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Viscose force
Equation
When a liquid with viscosity $\eta$ flows between two surfaces $S$ at a distance $dz$ with a velocity difference $dv_x$, it experiences a viscous force $F_v$ given by the Newton's law of viscosity:
 $ F_v =- S \eta \displaystyle\frac{ \Delta v }{ \Delta z }$ |
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Viscose force, cylinder case
Equation
A viscose force ($F_v$) generated by a liquid with viscosity ($\eta$) between some parallel surfaces ($S$) and a distance between surfaces ($\Delta z$), along with a speed difference between surfaces ($\Delta v$) and the speed on a cylinder radio ($v$), is calculated as
| $ F_v =- S \eta \displaystyle\frac{ \Delta v }{ \Delta z }$ |
In the case of a cylinder, the surface is defined by tube length ($\Delta L$), 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 }$ |
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, 0)
Change in length
Equation
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 $ |
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Einstein Viscosity Model of the Blood
Equation
$\eta_n=\eta_p\left(1+\displaystyle\frac{2}{5}Ht\right)$
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Speed profile of flow in a cylinder
Equation
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 cylinder radio ($R$):
| $ v = v_{max} \left(1-\displaystyle\frac{ r ^2}{ R ^2}\right)$ |
When a the pressure difference ($\Delta p$) acts on a section with an area of $\pi R^2$, with the cylinder radio ($R$) 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 cylinder radio ($R$) (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.
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ID:(3627, 0)
Maximal speed of flow in a cylinder
Equation
The value of the maximum flow rate ($v_{max}$) at the center of a cylinder depends on the viscosity ($\eta$), the cylinder radio ($R$), and the gradient created by the pressure difference ($\Delta p$) 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, 0)
Viscosity with Fahraeus-Lindqvist Effect
Equation
$\eta_{fl}=\displaystyle\frac{\eta_n}{\left(1-\displaystyle\frac{d}{R}\right)^4}$
ID:(3638, 0)
Viscosity of Deformed Hemocytes
Equation
$\eta_d=\displaystyle\frac{\eta_n}{1+C_{\sigma}\displaystyle\frac{d\sigma}{dt}}$
ID:(3637, 0)
Hagen Poiseuille Equation
Equation
If we examine the profile of the speed on a cylinder radio ($v$) for a fluid within a cylindrical channel of radius cylinder radio ($R$), in which the speed on a cylinder radio ($v$) varies as a function of ($$), we can integrate it across the entire cross-section of the channel:
$J_V= \pi \displaystyle\int_0^Rdr r v(r)$
This leads to the Hagen-Poiseuille law with parameters the volume flow ($J_V$), the viscosity ($\eta$), the pressure difference ($\Delta p$), and the tube length ($\Delta L$):
| $ J_V =-\displaystyle\frac{ \pi R ^4}{8 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
If we consider the profile of speed on a cylinder radio ($v$) for a fluid in a cylindrical channel, where the speed on a cylinder radio ($v$) varies with respect to radius of position in a tube ($r$) according to the following expression:
| $ v = v_{max} \left(1-\displaystyle\frac{ r ^2}{ R ^2}\right)$ |
involving the cylinder radio ($R$) 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 radius of position in a tube ($r$) from $0$ to cylinder radio ($R$). 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 }$ |
The original papers that gave rise to this law with a combined name were:
• Gotthilf Hagen: "Ueber die Gesetze, welche des der Strom des Wassers in röhrenförmigen Gefässen bestimmen" (On the laws governing the flow of water in cylindrical vessels), Annalen der Physik und Chemie 46:423442 (1839).
• Jean-Louis-Marie Poiseuille: "Recherches expérimentales sur le mouvement des liquides dans les tubes de très-petits diamètres" (Experimental research on the movement of liquids in tubes of very small diameters), Comptes Rendus de l'Académie des Sciences 9:433544 (1840).
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