Viscous fluid flow

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

A steady two-dimensional purely radial incompressible Newtonian fluid flow has $u_r=g(\theta )/r$ by the continuity equation. The polar Navier-Stokes equation givesEquality of the mixed pressure derivatives yields the displayed third-order equation. On fixed walls $\theta =\pm \alpha$, no-slip boundary conditions require $g(\pm \alpha )=0$. A source with volume flux $Q$ per unit out-of-plane length requires ${\int }_{-\alpha }^{\alpha }gd\theta =Q$. Consequently $g=QG$ leaves the single inertial parameter $ϵ=Q/\nu$ in $G^{(3)}+4G^{\prime }+2ϵG{G}^{\prime }=0$. The origin is a source singularity; the radial ansatz describes the wedge away from that point, not the detailed source apparatus.

On the symmetric small-$Q/\nu$ branch of radial viscous flow in a wedge, the leading Stokes flow solves $G_0^{(3)}+4G_0^{\prime }=0$, zero wall values and unit integral. HenceFor $\pi /2<\alpha <{\alpha }_c$, where ${\alpha }_c\simeq 2.24670$, this is positive in $|\theta |<\pi -\alpha$ and negative near the walls. The normalization denominator tends to zero at the stated root. At the root the homogeneous angular mode has zero net flux, so it cannot be rescaled to carry a fixed nonzero source flux.

Approaching that angle makes large outward and reverse contributions cancel to leave the prescribed flux. The leading amplitude diverges and the nonlinear inertial correction ceases to be perturbative, however small its coefficient is at fixed angle. This is a failure of a regular expansion, not a proof of infinite speed in the full Navier-Stokes equation. Distinct inner scalings or different steady branches require separate analysis.

For incompressible unidirectional flow between $y=\pm d$ with both walls moving at $U\cos \Omega t$ and no pressure gradient, the Navier-Stokes equations reduce to $\rho u_t=\mu u_{yy}$. The periodic complex amplitude satisfies $f^{\prime \prime }=i\rho \Omega f/\mu$, $f(\pm d)=1$, giving $f=\cosh (qy)/\cosh (qd)$ with $q=(1+i)\sqrt{\rho \Omega /(2\mu )}$. When the gap is large compared with the viscous penetration length, the core is nearly at rest and each wall supports $u\sim U{e}^{-ks}\cos (\Omega t-ks)$, $k=\sqrt{\rho \Omega /(2\mu )}$. Without initial data this specifies the periodic response; decaying zero-wall transients may be superposed.

For a straight pipe with cross-section $0<x<a$, $|y|<x/\sqrt{3}$, steady unidirectional incompressible flow reduces the Navier-Stokes equations to $\mu {\nabla }_{\perp }^{2}w=p_z$, with constant axial pressure gradient. The displayed cubic polynomial vanishes on every wall, and its transverse Laplacian is $p_z/\mu$. The axial drag per unit length on each of the three equal walls is $-p_z{a}^2/(3\sqrt{3})$. This follows by integrating the viscous traction on one wall; summing the three forces equals the pressure gradient times the triangular area.

Stokes' second problem describes the oscillatory shear flow generated by a tangentially oscillating flat plate in a semi-infinite viscous fluid flow. The velocity satisfies $u_t=\nu u_{xx}$, with $u(0,t)=U_0\cos \omega t$ and decay away from the plate. For $\omega >0$, the displayed time-periodic solution follows by taking the real part of $U_0{e}^{i\omega t-(1+i)x/\delta }$. The length $\delta$ is the penetration depth: both amplitude attenuation and phase lag grow with $x/\delta$. Boundary data alone also allow homogeneous transients; the periodic solution describes the established oscillation after those transients decay.

The viscous penetration depth is the distance over which a time-periodic shear velocity driven by an oscillating boundary decays by $1/e$ in a semi-infinite Newtonian fluid. The momentum diffusion equation gives $u=U_0{e}^{-z/\delta }\cos (\omega t-z/\delta )$. Both attenuation and phase lag are controlled by $z/\delta$. The corresponding fluid-displacement amplitude has the same spatial decay, because dividing each velocity amplitude by the common angular frequency does not change their ratio.

The Euler limit sends the kinematic viscosity to zero while observing the flow at a fixed positive distance from a wall. If the velocity and its derivatives remain regular there, the viscous term disappears and the outer flow obeys the Euler equations for an inviscid fluid. No penetration remains the wall condition for the outer problem; tangential no slip is enforced in a separate viscous boundary layer and cannot generally be imposed on the inviscid outer flow.

An inviscid core is the part of a high-Reynolds number flow outside thin wall or shear layers, where viscosity is negligible at the retained asymptotic order. It obeys the Euler equations for an inviscid fluid. Its tangential velocity need not satisfy no slip at a material wall; adjoining viscous boundary layers supply that condition and can influence the core through displacement and pressure matching.

