Many electrorheological fluids (ERFs) as fluids with microstructure demonstrate viscoplastic behaviours. Rheometric measurements indicate that some flows of these fluids may be modelled as the flows of a Herschel-Bulkley fluid. In this paper, the flow of a Herschel-Bulkley ER fluid—with a fractional power-law exponent—in a narrow clearance between two fixed surfaces of revolution with common axis of symmetry is considered. The flow is externally pressurized, and it is considered with inertia effect. In order to solve this problem, the boundary layer equations are used. The influence of inertia forces on the pressure distribution is examined by using the method of averaged inertia terms of the momentum equation. Numerical examples of externally pressurized ERFs flows in the clearance between parallel disks and concentric spherical surfaces are presented.

In recent years, the study of fluids with microstructures has gained much importance because of its numerous applications in various engineering disciplines such as chemical engineering, polymer processing, plastic forming foundry engineering, and engineering of lubrication [

In machines and mechanisms systems of many industrial processes, the phenomena of a flow of viscoplastic fluids are used. One of these phenomena is a slide bearing lubrication [

Most substances used in the lubrication technology are polymer solutions, thus, the characteristics of the bearings change when such rheological substances, known as non-Newtonian fluids, are used as lubricants. Several constitutive relations applied were used to model the non-Newtonian characteristics exhibited by some lubricants [

Another ones of these phenomena are processes of vibration control and torque transmission. In the last years, the electrorheological fluids (abbreviated to ERFs) have acquired a great relevance for supporting vibration control and torque transmission devices, based on the characteristic dependence of their viscosity on applied electric field strength. Since their initial discovery by Winslow [

To describe the rheological behaviour of viscoplastic fluids in complex geometries, the Bingham model is used [

The most popular model of the ERF is the Bingham model for which

The purpose of this paper is a study of pressure distribution in a flow of the Herschel-Bulkley ERF—with a fractional power-law exponent [

The geometry of a curvilinear clearance between two surfaces of revolution.

The yield shear stress for the ERF varies with respect to the electric field. According to the experimental results reported in Shulman et al. [

The three-dimensional constitutive equation for the Herschel-Bulkley ERF has the form [

The general equations of motion of the Herschel-Bulkley fluid have a form:

equation of continuity:

equation of momentum

Let us consider the Herschel-Bulkley ERF in a clearance between fixed surfaces of revolution. The flow configuration is shown in Figure

The physical parameters of the lubricant flow are the velocity components

The assumption typical for the flow in a narrow clearance [

A further simplification comes by noting that—in accordance to the lubrication approximation—the most important changes in an annular channel occur in the normal (to the channel median) direction. This leads to the assumption that the flow is nearly parallel to the surfaces bounding the clearance, so that

If some asymptotic transformations are made, the same as in (Falicki [

The order-of-magnitude arguments show that

For a majority of greases, molten polymers, mush metals, and ERFs, the values of a yield shear stress are contained in the limits:

In the flow of a fluid with the yield shear stress, there exists a quasi-solid core flow bounded by surfaces laying at

Taking into account (

Its analytical solution exists only for large

For intermediate values of

Curves illustrating the progression of

Curves illustrating the progression of

The functions

for large values of

for large values of

for small values of

for large values of

for large values of

for small values of

Taking into account the results obtained in the previous section we will present the pressure distribution in the clearance of constant thickness between two parallel disks as shown in Figure

Clearance between two parallel disks.

Nondimensional formula for pressure distribution in ER flow of the Herschel-Bulkley fluid has the form

Note that for

Pressure distributions

Pressure distributions

Pressure distributions

Pressure distributions

Pressure distributions

Let us consider now the pressure distribution in the clearance of constant thickness between two concentric spheres surfaces shown in Figure

Clearance between concentric spherical surfaces.

The pressure distributions for the Herschel-Bulkley ERF flow in a clearance between concentric spherical surfaces are presented in Figures

Pressure distributions

Pressure distributions

Pressure distributions

Pressure distributions

From the general considerations, formulae and graphs presented here for the Herschel-Bulkley ERF flows in the narrow clearance of constant thickness between parallel disks and concentric spherical surfaces shown in Figures

decrease with the increase of the modified Reynolds number

increase with the decrease of the nonlinearity index

increase with the decrease of the de Saint-Venant ER number

are larger between concentric spherical surfaces than these ones between parallel disks for

for

For small values of the de Saint-Venant ER number

Generally, it may be concluded that for the Herschel-Bulkley ERF flows in the clearance between two surfaces of revolution the pressure values

are larger for the flow in the clearances with curvilinear generating lines than these ones for the flow in clearances with rectilinear generating lines for the nonlinearity index

Note that the results obtained here for

The formulae for the pressure distribution obtained previously may be used to model its distribution in slide thrust bearings—of arbitrary curvilinear shapes—lubricated by ER lubricants.

Auxiliary function defined by formulae

(_{1} or (_{1}

Auxiliary function defined by formula

(_{2}

Auxiliary function defined by formula

(_{2} or

Half of the fluid film thickness

Auxiliary function defined by formula

(_{8}

Auxiliary function defined by formula

(_{6}

Auxiliary function defined by formula

(_{3}

Plasticity number

De Saint-Venant plasticity number

Power-law index

Pressure

Flow rate

Local radius

Inlet radius to the bearing clearance

Outlet radius from the bearing clearance

Modified Reynolds number

Signum function

Auxiliary function defined by formulae

(_{1},

Component of the shear stress tensor

Applied voltage

Velocity components

ER experimental constants

Fluid density.

Central angle of spherical surface

Shear stress

Yield shear stress.