Laminar Flow Fluid Element Analysis

To analyze a fluid element flowing through a cylindrical pipe under fully developed laminar flow, we apply Newton’s Second Law to a cylindrical volume of fluid.

In the fully developed laminar flow, The pressure difference ($p_1-p_2$) across the length ($\ell$) of the fluid element creates a net driving force. This pressure force is balanced by shear stress $\tau$ acting on the walls of the pipe.

This force balance is written as

$$(p_1-p_2)\pi r^2=\tau \cdot 2\pi r\ell$$

where $r$ is the radius at any point within the pipe, $l$ is the length of the cylindrical element, and $\tau$ is the shear stress.

The relationship rearranges to express shear stress as a function of the radius:

$$\frac{\Delta p}{\ell }=\frac{2\tau }{r}⟹\tau =C\cdot r$$

which shows that shear stress linearly proportional to the radius $r$. The constant $C=\frac{2{\tau }_w}{D}$ is introduced to express shear stress independently of $r$. At the centerline, $r=0$, which gives

$$\tau =\frac{2{\tau }_w}{D}\cdot r=0$$

At the pipe wall, $r=D/2$,

$$\tau =\frac{2{\tau }_w}{D}\cdot \frac{D}{2}={\tau }_w$$

showing that shear stress indeed reaches its maximum value of ${\tau }_w$.

We can also connect the maximum shear stress at the wall to the overall pressure drop along the pipe’s length:

$$\Delta p=\frac{4l{\tau }_w}{D}$$

Velocity Profile Derivation

The velocity profile for fully developed laminar flow is derived from the derivation of shear stress in terms of velocity gradient:

$$\tau =\mu \frac{dy}{dy}=-\mu \frac{du}{dr}$$

or

$$\begin{array}{rl}\tau =-\mu \frac{du}{dr}& =\frac{\Delta pr}{2l}\\ \frac{du}{dr}& =-\frac{\Delta pr}{2\mu l}\\ \int du& =-\frac{\Delta pr}{2\mu l}\int rdr\\ u& =-\frac{\Delta p}{4\mu l}{r}^2+C_1\end{array}$$

This equation represents a parabolic boundary profile. We can use $\mu =0$ and $r=D/2$ as boundary conditions:

$$\begin{array}{rl}{C}_1& =\frac{\Delta p{D}^2}{16\mu l}\\ \mu & =\frac{\Delta p{D}^2}{16\mu l}\left[1-{\left(\frac{2r}{D}\right)}^2\right]\\ V_c& =\frac{\Delta p{D}^2}{16\mu l}\end{array}$$

where $V_C$ is the maximum velocity, occurring at the centerline.

The velocity profile at any point is then by:

$$\begin{array}{rl}\mu & =V_C\left[1-{\left(\frac{r}{R}\right)}^2\right]\\ & =\frac{{\tau }_{w}D}{4\mu }\left[1-{\left(\frac{r}{R}\right)}^2\right]\end{array}$$

Flow Rate Calculation

The volumetric flow rate $Q$ is determined by integrating the velocity profile across the pipe’s cross-sectional area:

$$\begin{array}{rl}Q& ={\int }_0^{R}u\cdot 2\pi rdr\\ & =\frac{\pi D^4\Delta p}{128\mu \ell }\end{array}$$

This represents the volume of fluid passing through the pipe per unit time. The average velocity is defined by the flow rate divided by the cross sectional area:

$$\overline{V}=\frac{Q}{A}=\frac{V_C}{2}=\frac{\Delta p{D}^2}{32\mu l}$$

In this way we can also write the pressure drop over a given pipe length as

$$\Delta p=\frac{16\mu l\overline{V}}{D^2}$$

Darcy Friction Factor

For laminar flow, the Darcy Friction factor $f$ quantifies the friction resistance

$$f=\frac{\delta p}{\frac{1}{2}\rho {\overline{V}}^2}\frac{D}{\ell }$$

or

$$\delta p=f\frac{l}{D}\frac{\rho {\bar{V}}^2}{2}$$

For fully developed laminar flow in a pipe, the friction factor simplifies to:

$$f=\frac{64}{\text{Re}}$$