# Tikaram's work

### Effective viscosity of the suspension

Fluid is a collection of many molecules that are randomly distributed and attached by weak cohesive forces. It isn't easy to study the motion of every individual particle of the fluid as a function of time. So, we describe the properties of the fluid at each point as a function of time. The motion of the fluid is divided into two groups: laminar and turbulent flow. Laminar flow is defined by $\Delta\times v=0$, which tells that the rotation of particles is zero and flow is along streamlines.
For the turbulent, the curl of the velocity field is non-zero; there is a rotation of streamlines and production of vortices.
Reynolds number is the dimensionless quantity that gives the information about the flow characteristics
Reynolds number is defined by the ratio of inertia force to viscous force;
\begin{equation}
Re=\frac{\rho v d}{\mu}
\end{equation}
where $d$ is the characteristic dimension and $\mu$
is the dynamic viscosity of the fluid.

#### Newtons law of viscosity
Viscosity is defined as the fluid resistance to a shear force. Newton’s
law of viscosity is defined as the
shear stress = $\mu \times$velocity gradient
or
\begin{equation}
\tau=\mu\frac{du}{dy}
\end{equation}
the above relation is valid for the newtonian fluid.

### viscosity of suspension

Suspension is the heterogeneous mixture of fluid having suspended rigid particles on it. The properties of suspension also depend on the suspended particles.
##### Various available relation for calculation of suspension viscosity

Einstein , who derived for the relative viscosity of suspensions of rigid spheres the equation:
\begin{equation}
\mu_r = \frac{\mu}{\mu_o} = 1 + \zeta \phi
\end{equation}
\&
\begin{equation}
\phi=\frac{4/3 \pi R^3N}{V_{fluid}}
\end{equation}
where $\mu$ is the viscosity of the suspension, $\mu_o$ is the viscosity of the pure liquid,$\zeta$ is the fitting constant determined experimently the value
$\zeta = 2.5$ is valid for lower volume fraction, and $\phi$ is the volume fraction . The derivation introduces some simplifications which are valid only for dilute limit.
In the dilute limit, the velocity disturbance does not affect the other suspended particle velocity.
Einstein's Equation is only valid for the low volume fraction of particles, another researcher Mooney considered higher volume fractions. Mooney quickly realized that, at higher concentrations, the empirical data for relative viscosity appear to follow more of an exponential function of volume fraction. Later Batchelor expanded newtons Equation to include second-order terms ( $\phi^2$). Again, he brought to light the fact that the particles interact with each other, and those interactions will affect the viscosity of the entire system.
so according to his empirical data Batchelor modifies the einstein's equation as follows
\begin{equation}
\mu_r=1+\zeta \phi +\alpha \phi^2
\end{equation}
The term $\alpha$ is the fitting parameter calculated based on the probability density, p(r, T) determined by three-sphere interactions, Brownian motion, or the assumption of some particle's initial state. The probability density represents the vector $r$separating the centers of the two particles. The values of alpha can range from $5.2 < \alpha < 7.6$. The linear coefficient $\zeta =2.5$ is taken to agree with the Einstein relation.