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For constant injected power $\dot{E}$ and ambient mass density ${\rho }_1$, dimensional analysis gives $R=\alpha (\dot{E}/{\rho }_1{)}^{1/5}{t}^{3/5}$. In a similarity solution, set $\eta =r/R$, $\rho ={\rho }_{1}f(\eta )$, $p={\rho }_1{\dot{R}}^{2}g(\eta )$ and $u=\dot{R}h(\eta )$. The strong-shock values on the inner side of $\eta =1$ are $f=(\gamma +1)/(\gamma -1)$ and $g=h=2/(\gamma +1)$.

Let $y$ solve the Airy ordinary differential equation $y^{\prime \prime }(s)=sy(s)$. For $t>0$, set $s=x/(3t{)}^{1/3}$ and define the displayed similarity solution. Differentiating the profile equation gives $y^{\prime \prime \prime }=y+s{y}^{\prime }$, so the chain rule yieldsTheir sum is zero, proving the Airy equation $u_t+u_{xxx}=0$. Every real linear combination of the two standard Airy functions gives such a solution. This is a solution of the linear dispersive equation; the nonlinear Korteweg-De Vries equation includes an extra term and does not follow from the same arbitrary profile construction.

The Barenblatt solution is a mass-preserving similarity solution of the porous medium equation. For $u_t=\beta (u{u}_x{)}_x$ in one dimension it has the form

A two-phase diffusion model with constant properties has similarity solutions built from the complementary error function in each half-line. Equilibrium gives $T_a=-m{C}_a$ and $C_s(a)=k_D{C}_a$. The Stefan condition and interfacial solute balance determine $\lambda$ and $C_a$; finite diffusion on both sides must be retained when constructing the narrow transition near $\lambda =0$.

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 the planar diffusion equation $n_t=D_0\nabla \cdot [(n/n_0{)}^2\nabla n]$, a mass-preserving radial similarity solution has $n=n_0{\lambda }^{-2}F(r/(r_0\lambda ))$. Substitution reduces the equation to $(\xi F^2{F}^{\prime }{)}^{\prime }=-({\xi }^{2}F{)}^{\prime }$ when ${\lambda }^5\dot{\lambda }=D_0/r_0^2$. Zero central flux and $F(0)=1$ give the compact profile shown above, with ${\lambda }^6=6D_{0}t/r_0^2$. Its height decays as $t^{-1/3}$ and its radius grows as $t^{1/6}$; its total mass is $2\pi n_0{r}_0^2/3$.

The symmetric similarity solution for a localized fixed volume $V$ in a V-shaped channel gravity current has $h(x,t)=[3(X^2-x^2)/(7Ct){]}_{+}^{1/3}$. Integrating the similarity conservation of mass equation once gives $C{h}^4{h}_x=-2x{h}^2/(7t)$, and vanishing thickness at the two noses gives the profile. Its volume fixes $X=[V/(I\tan \alpha ){]}^{3/7}(7Ct/3{)}^{2/7}$, where $I={\int }_{-1}^1(1-s^2{)}^{2/3}ds$. At any fixed nonzero position, the thickness first rises after arrival and then falls; its maximum occurs when $X^2=7x^2/3$.

For a one-sided draining gravity current with injected volume per unit width $V_0(t/\tau {)}^3$, defineThe similarity solution satisfieswhere $K=k[g^3{\tau }^3/({\nu }^3{V}_0){]}^{2/5}$. The inlet condition is $-H(0{)}^3{H}^{\prime }(0)=9$, the integral of $H+\varphi L$ is one, and both depths vanish at the advancing front. The endpoint ${\xi }_N$ gives $x_N={\xi }_{N}D{t}^2$ and depends on $K$ and $\varphi$.

