Viscous fluid flow

Viscous fluid flow includes shear stresses proportional to velocity gradients for a Newtonian fluid.

A fully developed parallel flow has a velocity profile independent of the streamwise coordinate. In a planar channel a representative profile is $\mathbf{u}=u(z)\hat{\mathbf{x}}$; wall conditions and translational symmetry remove cross-stream motion. The streamwise pressure gradient may remain nonzero, while the convective acceleration vanishes. Fully developed flow is a modeling branch, not a uniqueness assertion for all flows in that geometry.

A velocity profile records the variation of velocity across a spatial coordinate, for example $u(z)$ across a parallel viscous layer. Its derivative sets the tangential shear stress in a Newtonian fluid, while its integral gives the volume flux.

A viscous boundary layer is a boundary layer next to a surface in which viscosity remains important even when the outer flow has a large Reynolds number. It reconciles the outer inviscid velocity with the no-slip boundary condition.

For a steady two-dimensional incompressible boundary layer with tangential velocity $u(x,y)$, normal velocity $v(x,y)$, outer tangential velocity $U(x)$, and kinematic viscosity $\nu$,The pressure is uniform across the layer, and the outer Euler equation supplies $-p_x/\rho =UdU/dx$.

The unsteady form of the Prandtl boundary-layer equation evolves the tangential boundary-layer velocity under advection, outer pressure forcing and normal viscous diffusion. Incompressibility supplies the normal velocity. A stationary wall imposes the no-slip boundary condition, and tangential velocity matches the prescribed outer slip velocity far from the wall. The normal far-field disturbance need not vanish.

For a fixed outer pressure and slip velocity, retain terms linear in a disturbance of a base flow $(U,V)$. The tangential equation has the displayed form and continuity is ${\tilde{u}}_x+{\tilde{v}}_y=0$. Disturbances vanish at the wall and the tangential disturbance vanishes in the far field. A local parallel-profile approximation may freeze the coefficients even when the profile is not an exact global steady solution.

For a frozen real parallel profile and modes proportional to $e^{i\alpha (x-ct)}$, continuity eliminates tangential velocity and gives this third-order equation. Wall conditions are $v=v^{\prime }=0$; outer tangential matching gives $v^{\prime }\to 0$, with bounded $v$. Real coefficients give the spectral symmetry $(\alpha ,c)\mapsto (-\alpha ,c^{\ast })$. For positive real wavenumber, growth means $\mathrm{Im}c>0$.

For streamwise length $L$, outer speed $U$, and layer thickness $\delta$, balancing $u{u}_x\sim U^2/L$ with $\nu u_{yy}\sim \nu U/{\delta }^2$ gives

Momentum diffuses across a distance $L$ in the characteristic time $L^2/\nu$, where $\nu$ is the kinematic viscosity.

Couette flow is viscous shear flow driven by relative tangential motion of parallel boundaries. In the steady no-pressure-gradient case its velocity profile is linear.

Between a resting wall at $y=0$ and a wall moving at speed $U$ at $y=h$, an adverse pressure gradient $p_x=G$ givesZero volume flux requires $G=6\mu U/h^2$, yielding $u/U=3(y/h{)}^2-2y/h$. Reverse flow below $2h/3$ balances the positive flux near the moving wall. The minimum is $-U/3$ at $h/3$.

In two-layer Couette flow, each Newtonian-fluid layer has a linear velocity profile. Continuity of velocity and shear stress at the interface makes the velocity drop across each layer proportional to its thickness divided by its dynamic viscosity.

A steady film down a wall has a parabolic velocity profile set by gravity, ambient pressure gradient, and surface shear.

For a film of thickness $h$ flowing down a slope of angle $\alpha$, an upslope surface stress of magnitude $\tau$ givesThe surface velocity, total flux, and basal shear reverse atrespectively.

At a fluid interface, normal and tangential tractions satisfy the imposed stress balance.

A clean fluid interface with negligible exterior viscosity has zero tangential traction. Its normal traction is determined by exterior pressure and surface tension.

The no-penetration boundary condition sets the relative normal velocity of a fluid and an impermeable boundary to zero. Tangential slip may still be permitted.

No slip requires fluid velocity to equal solid-boundary velocity.

