Storyboard

In saturated soil, there can be situations where pressure variations occur. These variations, in turn, generate flow that, in this case, should occur within the soil pores. Since these pores are on the order of microns or tens of microns, the flow tends to be laminar due to the low Reynolds numbers.

>Model

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Concept

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Concept

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The solution obtained for the height and the parameters the flow at a reference point ($j_{s0}$) and the reference height of the water column ($h_0$) shows us that the flux density ($j_s$) is equal to:

We can graphically represent the flux density ($j_s$) in terms of the additional factors $j_s/j_{s0}$ and $x/x_0$ as follows:

the flux density ($j_s$) continues to increase as we approach the channel, as the height of the water column on the ground ($h$) decreases. This increase is necessary to maintain the flow velocity in the flux density ($j_s$) or, alternatively, to increase it.

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Concept

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When a tube filled with liquid with a viscosity of viscosity ($\eta$) is exposed to the pressure in the initial position ($p_i$) at the position at the beginning of the tube ($L_i$) and the pressure in end position (e) ($p_e$) at the position at the end of the tube ($L_e$), it generates a pressure difference ($\Delta p$) along the tube length ($\Delta L$), resulting in the profile of the speed on a cylinder radio ($v$):

In flows with low values of the number of Reynold ($Re$), where viscosity is more significant than the inertia of the liquid, the flow develops in a laminar manner, meaning without the presence of turbulence.

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Concept

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In laminar flow, adjacent layers move, and there exists a force generated by viscosity between them. The faster layer drags its slower neighbor, while the slower one restricts the advancement of the faster one.

Therefore, the force the viscose force ($F_v$) generated by ($$) over the other is a function of ($$), ($$), and ($$), as depicted in the following equation:

illustrated in the following diagram:

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Concept

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Laminar flow around a cylinder can be represented as multiple cylindrical layers sliding under the influence of adjacent layers. In this case, the viscose force ($F_v$) with the tube length ($\Delta L$), the viscosity ($\eta$), and the variables the cylinder radial position ($r$) and the speed on a cylinder radio ($v$) is expressed as:

The layer at the boundary at ($$) remains stationary due to the boundary effect and, through the viscosity ($\eta$), slows down the adjacent layer which does have velocity.

The center is the part moving at the maximum flow rate ($v_{max}$), dragging the surrounding layer. In turn, this layer drags the next one, and so on until reaching the layer in contact with the cylinder wall, which is stationary.

Thus, the system transfers energy from the center to the wall, generating a velocity profile represented by:

with:

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Concept

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The profile of the speed on a cylinder radio ($v$) in the radius of position in a tube ($r$) allows us to calculate the volume flow ($J_V$) in a tube by integrating over the entire surface, which leads us to the well-known Hagen-Poiseuille law.

The result is an equation that depends on cylinder radio ($R$) raised to the fourth power. However, it is crucial to note that this flow profile only holds true in the case of laminar flow.

Thus, from the viscosity ($\eta$), it follows that the volume flow ($J_V$) before ($$) and ($$), the expression:

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Concept

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During ($$), the fluid with ($$) moves ($$). If the section ($S$) is the amount of fluid that passes through the section ($S$) in the infinitesimal time ($dt$), it is calculated as:

$dV = S ds = Sv dt$

This equation indicates that the volume of fluid flowing through the section ($S$) in ($$) is equal to the product of the cross-sectional area and the distance traveled by the fluid in that time. This allows for the calculation of the amount of liquid flowing through the channel within a specific time interval.