A steady unidirectional incompressible viscous flow in a rigid channel, driven by $G=-dp/dz$, reduces the axial momentum equation to the displayed Poisson equation, with $w=0$ on the cross-sectional boundary. Circular-pipe flow is the Hagen-Poiseuille flow special case; an arbitrary cross section need not have that parabolic radial profile.

Rectangular duct flow is steady flow of a Newtonian fluid down a straight rectangular pipe under a constant axial pressure gradient $-G$. For stationary walls and unidirectional flow, incompressibility eliminates axial dependence and the Navier-Stokes equation becomes a two-dimensional Poisson equation with zero Dirichlet boundary conditions. Subtracting a plane Poiseuille flow profile leaves a harmonic function that repairs the side-wall values using separation of variables. Uniqueness follows from Green's first identity: the difference of two solutions has zero boundary values and zero integral of the squared gradient, and hence vanishes.

In a symmetric wedge $|\theta |<\alpha$, a constant Poisson forcing admits the particular solution $r^2(\cos 2\theta /\cos 2\alpha -1)/4$, except at resonant angles. Zero-wall harmonic corrections have exponents ${\lambda }_n=(2n+1)\pi /(2\alpha )$. If the smallest exponent is below two, the global boundary data make that homogeneous mode dominate the local quadratic forcing. If it is below one, the velocity gradient can be singular although the velocity vanishes. When an exponent equals two, the appropriate radial solution contains $r^2\log r$, since $\Delta (r^2\log r\cos 2\theta )=4\cos 2\theta$.

For a wall shear $u\simeq sz$, heating over streamwise length $L$ gives an outer temperature rise proportional to $L/(sz)$. Its transverse thermal conduction grows as $z^{-3}$ relative to the local source. If $\chi$ is thermal diffusivity, comparing conduction with heating gives $\delta \sim (\chi L/s{)}^{1/3}$, up to a numerical constant depending on the precise comparison. The same scale follows by comparing transverse diffusion time $z^2/\chi$ with residence time $L/(sz)$.

For a unidirectional incompressible flow between stationary plates a distance $H>0$ apart, removing the driving pressure gradient leaves $u_t=\nu u_{yy}$ when the kinematic viscosity $\nu >0$ is constant. The no-slip boundary condition gives $u(0,t)=u(H,t)=0$. For an initial profile $g$ in L2 space, separation of variables and the Fourier sine series give the displayed solution, withEach normal mode decays at a rate proportional to $n^2$. The lowest mode present dominates at late times; when $b_1\ne 0$, its time constant is $H^2/(\nu {\pi }^2)$, consistent with the viscous diffusion time. The solution approaches the initial data in L2 space; compatible smooth profiles also give the familiar pointwise initial condition. This is a heat equation for momentum diffusion, rather than temperature.

At large Reynolds number $\mathcal{R}$, a long indentation of relative height $ϵ$ has nonlinear near-wall interaction when the streamwise scale ratio satisfies $\lambda /\mathcal{R}={ϵ}^3$. The regime ${\mathcal{R}}^{1/7}\ll \lambda \ll \mathcal{R}$ gives a thin viscous layer and an effectively common pressure across the core. The core displacement is $u_1=A(x)U_0^{\prime }(y)$ and $v_1=-A^{\prime }(x)U_0(y)$. A lower wall displaced to $ϵF(x)$ produces layer displacement $F+A$, while the upper wall produces $-A$. Uniqueness for their common pressure forcing and upstream data makes the two layer displacements equal, so $A=-F/2$.

For indentation $F=b{x}^{1/3}$, the common layer displacement is $H=b{x}^{1/3}/2$. The boundary layer stream function $\psi =x^{2/3}G(z{x}^{-1/3})$ gives velocity $U=x^{1/3}{G}^{\prime }$ and pressure $P=\bar{p}{x}^{2/3}$. Substitution into the Prandtl boundary-layer equation yields the displayed equation, with $G(0)=G^{\prime }(0)=0$ and $G^{\prime }\sim 6\eta +3b$. This is a conditional similarity construction: the constant $\bar{p}$ must permit a solution of the boundary value problem.

Two viscous layers occupy $-a<y<0$ and $0<y<a$, with viscosities ${\mu }_{-}$ and ${\mu }_{+}$. A uniform driving pressure gradient $G=-p_x>0$, a stationary flat interface and no-slip outer walls give the displayed plane Poiseuille flow, where $U_0=G{a}^2/({\mu }_{-}+{\mu }_{+})$ and $C=Ga({\mu }_{-}-{\mu }_{+})/[2({\mu }_{-}+{\mu }_{+})]$. These constants follow from continuity of velocity and shear stress at the interface and zero wall velocities. The upper-to-lower volume-flux ratio is $(1+7r)/[r(7+r)]$, with $r={\mu }_{+}/{\mu }_{-}$. As $r\to \infty$, the interface becomes effectively stationary, the upper flux vanishes, and the lower layer approaches flow between two stationary plates a distance $a$ apart. The upper wall stress remains finite despite its vanishing velocity gradient.

