Physiological fluid dynamics

Physiological fluid dynamics studies the motion of fluids in biological systems, including blood flow in elastic arteries, flow through deformable vessels, and liquid layers lining airways. Wall compliance couples pressure to geometry, so wave propagation and mechanical feedback are as important as viscosity.

For a freely translating compliant body separated from a cylindrical wall by a thin lubrication gap $h$, the body frame has wall speed $-U$ and body surface speed zero. The local axial flow is $w=-U(1-y/h)+p_{x}y(y-h)/(2\mu )$, so its flux per circumference is $-Uh/2-p_x{h}^3/(12\mu )=-Q$. Rearranging proves the displayed pressure gradient. A linear compliance $p=p_0+\alpha (r_0-r)$ couples this flow to deformation. Specified end pressures and zero net axial body force close the problem.

The pressure force on a plug-and-film control volume is $\pi a^2\Delta p$. Its opposing wall traction is $2\pi a\mu \int w_y(0)dx$, where $w_y(0)=4U/h-6Q/h^2$. Equating them for a freely moving body proves the displayed formula. Writing $s=2Q/U$, $h=sH$ and $x=\sqrt{s/\kappa }X$ converts it into $H(-\tilde{L})-H(\tilde{L})=C\lambda \int [4/(3H)-1/H^2]dX$, with $C=s/a$ and $\lambda =6\mu U/(\alpha \sqrt{\kappa }{s}^{5/2})$.

For $H_X+\lambda (H^{-2}-H^{-3})=X$, small $\lambda$ gives $H_0=(b^2+X^2)/2$. In the large-domain limit its inverse-power integrals are $I_1=2\pi /b$, $I_2=2\pi /b^3$ and $I_3=3\pi /b^5$. The force-free condition is $I_2-I_3=C(4I_1/3-I_2)$, or $2b^2-3=C(8b^4/3-2b^2)$. Expanding the root near $b^{-2}=2/3$ proves the displayed minus sign and leading pressure drop. The absolute downstream pressure relative to the elastic reference pressure fixes $s$ and therefore closes the speed-pressure relation.

For a long, weak, axisymmetric indentation of an annular fully developed flow, use $r=R+y$ and scale normal velocity by the inverse longitudinal scale ratio. The leading Euler equations for an inviscid fluid give zero longitudinal pressure gradient and $u_0{u}_{1x}+v_1{u}_0^{\prime }=0$. Together with $u_{1x}+(r{v}_1{)}_y/r=0$, this yields ${\psi }_1=A{u}_0$, hence the displayed displacement fields. Thin viscous boundary layers enforce tangential wall conditions separately.

A distinguished annular indentation limit has ${\lambda }^{-2}=O(ϵ)$, $\lambda St=O(ϵ)$ and $Re\gg \lambda {ϵ}^{-3}$. Define $\beta =\lambda St/ϵ$, $\sigma =({\lambda }^2ϵ{)}^{-1}{\int }_0^1{u}_0^2/(R+y)dy$, ${\alpha }_1={\gamma }_0/R+{\gamma }_1/(R+1)$ and ${\alpha }_2={\gamma }_0^2/R^2-{\gamma }_1^2/(R+1{)}^2$. Radial inertia supplies the pressure difference $\sigma A_{xx}$ between the walls. The axial momentum limits and the moving-wall condition supply its derivative, giving the displayed forced KdV equation. This reduction assumes the viscous boundary layers remain thin and their displaced flux is smaller than $O({ϵ}^2)$.

With no forcing, the linear dispersion relation has positive phase velocity $\sigma k^2/(\beta {\alpha }_1)$ and group velocity three times this. A decaying permanent wave with $\xi =x+ct$ instead obeys $\sigma (A^{\prime }{)}^2=A^2(\beta {\alpha }_{1}c+{\alpha }_{2}A/3)$. For positive coefficients a nonzero homoclinic wave requires $c>0$ and is negative, travelling upstream. Its inverse width is $k_s=\frac{1}{2}\sqrt{\beta {\alpha }_{1}c/\sigma }$. The opposite directions reflect the nonlinear amplitude-dependent propagation speed.