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Concept

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Selected parameter

Calculations

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Equation

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The key criterion for determining whether a medium is laminar or turbulent is the Reynolds number, which compares the energy associated with inertia to that associated with viscosity. The former depends on the liquid density ($\rho_w$), maximum Speed ($v_{max}$), and the typical Dimension of the System ($R$), while the latter depends on the viscosity ($\eta$), defining it as:

$ Re =\displaystyle\frac{ \rho R v }{ \eta }$

Conditions

Variables

$\rho$

Liquid density

$kg/m^3$

$v$

Maximum Speed

$m/s$

$Re$

Number of Reynold

$-$

$R$

Typical Dimension of the System

$m$

$\eta$

Viscosity

$Pa s$

Relation

Development

The inertia of a medium can be understood as proportional to the density of kinetic energy, given by:

$\displaystyle\frac{\rho_w}{2}v^2$

where the liquid density ($\rho_w$) and the mean Speed of Fluid ($v$) are.

If we consider the viscose force ($F_v$) as:

$F_v=S\eta\displaystyle\frac{v}{R}$

where the section or Area ($S$), the viscosity ($\eta$), the mean Speed of Fluid ($v$), and the typical Dimension of the System ($R$) are properties of the medium.

Let's recall that energy equals the viscose force ($F_v$) multiplied by the distance traveled ($l$). The density of energy lost due to viscosity will be equal to the force multiplied by the distance divided by the volume $S l$:

$\displaystyle\frac{F_vl}{Sl}=S\eta\displaystyle\frac{v}{R}\displaystyle\frac{l}{Sl}=\eta\displaystyle\frac{v}{R}$

Therefore, the relationship between the density of kinetic energy and the density of viscous energy is equal to a dimensionless number known as the number of Reynold ($Re$). If the number of Reynold ($Re$) is several orders of magnitude greater than one, inertia dominates over viscous force and the flow becomes turbulent. On the other hand, if the number of Reynold ($Re$) is small, viscous force dominates and the flow is laminar.

$ Re =\displaystyle\frac{ \rho R v }{ \eta }$

The original article in which Osborne Reynolds introduces the number bearing his name is:

"An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels," authored by Osborne Reynolds, published in Philosophical Transactions of the Royal Society of London, Vol. 174, pp. 935-982 (1883).

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Equation

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When the pressure in the initial position ($p_i$) and the pressure in end position (e) ($p_e$) are connected, a the pressure difference ($\Delta p$) is created, which is calculated using the following formula:

$ \Delta p = p_e - p_i $

Conditions

Variables

$\Delta p$

Pressure difference

$Pa$

$p_e$

Pressure in end position (e)

$Pa$

$p_i$

Pressure in the initial position

$Pa$

Relation

the pressure difference ($\Delta p$) represents the pressure difference that will cause the liquid to flow from the taller column to the shorter one.

ID:(14459, 0)

Equation

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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 $

Conditions

Variables

$\Delta L$

Body length

$m$

$L_i$

Position at the beginning of the tube

$m$

$L_e$

Position at the end of the tube

$m$

Relation

ID:(3802, 0)

Equation

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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 }$

Conditions

Variables

$\Delta z$

Distance between surfaces

$m$

$S$

Section

$m^2$

$\Delta v$

Speed difference between surfaces

$m/s$

$F_v$

Viscose force

$N$

$\eta$

Viscosity

$Pa s$

Relation

ID:(3622, 0)

Equation

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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

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 }$

Conditions

Variables

$r$

Cylinder radial position

$m$

$R$

Cylinder radio

$m$

$\pi$

Pi

3.1415927

$rad$

$v$

Speed on a cylinder radio

$m/s$

$\Delta L$

Tube length

$m$

$F_v$

Viscose force

$N$

$\eta$

Viscosity

$Pa s$

Relation

Development

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)

Equation

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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)$

Conditions

Variables

$r$

Cylinder radial position

$m$

$R$

Cylinder radio

$m$

$v_{max}$

Maximum flow rate

$m/s$

$v$

Speed on a cylinder radio

$m/s$

Relation

Development

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.

.