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.

The steady planar stagnation point boundary layer with outer velocity $(Ax,-Ay)$ has the similarity solution $u=Ax{f}^{\prime }(\eta )$, $v=-\sqrt{A\nu }f(\eta )$, $\eta =y\sqrt{A/\nu }$. The Prandtl boundary-layer equation becomes $f^{\prime \prime \prime }+f{f}^{\prime \prime }-(f^{\prime }{)}^2+1=0$, with $f(0)=f^{\prime }(0)=0$ and $f^{\prime }(\infty )=1$. Its thickness is of order $\sqrt{\nu /A}$, independent of horizontal distance. Normal dynamic pressure corrections are of smaller order than the tangential driving pressure.

The Prandtl limit observes a high-Reynolds number flow within a shrinking wall-normal layer. For streamwise distance $x$ and outer speed $U$, rescale the normal distance by $\delta =\sqrt{\nu x/U}$ and the normal velocity by $U\delta /x$. Convection and normal viscous diffusion then remain comparable, while streamwise viscous diffusion is smaller. The leading normal momentum equation makes pressure independent of the rescaled normal coordinate, and the tangential equation is the Prandtl boundary-layer equation. This laminar asymptotic balance is local and need not remain valid through separation or instability.

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 $-p_x/\rho =UdU/dx$.

For outer tangential speed $U_s=A{e}^{\alpha x}$ in steady Prandtl boundary-layer equations, the displayed similarity solution makes the streamwise convection, pressure gradient and transverse viscous diffusion have the same exponential factor. Its profile equation is $f^{\prime \prime \prime }+(\alpha /2)f{f}^{\prime \prime }+\alpha (A^2-(f^{\prime }{)}^2)=0$, with $f(0)=f^{\prime }(0)=0$ and $f^{\prime }(\infty )=A$. Attached far-field matching requires $\alpha A>0$: the outer speed increases along the direction of motion. Linearizing $g=f^{\prime }-A$ at large $\eta$ gives $g^{\prime \prime }+(\alpha A/2)\eta g^{\prime }-2\alpha Ag\simeq 0$. For positive $\alpha A$ there is a Gaussian-decaying mode, while negative $\alpha A$ leaves only growing nontrivial asymptotic modes. Local thinness additionally requires a large Reynolds number based on the speed and its streamwise variation length.

For a steady incompressible viscous boundary layer over a flat plate with constant outer speed $U$, the stream function $\psi =\sqrt{\nu Ux}f(y\sqrt{U/(\nu x)})$ reduces the Prandtl boundary-layer equation to the displayed ordinary differential equation. The wall and outer conditions are $f(0)=f^{\prime }(0)=0$ and $f^{\prime }(\infty )=1$. The wall shear is $\rho U\sqrt{\nu U/x}{f}^{\prime \prime }(0)$, so its integral on one plate face of length $L$ is $2\rho U^{2}L{f}^{\prime \prime }(0)/\sqrt{UL/\nu }$. Both equally exposed faces contribute twice this value.

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.

A rotating inner cylinder of radius $R$ and height $H$ has a uniform gap $h$ from a fixed housing at its side and both ends. For $h\ll R,H$, no-slip boundary condition and laminar Couette flow give side torque $2\pi \eta \omega R^{3}H/h$. Each end contributes ${\int }_0^R(\eta \omega s/h)(2\pi sds)s=\pi \eta \omega R^4/(2h)$. Summing both end faces proves the displayed expression. Shaft area, corner regions, gas slip and viscous heating are neglected.

In the thin-gap, laminar approximation, a rotating disc of radius $a$ separated by $d$ from a stationary disc shears a Newtonian fluid. The resisting torque is $\pi \eta \omega a^4/(2d)$, neglecting edge effects and secondary flows. This gives a way of inferring dynamic viscosity from a measured torque.

Taylor-Couette flow is the flow between coaxial rotating cylinders. Its simplest steady homogeneous purely azimuthal viscous state has steady circular Couette profile $v_{\varphi }=Ar+B/r$. Stability of that base state depends on rotation rates, geometry and viscosity; the inviscid Rayleigh centrifugal stability criterion addresses axisymmetric centrifugal disturbances, not every viscous instability.

