(ID 939)

If we consider the profile of ERROR:5449,0 for a fluid in a cylindrical channel, where the speed on a cylinder radio ($v$) varies with respect to ERROR:10120,0 according to the following expression:

involving the tube radius ($R$) and the maximum flow rate ($v_{max}$). We can calculate the maximum flow rate ($v_{max}$) using the viscosity ($\eta$), the pressure difference ($\Delta p$), and the tube length ($\Delta L$) as follows:

If we integrate the velocity across the cross-section of the channel, we obtain the volume flow ($J_V$), defined as the integral of $\pi r v(r)$ with respect to ERROR:10120,0 from $0$ to ERROR:5417,0. This integral can be simplified as follows:

$J_V=-\displaystyle\int_0^Rdr \pi r v(r)=-\displaystyle\frac{R^2}{4\eta}\displaystyle\frac{\Delta p}{\Delta L}\displaystyle\int_0^Rdr \pi r \left(1-\displaystyle\frac{r^2}{R^2}\right)$

The integration yields the resulting Hagen-Poiseuille law:

(ID 3178)

The volume flow ($J_V$) can be calculated from the hydraulic conductance ($G_h$) and the pressure difference ($\Delta p$) using the following equation:

Furthermore, using the relationship for the hydraulic resistance ($R_h$):

results in:

(ID 3179)

One way to model a tube with varying cross-section is to divide it into sections with constant radius and then sum the hydraulic resistances in series. Suppose we have a series of the hydraulic resistance in a network ($R_{hk}$), which depends on the viscosity ($\eta$), the cylinder k radio ($R_k$), and the tube k length ($\Delta L_k$) via the following equation:

In each segment, there will be a pressure difference in a network ($\Delta p_k$) with the hydraulic resistance in a network ($R_{hk}$) and the volume flow ($J_V$) to which Darcy's Law is applied:

the total pressure difference ($\Delta p_t$) will be equal to the sum of the individual ERROR:10132,0:

therefore,

$\Delta p_t=\displaystyle\sum_k \Delta p_k=\displaystyle\sum_k (R_{hk}J_V)=\left(\displaystyle\sum_k R_{hk}\right)J_V\equiv R_{st}J_V$

Thus, the system can be modeled as a single conduit with the hydraulic resistance calculated as the sum of the individual components:

(ID 3180)

The parallel total hydraulic conductance ($G_{pt}$) combined with the hydraulic conductance in a network ($G_{hk}$) in

and along with the hydraulic resistance in a network ($R_{hk}$) and the equation

leads to the total hydraulic resistance in parallel ($R_{pt}$) via

(ID 3181)

(ID 3469)

(ID 3624)

(ID 3625)

(ID 3626)

Since the hydraulic resistance ($R_h$) is equal to the hydraulic conductance ($G_h$) as per the following equation:

and since the hydraulic conductance ($G_h$) is expressed in terms of the viscosity ($\eta$), the tube radius ($R$), and the tube length ($\Delta L$) as follows:

we can conclude that:

(ID 3629)

(ID 3632)

(ID 3635)

(ID 4256)

(ID 4257)

(ID 4347)

(ID 4502)

(ID 4503)

(ID 4833)

(ID 4834)