Let us consider a coaxial cylinder of radius r and thickness...

Let us consider a coaxial cylinder of radius $r$ and thickness $d r$, then force of friction or viscous force on this elemental layer, $F=2 \pi r \ln \frac{d v}{d r}$ This force must be constant from layer to layer so that steady motion may be possible. or, $\frac{F d r}{r}=2 \pi l \eta d v$ Integrating, $F \int_{R_{2}}^{r} \frac{d r}{r}=2 \pi l \eta \int_{0}^{v} d v$ or, $\quad F \ln \left(\frac{r}{R_{2}}\right)=2 \pi \ln v$ (2) Putting $r=R_{1}$, we get $F \ln \frac{R_{1}}{R_{2}}=2 \pi l \eta v_{0}$ From (2) by (3) we get, $v=v_{0} \frac{\ln r / R_{2}}{\ln R_{1} / R_{2}}$ Note : The force $F$ is supplied by the agency which tries to carry the inner cylinder with velocity $v_{0}$.

Let us consider a coaxial cylinder of radius $r$ and thickness $d r$, then force of friction or viscous force on this elemental layer, $F=2 \pi r \ln \frac{d v}{d r}$ This force must be constant from layer to layer so that steady motion may be possible.
or, $\frac{F d r}{r}=2 \pi l \eta d v$
Integrating,
$F \int_{R_{2}}^{r} \frac{d r}{r}=2 \pi l \eta \int_{0}^{v} d v$
or, $\quad F \ln \left(\frac{r}{R_{2}}\right)=2 \pi \ln v$
(2)
Putting $r=R_{1}$, we get
$F \ln \frac{R_{1}}{R_{2}}=2 \pi l \eta v_{0}$
From (2) by (3) we get, $v=v_{0} \frac{\ln r / R_{2}}{\ln R_{1} / R_{2}}$
Note : The force $F$ is supplied by the agency which tries to carry the inner cylinder with velocity $v_{0}$.

Key Concepts

Newton's Law of Viscosity

This fundamental principle in fluid mechanics states that the shear stress between adjacent fluid layers is directly proportional to the velocity gradient between those layers. It explains how the viscous force arises from the resistance of the fluid to deformation, providing the basis for relating the force per unit area to the rate of change of velocity in the fluid.

Shear Stress in Fluids

Shear stress represents the internal frictional force per unit area acting tangentially within a fluid due to layers moving at different velocities. In scenarios with concentric moving cylinders, it is key to relating the applied force to the velocity gradient across the fluid, thereby enabling the analysis of steady motion through a balance of viscous forces.

Boundary Conditions in Fluid Mechanics

Setting appropriate boundary conditions is critical in fluid flow problems to determine the unique solution. In the context of concentric cylinders, these conditions typically involve specifying the velocity at the surfaces of the cylinders, such as the inner cylinder moving with a constant velocity and the outer cylinder being stationary, ensuring a consistent and physically realistic velocity profile within the fluid.

Integration in Cylindrical Coordinates

When dealing with fluid flow in geometries like coaxial cylinders, integration in cylindrical coordinates becomes necessary to accumulate effects radially. This technique is used to derive relationships, such as the velocity profile, by integrating differential contributions of viscous forces across the radial distance, leading to expressions that often involve logarithmic terms which reflect the geometry of the system.

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