For a smooth incompressible Navier-Stokes equation trajectory with zero velocity at a fixed flat wall, the normal momentum equation determines the normal pressure gradient. In a perturbation model with buoyancy term ${\mathrm{Ri}}_B\theta \hat{\mathbf{y}}$, it gives ${\partial }_{n}p=\nu \mathbf{n}\cdot \Delta \mathbf{u}-{\mathrm{Ri}}_B\theta \mathbf{n}\cdot \hat{\mathbf{y}}$. An extra homogeneous pressure Neumann boundary condition is valid only when the right side vanishes; it does not follow from the no-slip boundary condition.

For a displaced wall $y=ϵh_1(x,t)$, write fluid velocity as $\mathbf{u}=ϵ{\mathbf{u}}_1+{ϵ}^2{\mathbf{u}}_2+\cdots$ and prescribed wall velocity as $ϵ{\mathbf{v}}_1+{ϵ}^2{\mathbf{v}}_2+\cdots$. A Taylor expansion about $y=0$ gives ${\mathbf{u}}_1(0)={\mathbf{v}}_1$ and ${\mathbf{u}}_2(0)={\mathbf{v}}_2-h_1{\partial }_y{\mathbf{u}}_1(0)$. The second-order correction converts first-order velocity gradients into a mean boundary flow.

A Newtonian fluid has viscous stress proportional to the local rate-of-strain tensor, with coefficients independent of that rate. For an isotropic fluid these coefficients are viscosities. Its Newtonian fluid stress tensor specifies the corresponding constitutive relation; the material and its stress tensor are distinct concepts.

For isotropic incompressible flow of a Newtonian fluid, the stress tensor is the displayed expression, with pressure $p$, dynamic viscosity $\mu$, velocity $\mathbf{u}$, and identity matrix $I$. In Couette flow, its off-diagonal component is the shear stress.

Shear stress is the tangential component of traction on a surface. For leading unidirectional flow $u(y)$ in a Newtonian fluid, $\tau =\mu du/dy$.

The shear velocity is the velocity scale $u_{\ast }=\sqrt{|\tau |/\rho }$ associated with a shear stress $\tau$ in a fluid of mass density $\rho$. It is a stress scale, not automatically the actual local fluid velocity at a grain.

The rate-of-strain tensor is the symmetric part of the velocity gradient,

The shear rate is the scalar rate at which adjacent material layers acquire shear strain. In simple shear $u_x=u_x(y)$, its signed value is $\dot{\gamma }=d{u}_x/dy$.

A simple shear flow has velocity $\mathbf{u}=(\dot{\gamma }y,0,0)$ in Cartesian coordinates. Its velocity gradient has one nonzero off-diagonal entry.

The spin tensor is the antisymmetric part of the velocity gradient, $\mathbf{\Omega }=(\nabla \mathbf{u}-(\nabla \mathbf{u}{)}^T)/2$. It describes local rigid-body rotation.

For a planar velocity field in polar coordinates,

For an incompressible Newtonian fluid of constant density and viscosity,

The Leray-Helmholtz projection $P_{\sigma }$ is the orthogonal projection of square-integrable vector fields onto divergence-free vector fields. Applying it to an incompressible momentum equation removes the pressure gradient.

The Stokes operator is $A=-P_{\sigma }\Delta$ on divergence-free vector fields, where $P_{\sigma }$ is the Leray-Helmholtz projection. On a periodic domain it equals $-\Delta$ on its domain, and its inverse gains two Sobolev derivatives.

The spectral projection $P_N$ retains the first $N$ Stokes eigenfunctions. On its finite-dimensional range, one derivative costs at most ${\lambda }_N^{1/2}$ and all Sobolev norms are equivalent.

For a periodic or boundary-tangent divergence-free vector field $u$, integration by parts givesConsequently $\int (u\cdot \nabla f)f=0$, the cancellation underlying many fluid energy estimates.

For a velocity field that need not be divergence free, the corrected transportsatisfies $⟨\tilde{B}(u,v),w⟩=-⟨\tilde{B}(u,w),v⟩$ and therefore preserves the quadratic energy cancellation.

