viscoelastic properties can be essential at high fluid speeds. Also, viscoelastic properties can be significant in the case of thin gaps. Maxwell's model is one of the most common models of viscoelastic materials. It combines the relative simplicity of constitutive equations with the ability to describe a stress relaxation. In addition, viscoelastic fluids also allow us to describe some effects that are missing in the case of viscous fluid. An example it is worth to mention the Weissenberg effect and a number of others. In particular, such effects can be used to increase the efficiency of the film carrier in the sliding bearings. Here we introduced characteristic assumptions on the form of the flow, allowing to significantly simplify the solution of the problem. We consider so-called self-similar solutions, which allows us to get a solution in an analytical form. As a result these assumptions, the formulae for pressure and friction forces are derived. Their dependency on time and Deborah number is analyzed. The limiting values of the flow characteristics were obtained. The latter can be used for steady state of the flow regime. Differences from the case of Newtonian fluid are discussed. It is shown that viscoelastic properties are most evident at the initial stage of flow, when the effects of non-stationarity are most important.

scholarly journals A MATHEMATICAL MODEL OF NONSTATIONARY MOTION OF A VISCOELASTIC FLUID IN ROLLER BEARINGS

2019 ◽  
Vol 81 (4) ◽  
pp. 500-511
Author(s):  
I.A. Zhurba Eremeeva ◽  
D. Scerrato ◽  
C. Cardillo ◽  
A. Tran
Keyword(s):  
Mathematical Model ◽  
Viscoelastic Fluid ◽  
Viscoelastic Properties ◽  
Fluid Model ◽  
Maxwell Fluid ◽  
Initial Boundary ◽  
Deborah Number ◽  
Nonstationary Motion ◽  
Friction Forces ◽  
Roller Bearings

Nowadays, the emergence of new lubricants requires an enhancement of the rheological models and methods used for solution of corresponding initial boundary-value problems. In particular, models that take into account viscoelastic properties are of great interest. In the present paper we consider the mathematical model of nonstationary motion of a viscoelastic fluid in roller bearings. We used the Maxwell fluid model for the modeling of fluid properties. The viscoelastic properties are exhibited by many lubricants that use polymer additives. In addition, viscoelastic properties can be essential at high fluid speeds. Also, viscoelastic properties can be significant in the case of thin gaps. Maxwell's model is one of the most common models of viscoelastic materials. It combines the relative simplicity of constitutive equations with the ability to describe a stress relaxation. In addition, viscoelastic fluids also allow us to describe some effects that are missing in the case of viscous fluid. An example it is worth to mention the Weissenberg effect and a number of others. In particular, such effects can be used to increase the efficiency of the film carrier in the sliding bearings. Here we introduced characteristic assumptions on the form of the flow, allowing to significantly simplify the solution of the problem. We consider so-called self-similar solutions, which allows us to get a solution in an analytical form. As a result these assumptions, the formulae for pressure and friction forces are derived. Their dependency on time and Deborah number is analyzed. The limiting values of the flow characteristics were obtained. The latter can be used for steady state of the flow regime. Differences from the case of Newtonian fluid are discussed. It is shown that viscoelastic properties are most evident at the initial stage of flow, when the effects of non-stationarity are most important.

scholarly journals On the flow of MHD generalized maxwell fluid via porous rectangular duct

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 989-1002
Author(s):  
Aamir Farooq ◽  
Muhammad Kamran ◽  
Yasir Bashir ◽  
Hijaz Ahmad ◽  
Azeem Shahzad ◽  
...  
Keyword(s):  
Fluid Model ◽  
Maxwell Fluid ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Rectangular Duct ◽  
Modified Darcy’S Law ◽  
Initial Boundary Value ◽  
Generalized Maxwell Fluid ◽  
Limiting Case ◽  
The Impact

Abstract The purpose of this proposed investigation is to study unsteady magneto hydrodynamic (MHD) mixed initial-boundary value problem for incompressible fractional Maxwell fluid model via oscillatory porous rectangular duct. Considering the modified Darcy’s law, the problem is simplified by using the method of the double finite Fourier sine and Laplace transforms. As a limiting case of the general solutions, the same results can be obtained for the classical Maxwell fluid. Also, the impact of magnetic parameter, porosity of medium, and the impact of various material parameters on the velocity profile and the corresponding tangential tensions are illuminated graphically. At the end, we will give the conclusion of the whole paper.