The downstream edge of a continuously fed porous gravity current with background flow has a transition width proportional to $\sqrt{a{h}_{\mathrm{int}}t}$ around drift position $ut$. Its similarity solution satisfies $(FF\prime )\prime +\eta F\prime /2=0$, $F(-\infty )=1$, and $F({\eta }_N)=0$. A finite front has $F\sim ({\eta }_N/2)({\eta }_N-\eta )$. A clipped linear ramp is a useful mass-preserving approximation, but not an exact solution of this nonlinear transition equation.

In a uniformly wind-forced polynya with normalized heat loss $f$, let $H=L+c_{p}m{C}_{\infty }$, $K=f/H$, $J=3S/(4\rho )$ and $r=\rho -{\rho }_s$. The dilute equilibrium model gives $b{u}^2=Jx$, $\varphi bu=Kx$ and $rg\varphi b{b}^{\prime }=S/4$. Hence $u=(rgKx/\rho {)}^{1/3}$, $b=Jx/u^2$ and $\varphi =(K/J)u$. This similarity solution is valid only while the frazil ice volume fraction is small and no exported crystal flux has been neglected improperly.

The drift $-x{u}_x/(2t)$ exactly cancels the coordinate-motion term of the similarity solution $u=t^{-1/2}f(\eta )$, $\eta =x/\sqrt{t}$. Direct differentiation givesConsequently the nonautonomous partial differential equation reduces to the autonomous ordinary differential equation $f^{\prime \prime }=f^3-f/2$. The balance fixes the same length scale $x\sim \sqrt{t}$ as diffusion, but removes the first-derivative term that usually appears in the profile equation. A first-integral argument can then determine a whole profile family without solving a nonautonomous second-order equation.

The outer similarity solution approaches a plate edge as $h/h_0\sim [(4C/5)(1-x/L){]}^{1/4}$, where $C=B(4/5,1/2)$. Its slope becomes order one atThis region lies outside lubrication theory because its vertical and horizontal scales are comparable. The outer zero-thickness boundary condition is consistent at leading order since the matching height is $O(\ell )\ll h_0$, while the actual edge flow requires an inner solution. A finite outward flux survives even though the outer thickness vanishes, because the slope diverges.

The porous medium equation $h_t=\lambda (h{h}_x{)}_x-\Gamma h$ with constant $\Gamma >0$ becomes $H_{\tau }=\lambda (H{H}_x{)}_x$ under the displayed transformation. Substitution gives $h_t+\Gamma h=e^{-\Gamma t}{\tau }^{\prime }{H}_{\tau }$ and $\lambda (h{h}_x{)}_x=\lambda e^{-2\Gamma t}(H{H}_x{)}_x$, so ${\tau }^{\prime }=e^{-\Gamma t}$. This transformation applies to every solution with appropriately transformed initial and boundary conditions, not just a similarity solution. A one-sided constant-volume porous gravity current therefore has remaining physical volume $V{e}^{-\Gamma t}$ and a limiting front $[9\lambda V/(\varphi \Gamma ){]}^{1/3}$. The finite transformed time $\tau \to 1/\Gamma$ means that arbitrary initial profiles need not reach the universal large-time source shape before drainage removes them.

A localized amount $M=\int Cdx$ of insoluble surfactant spreads over a planar film of depth scale $h_0$. Neglect gravity, capillarity and diffusion. Then $C\sim M/\ell$ and Marangoni stress drives $u_s\sim AM{h}_0/(\mu {\ell }^2)$, giving $\ell \sim (AM{h}_{0}t/\mu {)}^{1/3}$. The thin-film equations with insoluble surfactant admit a similarity solution in which the film height and concentration depend on $x/\ell$, conserving total surfactant and fluid volume relative to the undisturbed layer.

For a one-sided fixed-volume gravity current in a unit-width channel with a closed rear wall, use front closure $\dot{L}=F\sqrt{g^{\prime }{h}_f}$ with $0<F<2$. An exact late-time similarity solution has $u=\dot{L}x/L$, $h=h_f[1-F^2/4+(F^2/4)(x/L{)}^2]$ and $h_{f}L=V/(1-F^2/6)$. Substituting into the shallow water equations proves the radius law. The Benjamin deep-ambient front condition has $F=\sqrt{2}$. The virtual time origin $t_v$ is fixed by matching to the finite release, not by volume alone.