Replacing the advecting velocity by a fixed finite Stokes projection $P_{N}u$ makes the three-dimensional nonlinearity controlled by ${\lambda }_N^{1/4}|u|\Vert v\Vert \Vert w\Vert$. This yields global existence and uniqueness at the weak energy level.

The steady equation $\nu Au+\tilde{B}(u,u)=f$ retains the energy cancellation without imposing incompressibility. Galerkin bounds, compactness, and elliptic bootstrapping produce an $H^2$ periodic solution for square-integrable forcing.

Rayleigh-Bénard convection is buoyancy-driven flow caused by heating a fluid from below. In an infinite-Prandtl-number reduction, velocity is determined instantaneously by a forced Stokes equation, while temperature satisfies an advection-diffusion equation.

For a fixed-temperature layer with stress-free boundary conditions, a vertical Fourier sine series has $a^2={\lambda }^2+n^2{\pi }^2$, where $\lambda$ is horizontal wavenumber. The coupled vertical velocity and temperature amplitudes satisfy the displayed quadratic for their growth rate $\sigma$. Its roots are real for positive Rayleigh number, while both roots have negative real parts for negative Rayleigh number. Setting $\sigma =0$ gives the free-slip convection neutral curve.

In nonrotating Rayleigh-Bénard convection with fixed-temperature, stress-free boundary conditions, the stress-free convection growth-rate polynomial has real roots on the destabilizing side. Consequently the first instability is stationary, rather than oscillatory. Minimizing the free-slip convection neutral curve selects ${\lambda }_c=\pi /\sqrt{2}$ and ${\mathrm{Ra}}_c=27{\pi }^4/4$.

Using layer depth $d$ and thermal diffusivity $\kappa$ gives time scale $d^2/\kappa$, velocity scale $\kappa /d$, and perturbation pressure scale ${\rho }_0{\kappa }^2/d^2$. With temperature scale $\Delta T$, the Linearized Boussinesq equations have viscous coefficient Prandtl number $\mathrm{Pr}=\nu /\kappa$ and buoyancy coefficient $\mathrm{Ra}\mathrm{Pr}$, where Rayleigh number $\mathrm{Ra}=g{\alpha }_T\Delta T{d}^3/(\nu \kappa )$.

The conductive state is the no-flow equilibrium in which heat conduction alone maintains a linear temperature profile across a layer with fixed boundary temperatures. Its hydrostatic pressure follows by integrating $P_b^{\prime }=-g{\rho }_b$, with mass density obtained from the equation of state. Under the Boussinesq approximation, deviations from this state are the variables used in the linear Rayleigh-Bénard convection problem.

A thermally driven fluid layer rotating about the vertical axis experiences a Coriolis force that couples vertical velocity to vertical vorticity. Its onset depends on the Rayleigh number, Prandtl number and Taylor number, as well as thermal and mechanical boundary conditions. The force does no mechanical work, but changes the linear mode balance and nonlinear convection roll competition.

Near an oscillatory onset of rotating Rayleigh-Bénard convection, opposite travelling roll waves have coupled Hopf bifurcation amplitudes. Symmetry gives the displayed cubic amplitude equations, with generally complex self-coupling and cross-coupling coefficients. Write $a_s=\mathrm{Re}{g}_s$ and $a_c=\mathrm{Re}{g}_c$. For $r>0,a_s>0$, a single travelling wave has intensity $r/a_s$ and is stable against its counterpropagating competitor when $a_c>a_s$. Equal standing-wave intensities are $r/(a_s+a_c)$; with $r>0$ and positive denominator they are stable to intensity differences when $a_s>a_c$. Phase symmetries give orbital, rather than strict phase-decay, stability. Spatial modulation, mean flows and oblique convection rolls require additional stability tests.

A saturated convection roll is not stable merely because its single-amplitude cubic coefficient is positive. Perturbations include phase sidebands, transverse bending, differently oriented convection rolls, oscillatory modes and large-scale Eulerian mean flow. Their growth rates follow from the corresponding enlarged amplitude equations or the full linearization about the convection roll. Küppers–Lortz instability and small-angle instability of rotating convection rolls are distinct orientation-dependent mechanisms.