Perturbation Solution of Non-Newtonian Lubrication With the Convected Maxwell Model

2005 ◽  
Vol 127 (2) ◽  
pp. 302-305 ◽  
Author(s):  
Rong Zhang ◽  
Xueming He ◽  
Simon X. Yang ◽  
Xinkai Li
Keyword(s):  
Viscoelastic Fluid ◽  
Reynolds Equation ◽  
Fluid Model ◽  
Maxwell Model ◽  
Deborah Number ◽  
Perturbation Solution ◽  
Two Dimensional ◽  
Sliding Velocity ◽  
Regular Perturbation ◽  
Leading Term

There are many studies on variations of the Maxwell model. Tichy (1996) discussed an admissible formulation of the Maxwell viscoelastic fluid model using a convected derivative and applied it to two-dimensional lubrication flow. Tichy obtained a solution using a regular perturbation in the Deborah number with the conventional lubrication solution as the leading term. This paper extends Tichy’s model by using a double regular perturbation to the convected Maxwell model. A correspondence solution can also be obtained. Our sliding velocity solution is different from Tichy’s solution; and a modified Reynolds equation is also different from that by Tichy.

On the Measurement of Dynamic Properties of Lubricants under E. H. L. Conditions Using a Disc Machine

1974 ◽  
Vol 16 (5) ◽  
pp. 331-338 ◽  
Author(s):  
J. M. Davies ◽  
K. Walters
Keyword(s):  
Viscoelastic Properties ◽  
Dynamic Properties ◽  
Elastohydrodynamic Lubrication ◽  
Deborah Number ◽  
Theoretical Treatment

Johnson and Roberts (1)‡ have recently shown how a simple adaptation of the conventional disc machine can be used to measure the viscoelastic properties of lubricants under conditions of elastohydrodynamic lubrication. The present paper contains a theoretical treatment of the problem and derives operating formulae pertinent to conditions of low, finite and high Deborah number.

Analysis of the Motion and Deformation of a Bubble Rising in a Viscoelastic Fluid

2006 ◽  
Author(s):  
Shriram Pillapakkam ◽  
Pushpendra Singh ◽  
Denis L. Blackmore ◽  
Nadine Aubry
Keyword(s):  
Velocity Field ◽  
Newtonian Fluid ◽  
Viscoelastic Fluid ◽  
Bubble Radius ◽  
Polymer Concentration ◽  
Deborah Number ◽  
Two Phase ◽  
Bubble Volume ◽  
Recirculation Zones ◽  
Bubble Rising

A finite element code based on the level set method is developed for performing two and three dimensional direct numerical simulations (DNS) of viscoelastic two-phase flow problems. The Oldroyd-B constitutive equation is used to model the viscoelastic liquid and both transient and steady state shapes of bubbles in viscoelastic buoyancy driven flows are studied. The influence of the governing dimensionless parameters, namely the Capillary number (Ca), the Deborah Number (De) and the polymer concentration parameter c, on the deformation of the bubble is also analyzed. Our simulations demonstrate that the rise velocity oscillates before reaching a steady value. The shape of the bubble, the magnitude of velocity overshoot and the amount of damping depend mainly on the parameter c and the bubble radius. Simulations also show that there is a critical bubble volume at which there is a sharp increase in the bubble terminal velocity as the increasing bubble volume increases, similar to the behavior observed in experiments. The structure of the wake of a bubble rising in a Newtonian fluid is strikingly different from that of a bubble rising in a viscoelastic fluid. In addition to the two recirculation zones at the equator of the bubble rising in a Newtonian fluid, two more recirculation zones exist in the wake of a bubble rising in viscoelastic fluids which influence the shape of a rising bubble. Interestingly, the direction of motion of the fluid a short distance below the trailing edge of a bubble rising in a viscoelastic fluid is in the opposite direction to the direction of the motion of the bubble, thus creating a “negative wake”. In this paper, the velocity field in the wake of the bubble, the effect of the parameters on the velocity field and their influence on the shape of the bubble are also investigated.