For a homogeneous Newtonian fluid with constant kinematic viscosity, a steady axisymmetric purely azimuthal flow obeys $v_{\varphi }^{\prime \prime }+v_{\varphi }^{\prime }/r-v_{\varphi }/r^2=0$. Writing $v_{\varphi }=r\Omega$ gives $(r^3{\Omega }^{\prime }{)}^{\prime }=0$, so $\Omega =A+B/r^2$. A domain containing the rotation axis requires $B=0$ for regularity; rotating walls in an annulus can sustain $B\ne 0$. An arbitrary time-dependent profile instead evolves by ${\partial }_t\Omega =\nu ({\Omega }^{\prime \prime }+3{\Omega }^{\prime }/r)$ within this ansatz.

Couette-Poiseuille flow superposes wall-driven shear and pressure-driven parabolic flow between no-slip parallel plates. With one plate moving at $U$, separation $a$, and $dp/dx=-\rho \nu G$, the displayed profile has volume flux $Ua/2+G{a}^3/12$ per unit width. Requiring zero flux gives $G=-6U/a^2$ and $u=U[3(y/a{)}^2-2y/a]$, with a pressure gradient induced by return-flow constraints.

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.

An inverted falling film flows on the underside of an inclined solid surface. Taking the normal coordinate from the solid into the liquid reverses the normal component of gravity relative to an upper-surface film. In the Benney equation, the hydrostatic contribution to the volume flux per unit width therefore changes from $-2\epsilon h^3{h}_x\cot \alpha /3$ to its negative. Normal gravity then amplifies disturbances of long wavelengths, while surface tension resists short wavelengths.

For downslope length scale $L$, film thickness scale $h_0$ and surface velocity scale $U_0=\rho g{h}_0^2\sin \alpha /(2\mu )$, this normalization makes the capillary part of the Benney equation flux $\epsilon S{h}^3{h}_{xxx}$. The dimensionless capillary pressure is $-3S{h}_{xx}/2$. Keeping $S=O(1)$ specifies a distinguished long-wave approximation in which surface tension remains relevant.

The Benney equation is a long-wave approximation for a gravity-driven thin film that expresses its volume flux per unit width locally in terms of height and its derivatives. For surface-velocity scaling on a plane inclined by $\alpha$, one normalization isHere $\epsilon$ is the film aspect ratio, $\mathrm{Re}$ the thickness-based Reynolds number, and $S$ the capillary parameter for a falling film. The leading velocity is parabolic. Its first inertial correction uses the leading mass conservation equation to eliminate the time derivative of the height. This local closure requires the correction to remain small; it does not describe arbitrary large-amplitude short-wave dynamics.

A uniform steady film of thickness $d$ on a plane inclined by $\theta$ above horizontal has downslope coordinate $s$ and outward normal coordinate $x$. Neglecting air shear, the pressure is $p_{\mathrm{atm}}+\rho g\cos \theta (d-x)$ and its downslope gradient vanishes. The Newtonian fluid momentum equation is $\eta v^{\prime \prime }+\rho g\sin \theta =0$. The no-slip boundary condition $v(0)=0$ and shear-free free surface $v^{\prime }(d)=0$ give the displayed parabolic profile. Integrating over thickness gives volume flux per unit width $q$. The wall shear $\rho gd\sin \theta$ balances the film's downslope weight, and the vertical-wall case is $\theta =\pi /2$.

Let the lower fluid have thickness $h$ and viscosity $\mu$, and the upper fluid thickness $\alpha h$ and viscosity $\beta \mu$, with common density $\rho$. For steady flow down a plane inclined at angle $\theta$, set $H=(1+\alpha )h$, $A=\rho g\sin \theta /\mu$. The common hydrostatic pressure is $p_{\mathrm{atm}}+\rho g\cos \theta (H-y)$. No slip at the wall, continuous velocity and shear stress at $y=h$, and zero upper surface shear give $u_1=A(Hy-y^2/2)$ and $u_2(h+z)=A{h}^2(\alpha +1/2)+(A/\beta )(\alpha hz-z^2/2)$. These follow by integrating ${\mu }_i{u}_i^{\prime \prime }=-\rho g\sin \theta$ in each layer. The stress applied to the lower fluid equals the downslope weight of the upper layer, independent of its viscosity; the upper viscosity affects its shear rate but not that transmitted stress.

A falling free surface does not instantaneously dry previously wetted rigid walls. Thin films remain above the bulk liquid and drain under the component of gravitational acceleration parallel to the walls. For a film of thickness $b$ over a wetted distance $\ell$, falling film flow gives a speed of order $g{b}^2/\nu$, hence $b\sim (\nu \ell /(gt){)}^{1/2}$ at late times for an order-one wall angle. These films can be visible in a cross-section while holding asymptotically negligible volume. In a V-shaped channel gravity current, local bulk depth falls only as $t^{-1/7}$; at a fixed nonzero station with a finite previous maximum wetted height, film volume decays as $t^{-1/2}$ while bulk-sectional volume decays as $t^{-2/7}$. Thus the film does not change the leading spreading law.