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Efecto de la Viscosidad
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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:
| $ v = v_{max} \left(1-\displaystyle\frac{ r ^2}{ R ^2}\right)$ |
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:
| $ v_{max} =-\displaystyle\frac{ R ^2}{4 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
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:
| $ J_V =-\displaystyle\frac{ \pi R ^4}{8 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
(ID 3178)
As the viscous force is
| $ F_v =- S \eta \displaystyle\frac{ \Delta v }{ \Delta z }$ |
and the surface area of the cylinder is
$S=2\pi R L$
where $R$ is the radius and $L$ is the length of the channel, the viscous force can be expressed as
| $ F_v =-2 \pi r \Delta L \eta \displaystyle\frac{ dv }{ dr }$ |
where $\eta$ represents the viscosity and $dv/dr$ is the velocity gradient between the wall and the flow.
(ID 3623)
When a the pressure difference ($\Delta p_s$) acts on a section with an area of $\pi R^2$, with the tube radius ($R$) as the curvature radio ($r$), it generates a force represented by:
$\pi r^2 \Delta p$
This force drives the liquid against viscous resistance, given by:
| $ F_v =-2 \pi r \Delta L \eta \displaystyle\frac{ dv }{ dr }$ |
By equating these two forces, we obtain:
$\pi r^2 \Delta p = \eta 2\pi r \Delta L \displaystyle\frac{dv}{dr}$
Which leads to the equation:
$\displaystyle\frac{dv}{dr} = \displaystyle\frac{1}{2\eta}\displaystyle\frac{\Delta p}{\Delta L} r$
If we integrate this equation from a position defined by the curvature radio ($r$) to the edge where the tube radius ($R$) (taking into account that the velocity at the edge is zero), we can obtain the speed on a cylinder radio ($v$) as a function of the curvature radio ($r$):
| $ v = v_{max} \left(1-\displaystyle\frac{ r ^2}{ R ^2}\right)$ |
Where:
| $ v_{max} =-\displaystyle\frac{ R ^2}{4 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
is the maximum flow rate ($v_{max}$) at the center of the flow.
(ID 3627)
Examples
The viscose force ($F_v$) can be calculated from the parallel surfaces ($S$), the viscosity ($\eta$), the speed difference between surfaces ($\Delta v$), and the distance between surfaces ($\Delta z$) using the following method:
| $ F_v =- S \eta \displaystyle\frac{ \Delta v }{ \Delta z }$ |
(ID 3622)
In the case of a cylinder, the surface is defined by ERROR:5430,0, and by the perimeter of each of the internal cylinders, which is calculated by multiplying $2\pi$ by the radius of position in a tube ($r$). With this, the cylinder resistance force ($F_v$) is calculated using the viscosity ($\eta$) and the variation of speed between two radii ($dv$) for the width of the cylinder the radius variation in a tube ($dr$), resulting in:
| $ F_v =-2 \pi r \Delta L \eta \displaystyle\frac{ dv }{ dr }$ |
(ID 3623)
To describe the flow, a coordinate system is defined in which the liquid flows from the position at the beginning of the tube ($L_i$) to the position at the end of the tube ($L_e$), indicating that the pressure at the pressure in the initial position ($p_i$) is greater than at the pressure in end position (e) ($p_e$). This movement depends on the tube length ($\Delta L$), which is calculated as follows:
| $ \Delta L = L_e - L_i $ |
(ID 3802)
$\eta_n=\eta_p\left(1+\displaystyle\frac{2}{5}Ht\right)$
(ID 3636)
When solving the flow equation with the boundary condition, we obtain the speed on a cylinder radio ($v$) as a function of the curvature radio ($r$), represented by a parabola centered at the maximum flow rate ($v_{max}$) and equal to zero at the tube radius ($R$):
| $ v = v_{max} \left(1-\displaystyle\frac{ r ^2}{ R ^2}\right)$ |
.
(ID 3627)
The value of the maximum flow rate ($v_{max}$) at the center of a cylinder depends on the viscosity ($\eta$), the tube radius ($R$), and the gradient created by the pressure difference ($\Delta p_s$) and the tube length ($\Delta L$), as represented by:
| $ v_{max} =-\displaystyle\frac{ R ^2}{4 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
The negative sign indicates that the flow always occurs in the direction opposite to the gradient, meaning from the area of higher pressure to the area of lower pressure.
(ID 3628)
(ID 1896)
$\eta_{fl}=\displaystyle\frac{\eta_n}{\left(1-\displaystyle\frac{d}{R}\right)^4}$
(ID 3638)
(ID 1895)
$\eta_d=\displaystyle\frac{\eta_n}{1+C_{\sigma}\displaystyle\frac{d\sigma}{dt}}$
(ID 3637)
(ID 1695)
The volume flow ($J_V$) can be calculated with the Hagen-Poiseuille law that with the parameters the viscosity ($\eta$), the pressure difference ($\Delta p$), the tube radius ($R$) and the tube length ($\Delta L$) is:
| $ J_V =-\displaystyle\frac{ \pi R ^4}{8 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
(ID 3178)
ID:(328, 0)