On the Flow of a Polymer Melt Passing Over a Transverse Slot

1999 ◽  
Vol 19 (3) ◽  
pp. 175-196 ◽  
Author(s):  
G.H. Wu ◽  
C.K. Chen ◽  
S.H. Ju
Keyword(s):  
Fluid Model ◽  
Polymer Melt ◽  
Maxwell Fluid ◽  
Creeping Flow ◽  
Analytical Prediction ◽  
Hole Pressure ◽  
Fluid Elasticity ◽  
Conduit Wall ◽  
Liquid Flows ◽  
Transverse Slot

Abstract The phenomenon of hole pressure occurs whenever a polymeric or viscoelastic liquid flows over a depression in a conduit wall. Numerical simulations undertaken for the flow of an aqueous polyacrylamide melt passing over a transverse slot arc considered here. The fluid model used for this study is a White-Metzner constitutive equation describing the non-Newtonian behavior of the melt. The results were computed by an elastic-viscous split-stress finite clement method (EVSS-FEM). a mixed finite clement method incorporating the non-consistent streamline upwind scheme. For verification, the numerical algorithm was first applied to compute the corresponding flow of the upper-convected Maxwell fluid model, a special case of the Whitc-Metzner model characterized by constant viscosity and relation time. The resulting hole pressure (Ph) was evaluated for various Deborah numbers (De) and compared with the analytical prediction derived from the Higashitani-Pritchard (HP) theory. The agreement was found to be satisfactory for creeping flow in the low De range, for which the HP theory is valid. Subsequently, the hole pressure of this flow problem was predicted. The streamlines and pressure distribution along the channel walls arc also presented. Furthermore, the effects of fluid elasticity, shear thinning, the exponent in the viscosity function and the relaxation-time function, and slot geometry on the hole pressure were investigated.

Estimating the Interaction Kernel in a Mathematical Model for a Beam With Internal Damping Mechanism

1995 ◽  
Author(s):  
Tao Lin ◽  
David L. Russell
Keyword(s):  
Mathematical Model ◽  
Fiber Composite ◽  
Mass Density ◽  
Integral Kernel ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Transverse Displacement ◽  
Internal Damping ◽  
Interaction Kernel ◽  
Initial Boundary Value

Abstract Beams formed by long fiber composite materials have certain internal damping torque. A mathematical model for the displacement of this type of beams in cantilever configuration is the following initial-boundary value problem of an integro-differential equation [1, 14]: (1) ρ ( x ) w t t ( x , t ) − 2 ( ∫ 0 L h ( x , y ) [ w t x ( x , t ) − w t x ( y , t ) ] d y ) x + ( E I w x x ( x , t ) ) x x = f ( x , t ) , (2) w ( 0 , t ) = 0 , w x ( 0 , t ) = 0 , (3) w x x ( L , t ) = b l 1 ( t ) , (4) − ( E I w x x ( x , t ) ) x | x = L + 2 ∫ 0 L h ( L , y ) [ w t x ( L , t ) − w t x ( y , t ) ] d y = b l 2 ( t ) , (5) w ( x , 0 ) = w 0 ( x ) , w t ( x , 0 ) = w 1 ( x ) , where L is length of the beam, w(x, t) is the transverse displacement of the beam at time t and position x, ρ(x) is the mass density, EI is the stiffness parameter. The interaction integral kernel h(x, ξ) is introduced in this model by considering a restoring torque which comes from spatially variable bending of the beam. This kernel h(x, ξ) depends on the material properties of the beam. Choosing a different material (different h(x, ξ)) can realize a different damping effect for the beam.

Numerical Simulation of Boiling Flows Using an Eulerian Multi-Fluid Model

2005 ◽  
Author(s):  
J. Han ◽  
D.-M. Wang ◽  
D. Filipi
Keyword(s):  
Mathematical Model ◽  
Fluid Model ◽  
Bubble Diameter ◽  
Interfacial Area ◽  
Nucleate Boiling ◽  
Industrial Applications ◽  
Turbulent Dispersion ◽  
Steady State Mode ◽  
Simple Method ◽  
Bubble Number

A mathematical model to simulate boiling flows in industrial applications is presented. Following the Eulerian multifluid framework, separate sets of mass, momentum, and energy conservation equations are solved for liquid and vapor phases, respectively. The interactions between the phases are accounted for by including relevant mass, momentum, heat exchanges and turbulent dispersion effects. Velocity-pressure coupling is achieved through a multiphase version of the SIMPLE method and the standard k-ε turbulence model is employed. In order to validate and assess the accuracy of the boiling model, subcooled nucleate boiling flows in a vertical annular pipe are simulated in the steady-state mode. The computed axial velocities, volume fractions, temperature profiles are compared with available experimental data (Roy et al., ASME J. of Heat Transfer, Vol. 119, 1997). The result obtained by assuming a constant value for the bubble diameter shows a reasonable agreement, but several limitations are observed in the details. A more advanced mathematical model incorporating separate transport equations for the bubble number density and the interfacial area is suggested.

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