Peristaltic pumping transports fluid by a travelling deformation or travelling external pressure acting on a tube. With wave coordinate $X=x-ct$, mass conservation makes the relative flux $Q=\alpha (u-c)$ constant. The laboratory flux is $\alpha u=c\alpha +Q$, so a negative relative flux does not rule out forward mean pumping.

For negligible inertia, periodic forcing $p_e=ϵP_e\sin kX$, resistance $R\propto {\alpha }^{-n}$ and prescribed mean area ${\alpha }_0$, write $Q=-c{\alpha }_0+{ϵ}^2{Q}_2+\cdots$. First-order area obeys ${\tilde{P}}_0^{\prime }{\alpha }_1^{\prime }+R_{0}c{\alpha }_1/{\alpha }_0=-k{P}_e\cos kX$. Its mean square is $P_e^2/[2({\tilde{P}}_0^{\prime 2}+(R_{0}c/(k{\alpha }_0){)}^2)]$. Averaging the second-order friction balance gives $Q_2=c(n+1)⟨{\alpha }_1^2⟩/{\alpha }_0$, proving the displayed forward pumping rate. Periodicity entails zero mean pressure drop; the mean-area convention fixes the expansion's otherwise free constant.

A one-dimensional collapsible tube model couples area $\alpha$, mean axial velocity $u$ and internal pressure $p$: ${\alpha }_t+(\alpha u{)}_x=0$ and $u_t+u{u}_x=-p_x-R(\alpha )u$. A wall relation specifies internal minus external pressure as a function of area. Decreasing area generally increases the friction coefficient $R$, while positive wall stiffness gives real small-amplitude wave speed.

For steady uniform-density inviscid flow with pressure law $p=p_0+k(r-R)$ and a prescribed thickness $h$, mass conservation and the Bernoulli equation give $h/R=1-s^{-1/2}+\lambda (1-s^2)/4$, where $s=u/V$ and $\lambda =2\rho V^2/(kR)$. The function has a unique maximum at $s={\lambda }^{-2/5}$, producing the displayed choking bound. Below it there are two positive roots; continuity from the upstream state selects the lower-speed root for $\lambda <1$ and higher-speed root for $\lambda >1$. At $\lambda =1$, the maximum is zero and any positive prescribed thickness is incompatible with this model.

A positive collapsible tube flow is subcritical when its characteristic speeds $u+c$ and $u-c$ have opposite signs. A downstream boundary can then send information upstream. For a quadratic pressure-area law and fixed dimensionless flux $q$, this corresponds to $A/A_0>\sqrt{q}$.

A positive collapsible tube flow is supercritical when both characteristic speeds $u\pm c$ are positive relative to the wall. Information then propagates downstream on both branches. For a quadratic pressure-area law and fixed dimensionless flux $q$, the condition is $A/A_0<\sqrt{q}$.

An elastic jump is a short transition from a fast, narrow supercritical collapsible tube flow to a slower, wider subcritical flow. Neglecting body force and distributed friction across the jump conserves flux and the momentum flux shown above. Distinct upstream and downstream areas therefore obey ${\alpha }_2^3+{\alpha }_1{\alpha }_2^2+{\alpha }_1^2{\alpha }_2=3q^2/{\alpha }_1$. The flux function has its unique minimum at $\alpha =\sqrt{q}$, so an upstream state below this value has one conjugate state above it. The jump dissipates mechanical energy; it does not conserve a Bernoulli constant.

For $p=P_0+\rho c_0^2{\alpha }^2/2$ and constant flux $c_0{A}_{0}q$, a downhill collapsible tube flow obeys the displayed steady equation. Uniform area satisfies $qR({\alpha }_1)=\beta$. Since $R^{\prime }<0$, linearizing about this root gives downstream exponential decay precisely when ${\alpha }_1^2<q$. This is spatial stability of a steady inlet-value problem, not a guarantee of temporal stability of the driven flow. The wave speed is $c_0\alpha$, so the stable spatial branch is supercritical.