For finite Prandtl number and stress-free boundaries, nearly parallel convection rolls can interact with a weakly damped large-scale Eulerian mean flow. The ordinary regular two-roll amplitude expansion becomes nonuniform as the angle tends to zero, so the mean-flow mode must be retained. This mechanism is not automatically the finite-angle Küppers–Lortz instability, and thresholds depend on the stated boundary and parameter regime.

For an existing convection roll of intensity $r/g_0$, the amplitude of a new orientation at relative angle $\vartheta$ grows at rate $r[1-g(\vartheta )/g_0]$. Rotation permits $g(\vartheta )\ne g(-\vartheta )$. If one oblique orientation has $g(\vartheta )<g_0$, the convection roll is unstable immediately above a supercritical stationary threshold. This is the Küppers–Lortz mechanism. Successive orientation replacements can produce time-dependent convection roll switching; a universal attractor does not follow from the two-mode calculation. The perturbing orientation requires three-dimensional disturbances even if the initial convection roll is two-dimensional.

For the free-slip plane layer, onset selects the smallest Rayleigh number across all admissible stationary and oscillatory normal modes. The first vertical mode wins because increasing vertical index raises viscous cost and reduces oscillatory admissibility. Oscillatory minimization replaces the right-hand side of the stationary wave-number equation by $P^2\mathrm{Ta}{\pi }^2/(1+P{)}^2$. For the oscillatory minimum also require positive angular frequency squared. Where both selected curves are admissible, at rapid rotation the ratio of the minimized thresholds is $R_{o,c}/R_{s,c}\sim 2P^{4/3}/(1+P{)}^{1/3}$. Equality gives $8P^4-P-1=0$, with positive root about $0.67660$; below it sufficiently rapid rotation can select oscillatory primary onset.

For stress-free impermeable fixed-temperature plates, thermal-time growth rate $s$, horizontal wavenumber $k$ and vertical index $n$, put $a=k^2+n^2{\pi }^2$ and $P=\mathrm{Pr}$. Vertical velocity, temperature and vertical vorticity amplitudes satisfy $a(s/P+a)W+\sqrt{\mathrm{Ta}}n\pi Z=R{k}^2\Theta$, $(s/P+a)Z=\sqrt{\mathrm{Ta}}n\pi W$, $(s+a)\Theta =W$. Taking their determinant givesThis is a cubic and remains valid without dividing by a potentially zero factor. Setting $s=0$ or $s=i\omega$ determines the stationary or admissible oscillatory neutral curves.

For $s=i\omega$, real-imaginary separation of the rotating-convection cubic givesThe curve is admissible only if ${\omega }^2>0$, so $P<1$ and sufficient rotation are necessary. A formal minimum with nonpositive angular frequency squared is not a Hopf bifurcation. At zero angular frequency this curve meets the stationary curve for that fixed wavenumber in a double-zero limit.

Substitution of zero growth rate into the rotating-convection cubic gives the displayed Rayleigh number. For the first vertical mode, setting $t=k^2$, $m={\pi }^2$, differentiating with respect to $t$ gives $(t+m{)}^2(2t-m)=\mathrm{Ta}m$. At rapid rotation, $k_c\sim (\mathrm{Ta}{\pi }^2/2{)}^{1/6}$ and $R_c\sim 3(\mathrm{Ta}{\pi }^2/2{)}^{2/3}$. Discrete allowed wavenumbers or different plate conditions require a different minimization.

For nonrotating convection under the Boussinesq approximation with fixed-temperature impermeable stress-free boundary conditions, a marginal vertical mode $\sin (n\pi z)$ has neutral Rayleigh number $(k^2+n^2{\pi }^2{)}^3/k^2$. The minimum is at $n=1$, $k=\pi /\sqrt{2}$, and $R_c=27{\pi }^4/4$. These follow by eliminating the temperature from $(D^2-k^2{)}^{2}w=R{k}^2\theta$ and $(D^2-k^2)\theta =-w$, then minimizing over $k$.

In thermal-time units, taking the Prandtl number large removes momentum inertia at leading order. The Stokes flow momentum equation then determines velocity instantaneously from the temperature perturbation, while the nonlinear heat equation retains its time derivative. Small Prandtl number does not generally give this model. An exact contrasting shear in a free-slip layer is $u=U_0{e}^{-\mathrm{Pr}{\pi }^{2}t}\cos \pi z$, $w=\theta =0$: it decays slowly as the Prandtl number tends to zero but is excluded by the instantaneous Stokes equation.