For fixed volume $V$ and symmetric spreading, conservation of mass and the cusp flow between touching cylinders equation give width proportional to $t^{-1/8}$ and support length proportional to $t^{3/8}$. The source-type similarity solution has the displayed profile and $z_N=(8Va/\pi {)}^{1/4}(8\gamma t/(21\mu a){)}^{3/8}$. It is a formal outer lubrication solution; its microscopic nose requires a local description when the axial slope becomes large.

Consider a nonlinear diffusion equation with uniform supply, $\varphi h_t=D_m(h^m{h}_x{)}_x+R$, on $x>0$, with an absorbing boundary $h(0,t)=0$ and initially $h=0$. Far from the boundary, $h=Rt/\varphi$. Balancing the time derivative, supply, and diffusion givesThe similarity solution satisfiesThe outward boundary volume flux per unit width is $Q=R\ell c_m$, where $c_m={\lim }_{\eta \downarrow 0}{F}^m{F}^{\prime }$. Integrating the equation givesThus the growing region of depleted storage fixes the discharge prefactor, and $Q\propto t^{(m+1)/2}$.

On $z>0$, consider $p_t+a{p}_z=D(z{p}_z{)}_z$ with $a,D>0$, zero scalar flux $ap-Dz{p}_z$ at the endpoints for $t>0$, and unit initial impulse at the origin. Put $r=a/D$ and $\eta =z/(Dt)$. The normalized similarity solution isThe ansatz $p=(Dt{)}^{-1}f(\eta )$ gives $[\eta f^{\prime }+(\eta -r)f{]}^{\prime }=0$. Endpoint decay sets this integrated constant to zero and hence $f\propto {\eta }^r{e}^{-\eta }$; the gamma function normalizes its integral to one. Its scale is $Dt$, so it converges weakly to a unit impulse as $t\downarrow 0$. Its gamma distribution shape is $1+r$, its maximum occurs at $z=at$, and its expected value is $(a+D)t$. For width proportional to $z$, the associated horizontally averaged plume concentration is proportional to $z^{r-1}{e}^{-z/(Dt)}$: its interior maximum is $(a-D)t$ when $a>D$, its supremum occurs at the origin when $a=D$, and it is singular there when $0<a<D$. The physical finite source regularizes this ideal-origin behaviour.

A wide, shallow drop with cross-sectional area $A$ has the gravity-driven volume flux $q=-g{h}^3{h}_x/(3\nu )$ when substrate no-slip and zero surface shear are imposed. Mass conservation yields a porous medium equation, $h_t=[g/(3\nu )](h^3{h}_x{)}_x$. Its symmetric source-type similarity solution has half-width $\ell$ in the display and height $h=[9\nu ({\ell }^2-x^2)/(10gt){]}_{+}^{1/3}$, where $I={\int }_0^{\pi /2}{\cos }^{5/3}\theta d\theta$. Integrating the profile recovers $A$. The contact-edge slope diverges, so the bulk lubrication approximation requires an inner edge regularization if contact-line microphysics or surface tension is retained.

When the smoothing width exceeds the capillary length, gravity dominates capillarity. Balancing hydrostatic flux $\rho g{h}_0^4/(\mu \Delta )$ against the Marangoni stress flux gives $\Delta \sim \rho g{h}_0^2{x}_N^2/(AM)\propto t^{2/3}$. The condition is ${\Delta }^2\gg {\gamma }_0/(\rho g)$ in physical units. The smoothing region catches up with the spreading pool at $x_N\sim AM/(\rho g{h}_0^2)$ and $t\sim \mu (AM{)}^2/[(\rho g{)}^3{h}_0^7]$, after which the sharp-front outer similarity solution ceases to apply.