Let $R(\alpha )\propto {\alpha }^{-n}$, with $n>0$, and use disturbances $e^{i(kx-\omega t)}$ about uniform velocity $U$. The intrinsic frequency is $\nu =\omega -kU$. Linearized mass conservation and the decrease in friction as area grows give the displayed dispersion relation. For nonzero $k$ and positive flow, one branch grows exactly when $nU>c_0$. At $nU=c_0$, one branch is neutral, $\nu =k{c}_0$, while the other remains damped, $\nu =-k{c}_0-i{R}_0$. The maintained external-pressure gradient supplies the energy; positive friction alone does not imply linear stability of a driven flow.

The pressure-area law of a collapsible tube describes transverse mechanical equilibrium of a deformable tube wall. On a stable branch ${\tilde{P}}^{\prime }(\alpha )>0$. Linearizing mass and momentum conservation about a stationary uniform tube gives the displayed wave speed. External pressure is prescribed separately and is not part of the elastic wall stiffness.

Linearizing area conservation and the axial momentum equation of a collapsible tube flow gives waves travelling relative to the fluid at speeds $\pm c$, where the displayed relation uses the area derivative at fixed external pressure. For $p=P_0+\rho c_0^2(A/A_0{)}^2/2$, it gives $c=c_{0}A/A_0$.

A small pressure disturbance in an elastic artery propagates through the coupled inertia of blood and compliance of the wall. If $\mathcal{A}$ is the undisturbed area, $\rho$ the mass density and $c$ the wave speed, the linearized equations give $c^2=\mathcal{A}/\rho dp/d\mathcal{A}$. A travelling pressure amplitude $p_{+}$ carries flow amplitude $\mathcal{A}{p}_{+}/(\rho c)$; the oppositely travelling wave has the opposite flow sign.

At a lossless vascular junction, pressure continuity and volumetric flow rate balance give $1+r=t$ and $Y_1(1-r)=Y_{d}t$, where $Y_d$ is the sum of daughter characteristic admittances of an arterial wave. Eliminating transmission amplitude $t$ gives the displayed reflection coefficient. A decrease in total area yields positive reflection if parent and daughter wave speeds are equal; differing stiffnesses require comparison of the admittances rather than areas alone.

For incident and reflected harmonic arterial pressure waves with real $0<r<1$, the local pressure and flow phasors are proportional to $1+r{e}^{-i\vartheta }$ and $1-r{e}^{-i\vartheta }$. Their maximum-time difference is the displayed phase lag, on the branch $0<\vartheta <\pi$. Here $\vartheta =2\omega d/c$ is the round-trip phase to the reflector. Its derivative has the sign of $\cos \vartheta$. Raising the wave speed reduces the lag when $0<\vartheta <\pi /2$, but increases it when $\pi /2<\vartheta <\pi$. This conditional harmonic model does not establish an age trend for arbitrary vascular geometry or measurement stations beyond the sole reflector.

A hydraulic resistance $R$ imposes $q=p/R$. Combining this with forward and backward arterial pressure waves, $p=p_{+}+p_{-}$ and $q=Y(p_{+}-p_{-})$, gives the displayed reflection coefficient. A matched load $RY=1$ has no reflected pressure wave, while a closed end has reflection coefficient one. At a network junction the combined downstream input admittance replaces $1/R$.

For time dependence $e^{i\omega t}$ and a segment of length $\ell$, let $\beta =\omega \ell /c$. Adding the forward and backward arterial pressure waves gives the displayed upstream-to-downstream matrix. The flow orientation is downstream at both ends. At a junction, pressure is continuous and incoming volumetric flow rates balance outgoing ones.

Consider two identical two-segment paths from an inlet to a distal junction, with equal resistive beds at the two intermediate junctions and the distal junction. The segment transfer matrix and junction flow balances determine all pressures. If one inlet branch is removed and the beds are weakly loading the waves, the remaining segments form a three-segment chain with an effectively closed far end. The altered reflection pattern can raise peripheral oscillatory flow amplitudes even though a branch was lost; the incident wave amplitude, rather than total inlet pressure or flow, is the prescribed forcing.

The characteristic admittance of an arterial wave relates the volumetric flow rate of a travelling arterial pressure wave to its pressure. For forward and backward amplitudes, $p=p_{+}+p_{-}$ and $q=Y(p_{+}-p_{-})$. It is the reciprocal of characteristic pressure-to-flow impedance, and should not be confused with the reciprocal hydraulic resistance of a steadily resistive segment.