For a nonzero horizontal Fourier mode $k$, solve $(D^2-k^2{)}^{2}w=R{k}^2\theta$, $w=w^{\prime \prime }=0$, to obtain $w={\mathcal{W}}_R\theta$. Under homogeneous temperature Dirichlet boundary conditions, the sine modes diagonalize this self-adjoint operator, with eigenvalues $R{k}^2/(k^2+n^2{\pi }^2{)}^2$. The temperature linear operator is ${\mathcal{L}}_R=D^2-k^2+{\mathcal{W}}_R$. At a marginal mode $g=\sin (n\pi z)$ it has ${\mathcal{W}}_{R}g=(k^2+n^2{\pi }^2)g$.

A damped-driven Navier-Stokes equation adds linear damping and external forcing. In two-dimensional steady vorticity form it reads

For a steady damped vorticity equation with $\gamma >0$, divergence-free transport, and bounded forcing, multiplication by $\omega |\omega {|}^{q-2}$ gives $\gamma \Vert \omega {\Vert }_q\le \Vert g{\Vert }_q$. Sending $q$ to infinity yields

A vanishing-viscosity limit studies solutions as the viscosity coefficient tends to zero. Uniform estimates and compactness can make the viscous term disappear weakly while retaining enough strong convergence to pass nonlinear transport terms to an inviscid equation.

Pipe flow is fluid motion through a closed cylindrical conduit. In fully developed axisymmetric flow, the axial velocity depends only on radius and the axial pressure gradient sets a shear stress linear in radius.

Steady pressure-driven flow through a circular pipe has the parabolic profile

For a fixed domain $\mathcal{D}$, body-force density $f$, and stress tensor $\sigma$, the kinetic energy satisfiesThe last term is the nonnegative rate at which viscosity converts mechanical energy into heat.

For a clean bubble of size $a$ and speed $U$ at large Reynolds number, the outer potential flow violates the stress-free condition by a strain of order $U/a$. A boundary layer of thickness $\delta \sim a{\mathrm{Re}}^{-1/2}$ needs a velocity correction only of order $U\delta /a$, so its dissipation is smaller than the outer-flow dissipation by order $\delta /a$. The leading drag power is therefore obtained from

For the zero-circulation flow around a circle, $e:e=8U^2{a}^4/r^6$. Integration over the exterior plane gives dissipation $8\pi \mu U^2$ per unit axial length and hence drag $D=8\pi \mu U$ per unit length.

Stokes flow is the zero-Reynolds-number limit in which viscous and pressure forces balance while fluid inertia is neglected.

The Brinkman equation augments Stokes flow with a local drag against a background matrix of velocity ${\mathbf{u}}_m$. The inverse screening length $\alpha$ has units of inverse length. With equal viscous coefficients the permeability is ${\alpha }^{-2}$. Taking the curl produces a screened vorticity equation.

Porous-matrix drag depends on fluid velocity relative to the matrix, so it is not invariant under adding a uniform fluid velocity while leaving the matrix fixed. In a swimmer frame with stationary laboratory matrix, the drag is proportional to $\mathbf{u}-U{\mathbf{e}}_x$. A constant shift can alternatively be absorbed into a linear pressure term, whose physical interpretation must then be retained.

A slender low-viscosity fluid column inside a much more viscous fluid rises under a density contrast. At negligible inertia, its axial flux is approximately $q=\pi a^4(\Delta \rho g-P_{i,z})/(8{\mu }_i)$. Radial expansion strains the outer fluid and gives the normal-stress pressure difference $P_i-P_o=2{\mu }_o{a}_t/a$ at leading order when inner normal viscous stress and interfacial tension are negligible. Coupling that pressure to volume conservation yields the conduit equation. This two-fluid viscous model differs from a turbulent entraining thermal plume.