ID:(3627, 0)

Equation

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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 }$

Conditions

Variables

$R$

Cylinder radio

$m$

$v_{max}$

Maximum flow rate

$m/s$

$\Delta L$

Tube length

$m$

$\Delta p$

Variación de la Presión

$Pa$

$\eta$

Viscosity

$Pa s$

Relation

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)

Equation

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The volume flow ($J_V$) corresponds to the quantity volume ($V$) that flows through the channel during a time ($t$). Therefore, we have:

$ J_V =\displaystyle\frac{ dV }{ dt }$

Conditions

Variables

$t$

Time

$s$

$V$

Volume

$m^3$

$J_V$

Volume flow

$m^3/s$

Relation

ID:(12713, 0)

Equation

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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 }$

Conditions

Variables

$R$

Cylinder radio

$m$

$\pi$

Pi

3.1415927

$rad$

$\Delta L$

Tube length

$m$

$\Delta p$

Variación de la Presión

$Pa$

$\eta$

Viscosity

$Pa s$

$J_V$

Volume flow

$m^3/s$

Relation

Development

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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Equation

>Top, >Model

If we examine the Hagen-Poiseuille law, which allows us to calculate the volume flow ($J_V$) from the cylinder radio ($R$), the viscosity ($\eta$), the tube length ($\Delta L$), and the pressure difference ($\Delta p$):

we can identify parameters related to geometry (the tube length ($\Delta L$) and the cylinder radio ($R$)) and the type of liquid (the viscosity ($\eta$)), which can be collectively referred to as a hydraulic conductance ($G_h$):

$ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$

Conditions

Variables

$R$

Cylinder radio

$m$

$G_h$

Hydraulic conductance

$m^4/kg s$

$\pi$

Pi

3.1415927

$rad$

$\Delta L$

Tube length

$m$

$\eta$

Viscosity

$Pa s$

Relation

Development

ID:(15102, 0)

Equation

>Top, >Model

With the introduction of the hydraulic conductance ($G_h$), we can rewrite the Hagen-Poiseuille equation with the pressure difference ($\Delta p$) and the volume flow ($J_V$) using the following equation:

$ J_V = G_h \Delta p $

Conditions

Variables

$G_h$

Hydraulic conductance

$m^4/kg s$

$\Delta p$

Pressure difference

$Pa$

$J_V$

Volume flow

$m^3/s$

Relation

Development

If we examine the Hagen-Poiseuille law, which allows us to calculate the volume flow ($J_V$) from the cylinder radio ($R$), the viscosity ($\eta$), the tube length ($\Delta L$), and the pressure difference ($\Delta p$):

$ J_V =-\displaystyle\frac{ \pi R ^4}{8 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$

we can introduce the hydraulic conductance ($G_h$), defined in terms of the tube length ($\Delta L$), the cylinder radio ($R$), and the viscosity ($\eta$), as follows:

$ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$

to arrive at:

$ J_V = G_h \Delta p $

ID:(14471, 0)

Equation

>Top, >Model

In the context of electrical resistance, there exists its inverse, known as electrical conductance. Similarly, what would be the hydraulic conductance ($G_h$) can be defined in terms of the hydraulic resistance ($R_h$) through the expression:

$ R_h = \displaystyle\frac{1}{G_h }$

Conditions

Variables

$G_h$

Hydraulic conductance

$m^4/kg s$

$R_h$

Hydraulic resistance

$kg/m^4s$

Relation

ID:(15092, 0)

Equation

>Top, >Model

Since the hydraulic resistance ($R_h$) is equal to the inverse of the hydraulic conductance ($G_h$), it can be calculated from the expression of the latter. In this way, we can identify parameters related to geometry (the tube length ($\Delta L$) and the cylinder radio ($R$)) and the type of liquid (the viscosity ($\eta$)), which can be collectively referred to as a hydraulic resistance ($R_h$):

$ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$

Conditions

Variables

$R$

Cylinder radio

$m$

$R_h$

Hydraulic resistance

$kg/m^4s$

$\pi$

Pi

3.1415927

$rad$

$\Delta L$

Tube length

$m$

$\eta$

Viscosity

$Pa s$

Relation

Development

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

$ R_h = \displaystyle\frac{1}{G_h }$

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

$ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$

we can conclude that:

$ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$

ID:(3629, 0)

Equation

>Top, >Model

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

it can be expressed in terms of the pressure difference ($\Delta p$). Considering that the inverse of the hydraulic resistance ($R_h$) is the hydraulic conductance ($G_h$), we arrive at the following expression:

$ \Delta p = R_h J_V $

Conditions

Variables

$R_h$

Hydraulic resistance

$kg/m^4s$

$\Delta p$

Variación de la Presión

$Pa$

$J_V$

Volume flow

$m^3/s$

Relation

Development

In the case of a single cylinder the hydraulic resistance ($R_h$), which depends on the viscosity ($\eta$), the tube length ($\Delta L$), and the cylinder radio ($R$), it is calculated using the following equation:

$ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$

On the other hand, Hagen-Poiseuille's law allows us to calculate the volume flow ($J_V$) generated by the pressure difference ($\Delta p$) according to the equation:

$ J_V =-\displaystyle\frac{ \pi R ^4}{8 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$

Combining both equations, we obtain Darcy's law:

$ \Delta p = R_h J_V $

which Henry Darcy formulated to model the general behavior of more complex porous media through which a liquid flows.

The genius of this way of rewriting the Hagen-Poiseuille law is that it demonstrates the analogy between the flow of electric current and the flow of liquid. In this sense, Hagen-Poiseuille's law corresponds to Ohm's law. This opens up the possibility of applying the concepts of electrical networks to systems of pipes through which a liquid flows.

This law, also known as the Darcy-Weisbach Law, was first published in Darcy's work:

• "Les fontaines publiques de la ville de Dijon" ("The Public Fountains of the City of Dijon"), Henry Darcy, Victor Dalmont Editeur, Paris (1856).

ID:(3179, 0)

Equation

>Top, >Model

The area the section ($S$) of a disk with a diameter of ($$) is calculated as follows:

$ S = \pi r ^2$

Conditions

Variables

$\pi$

Pi

3.1415927

$rad$

$r$

Radius of a circle

$m$

$S$

Section

$m^2$

Relation

ID:(3804, 0)

Equation

>Top, >Model

Flow is measured in the volume that passes through a section per time, which can finally be expressed as the section times an average flow velocity

$ j_s = \displaystyle\frac{ J_V }{ S }$

Conditions

Variables

$j_s$

Flux density

$m^3/s$

$S$

Section Flow

$m^2$

$J_V$

Volume flow

$m^3/s$

Relation

Development

Since the flow is defined as the volume \Delta V per time \Delta t is

$ J_V =\displaystyle\frac{ \Delta V }{ \Delta t }$

and the volume is equal to the S section along the path traveled \Delta x

$ dV = S ds $

As the path traveled dx by time dt corresponds to the speed

$ j_s =\displaystyle\frac{ \Delta s }{ \Delta t }$

you get that the flow is

$ j_s = \displaystyle\frac{ J_V }{ S }$

Hay que tener presente que en este modelamiento:

La densidad de flujo cumple el rol de una velocidad media sobre toda la sección del flujo.

ID:(4349, 0)

Equation

>Top, >Model

When analyzing the hydraulic conductance ($G_h$), it can be observed that in the numerator, the cross-sectional area of the tube is represented as $\pi R^2$, where the cylinder radio ($R$) corresponds to a property of the liquid, the viscosity ($\eta$) is related to the fluid's viscosity, and the tube length ($\Delta L$) pertains to the pressure gradient generated.

The remaining factor is referred to as the hydrodynamic permeability ($k$), known as

$ k = \displaystyle\frac{ R ^2}{8}$

Conditions

Variables

$R$

Cylinder radio

$m$

$k$

Hydrodynamic permeability

$m^2$

Relation

.

ID:(108, 0)