Question

Hair growth We consider a liquid (of density $p$) which wets a vertical surface located at $x = 0$ as in Figure 1. We speak of capillary rise. The objective is to determine the law linking the maximum height $h$ to the wetting angle $\theta$. air $R(s)$ $P_{atm}$ $\theta$ $ds = \sqrt{dx^2 + dz^2}$ liquid $x$ FIGURE 1 - Diagram of capillary rise. 1. The interface being curved, show that the pressure in the liquid at a point on the rib interface $z$ is written $P_L(z) = P_{atm} - \frac{\gamma}{R}$ (6) where $P_{atm}$ is the pressure in the air (assumed constant), $\gamma$ the surface tension and $R$ the radius of curvature of the interface at this point [in the ($x$, $z$) plane). 2. Write another equation for the pressure at this point (law of hydrostatics). 3. Show using the notations in Figure 1 that $ds = R \frac{dz}{sin \phi}$ (7) 4 By integrating $z$ between 0 and $h$ in the pressure equation, deduce that $h(\theta) = \ell_c \sqrt{2(1 - sin \theta)}$ (8) where we will express the capillary length $\ell_c$ as a function of the parameters $\gamma$, $p$ and the gravitational constant $g$. 5. Estimate its value for water. Comment. Radius of gyration We define the radius of gyration $R_g$ of a polymer of N monomers (N + 1 balls at positions $r_i$, connected by N links of length $a$) according to $R_g^2 = \frac{1}{N+1} \sum_{i=1}^{N+1} (r_i - r_{0})^2$ (9) where $r_0r$ is the barycenter.

          Hair growth
We consider a liquid (of density $p$) which wets a vertical surface located at $x = 0$ as
in Figure 1. We speak of capillary rise. The objective is to determine the law linking the
maximum height $h$ to the wetting angle $\theta$.
air
$R(s)$
$P_{atm}$
$\theta$
$ds = \sqrt{dx^2 + dz^2}$
liquid
$x$
FIGURE 1 - Diagram of capillary rise.
1. The interface being curved, show that the pressure in the liquid at a point on the rib interface
$z$ is written
$P_L(z) = P_{atm} - \frac{\gamma}{R}$ (6)
where $P_{atm}$ is the pressure in the air (assumed constant), $\gamma$ the surface tension and $R$ the radius of
curvature of the interface at this point [in the ($x$, $z$) plane).
2. Write another equation for the pressure at this point (law of hydrostatics).
3. Show using the notations in Figure 1 that
$ds = R \frac{dz}{sin \phi}$ (7)
4 By integrating $z$ between 0 and $h$ in the pressure equation, deduce that
$h(\theta) = \ell_c \sqrt{2(1 - sin \theta)}$ (8)
where we will express the capillary length $\ell_c$ as a function of the parameters $\gamma$, $p$ and the gravitational
constant $g$.
5. Estimate its value for water. Comment.
Radius of gyration
We define the radius of gyration $R_g$ of a polymer of N monomers (N + 1 balls at positions $r_i$,
connected by N links of length $a$) according to
$R_g^2 = \frac{1}{N+1} \sum_{i=1}^{N+1} (r_i - r_{0})^2$ (9)
where $r_0r$ is the barycenter.

Hair growth
We consider a liquid (of density p) which wets a vertical surface located at x = 0 as
in Figure 1. We speak of capillary rise. The objective is to determine the law linking the
maximum height h to the wetting angle θ.
air
R(s)
Patm
θ
ds = √(dx^2 + dz^2)
liquid
x
FIGURE 1 - Diagram of capillary rise.
1. The interface being curved, show that the pressure in the liquid at a point on the rib interface
z is written
PL(z) = Patm - (γ)/(R) (6)
where Patm is the pressure in the air (assumed constant), γ the surface tension and R the radius of
curvature of the interface at this point [in the (x, z) plane).
2. Write another equation for the pressure at this point (law of hydrostatics).
3. Show using the notations in Figure 1 that
ds = R (dz)/(sin ϕ) (7)
4 By integrating z between 0 and h in the pressure equation, deduce that
h(θ) = √(2(1 - sin θ)) (8)
where we will express the capillary length as a function of the parameters γ, p and the gravitational
constant g.
5. Estimate its value for water. Comment.
Radius of gyration
We define the radius of gyration Rg of a polymer of N monomers (N + 1 balls at positions ri,
connected by N links of length a) according to
Rg^2 = (1)/(N+1)∑i=1^N+1 (ri - r0)^2 (9)
where r0r is the barycenter.

Added by Amber F.

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