The conduit equation evolves the positive cross-sectional area $A$ of a viscous buoyant conduit. Its scaling uses axial length $a_0/\sqrt{8\lambda }$ and velocity $\Delta \rho g{a}_0^2/(8\lambda {\mu }_o)$, where $\lambda ={\mu }_i/{\mu }_o$. Linearization at $A=1$ gives $\omega =2k/(1+k^2)$: phase velocity is upward but group velocity changes sign at $|k|=1$. Nonlinear elevation waves obey the solitary-wave amplitude-speed relation for the conduit equation.

A positive travelling wave $f$ on a uniform background $f_0$ has first integral $c{f}^{\prime 2}/(2f^2)+V(f)=V(f_0)$, with $V(f)=\ln f+c/f+(f_0^2-c{f}_0)/(2f^2)$. Equating the crest and background potentials gives the displayed speed relation for crest ratio $\alpha >1$. The elevation-wave speed exceeds $2f_0$ and approaches that long-wave speed as $\alpha \to 1$. The formula requires the positive-area branch, rather than continuation through $f=0$.

Bodies immersed in one fluid affect each other through the flow that each produces. At large separation, a force-driven sphere generates a Stokeslet that modifies the other sphere's velocity. These interactions produce off-diagonal entries in the hydrodynamic mobility matrix.

Two spheres whose prescribed translations have a phase lag can exert a nonzero mean force on the fluid while each returns to its starting position. Their changing separation correlates the interaction strength with the other sphere's velocity. This pump is externally driven, so it is distinct from force-free swimming governed by the scallop theorem.

The centre separation for equal-amplitude oscillations has minimum ${\ell }_0-2\delta |\sin (\varphi /2)|$. A bound ${\ell }_0\ge \delta$ alone cannot guarantee nonoverlap. An asymptotic mobility calculation needs this minimum much larger than the sphere radii.

For equal oscillation amplitudes $\delta$, frequency $\omega$, mean separation ${\ell }_0$ and phase lag $\varphi$, the leading mean total force on the fluid is $9\pi \mu a_1{a}_2{\delta }^2\omega \sin \varphi /{\ell }_0^2$. Each sphere supplies half. The large-separation calculation expands the Stokeslet interaction; reciprocal in-phase and antiphase motions have zero mean force.

For well-separated spheres moving along their line of centres, the leading mobility matrix has diagonal entries $m_i=(6\pi \mu a_i{)}^{-1}$ and off-diagonal entries $h=(4\pi \mu \ell {)}^{-1}$. This is the axial Stokeslet interaction. Finite-size and repeated-reflection corrections enter at higher order.

The symmetric first moment of the traction exerted by a rigid inclusion, with the surface normal pointing into the inclusion. Its contraction with the background rate-of-strain tensor is the strain-related contribution to extra dissipation due to a rigid inclusion. A trace-free version is equivalent for an incompressible flow. This tensor is distinct from the axisymmetric force-dipole flow used as a far-field singularity.

Let the ambient velocity be $\mathbf{Ex}$, with $\mathbf{E}$ symmetric and trace free, and centre a rigid sphere of radius $a$ at the origin. It is force-free and torque-free, with zero translation and rotation. Its disturbance isThe total velocity vanishes at $r=a$, and the leading disturbance is a stresslet. Harmonic dipole and quadrupole potentials in the Papkovich–Neuber representation give the complete field.

For an incompressible Newtonian fluid with negligible inertia, the Stokes equations are

The Stokes equation is linear in velocity, pressure, body force, and boundary data. Solutions may therefore be superposed, and rigid-body velocities depend linearly on applied forces and torques.

With prescribed velocity on the boundary and suitable decay or far-field data, a Stokes flow is unique up to an additive pressure constant. Applying the viscous dissipation identity to the difference of two solutions makes its rate-of-strain tensor vanish; the homogeneous boundary data then eliminate the remaining rigid motion.

Let a viscous layer occupy $0<y<h(t)$, with no slip at $y=0$, zero tangential traction at $y=h$, and exterior pressure $p_0-{\rho }_a{E}^2{x}^2/2$. The Cartesian streamfunction ansatz $\psi =xf(y)$ reduces the Stokes equations to $f^{(4)}=0$. The resulting surface velocity isso the positive thinning rate is $-\dot{h}={\rho }_a{E}^2{h}^3/(3\mu )$.

A Stokeslet is the velocity field produced by a point force $\mathbf{F}$ in an unbounded three-dimensional Stokes fluid: