Governing equations for a rigid particle in a flowing liquid. Drag force

=k

T_i , (2.54)

where k is another instrument constant which can be determined using a liquid of known viscosity. the slope and used for evaluation of η for other liquids from similar plots [7].

The shear rate in the gap between the cylinders can be defined as 2 L r2

It means for a viscometer in which radius of outer cylinder is twice bigger than radius of inner cylinder there will be a fourfold variation in shear rate across the gap. For investigating the flow of colloidal suspensions where the viscosity is often a function of shear rate such a device would not be appropriated. That would make η be a function of the radius and the above analysis than is invalid. Taking account of a varying viscosity in the gap, it is possible to modify the analysis by keeping the gap width, d, as small as possible. The shear rate is then approximately constant and is determined as

d

⋅R

= Ω

γ . (2.56)

The radius of commercial viscometers is usually several centimeters and a gap width is 1 mm or less. Often these viscometers have an inner cylinder rotating, due to reasons of convenience, and it gives rise to instabilities for liquids of low viscosity like water, even at low speeds. It is necessary to make an end-correction because the flow regime in the annulus between the cylinders of the Couette viscometer is different from the flow in the bottom [7].

There are two general types of instruments. Some maintain a known speed of rotation (that is a constant shear rate) and have some kind of strain gauge which allows determining the shear stress. Others determine the consequent rotational speed by applying a constant stress. To the behaviour at small deformations the instruments of the second type are usually more sensitive.

Both kinds can display the results either as a meter reading or a digital readout and can be electronically automated [7].

2.10 Forces affecting on a particle in a shear flow

In equilibrium position, the forces acting on the bubble, particle – buoyancy, viscous drag, added mass, inertial (or pressure gradient) force and lift – exactly balance each other.

2.10.1 Governing equations for a rigid particle in a flowing liquid. Drag force

An investigating of the terminal falling velocity of particles in stationary and moving fluid flows is needed in a wide range of process engineering applications, for instance liquid – solid separations, fluidization and transportation of solids, falling ball viscometry, drilling applications. The size, shape, and density of particles, its orientation, properties of the liquid medium (density, rheology), size and shape of the fall vessels, and whether the liquid is

stationary or moving, that are variables which influence on the terminal falling velocity of particle [10].

A rigid spherical particle of diameter d, which is schematically represented in Figure 2.14, is moving relative to an incompressible fluid of infinite extent with a steady velocity υ.

Figure 2.14 A rigid particle moving in a liquid [10]

Consider the symmetrical two-dimensional flow. The velocity vector, υ, has φ-component which is equal to zero, and there is no variation of flow variables with φ; hence, one can be written

( )

Introduce a stream function, ψ, determined as

θ

The conservation of mass and the Newton’s second law of motion are two fundamental physical laws, which govern the steady motion of a sphere in an incompressible fluid under isothermal conditions. So-called continuity and momentum equations obtained from these laws to an infinitesimal control volume of a fluid are, respectively, written as [10]

=0 where p is the nongravitational pressure.

For this flow problem it is necessary to set the appropriate boundary conditions which are no-slip at the sphere surface, and the free stream velocity far away from the sphere. Assume a reference frame fixed to the particle with the origin at its center and write these boundaries as:

At r = R,

θ To define the problem completely a rheological equation of state has to be written in addition to the field equations and the boundary conditions (Eq. (2.59) – Eq. (2.61)) to relate the components of the extra stress tensor to that of the rate of deformation tensor for the fluid. This leads to the viscosity term appearing in the momentum equation (Eq. (2.60)) to be shown in terms of the relevant velocity components and their gradients.

This will enable the viscosity term appearing in the momentum Eq. (2.59) to be expressed in terms of the relevant velocity components and their gradients. However, it is possible to make some progress without choosing an equation of state at this stage. There are two unknowns the velocity and pressure fields (υ and p) in the equation. To solve the problem for these two unknowns Eq. 2.60 with the boundary conditions (Eq. (2.60) – Eq. (2.61)) are sufficient [10].

Knowing the pressure and velocity fields, the drag force acting on a moving particle can be estimated as

The two components are known as the form or pressure and the friction drag are represented, respectively, on the RHS of Eq. (2.62).

Reynolds number is determined as

ref

For developing a general formulation applicable to the flow of time-independent fluids a reference viscosity, μref, has been introduced here. Rewrite the drag force in dimensionless form by introducing a drag coefficient CD as

2

The simplest realistic class of materials is represented by the Newtonian fluid and Stokes was the first who considered the hydrodynamic behavior of rigid spheres in Newtonian media in 1851.

The rheological equation of state for an incompressible fluid can be written as [10]

ij

ij µ ε

τ =2⋅ ⋅ , i,j=r,θ,ϕ, (2.65)

where ε_ij represents the components of the rate of deformation tensor.

In a spherical coordinate system these are related to the two nonzero components of the velocity vector (υ_r,υ_θ) as following

r

Rewrite the momentum equation in terms of ψ^∗ using equations and obtain θ θ

The reference viscosity, μref, is equal to µ, the Newtonian viscosity in this case, and this leads to the regular definition of the Reynolds number.

Only approximate solutions are available for the Eq. (2.67). Due to its highly nonlinear form the general solution is impossible. For instance, the nonlinear inertial terms can be dropped by using creeping flow approximation and Eq. (2.67) can be reduced to the form

* 0

4 ⋅ψ =

E . (2.69)

Stokes obtained a solution for the fourth-order partial differential equation in 1851, and the resulting expression for the so-called Stokes drag is defined as

υ

4 , respectively. Eq. (2.71) is determined in a more familiar form as [10]

Re

= 24

CD ^. (2.71)