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the wetting flows ( where the triple solid - liquid - fluid contact line is present ) are important for many practical applications ranging from the metal coating to the medical treatment of the lung airways . the contact line statics and dynamics attracted a lot of attention from the scientific community during the last decades . it became clear that the hydrodynamics in the region of the liquid wedge close to the contact line ( which we will call clr - contact line region ) differs from the hydrodynamics in the bulk of the liquid . because of the contact line singularity @xcite , the fluid motion in presence of the contact line appears to be much slower than without it . this difference can be accounted for by the introduction of the anomalously large energy dissipation inside the clr @xcite .    for practical purposes , one needs to know the dynamics of the liquid surface influenced by this dissipation . this influence is especially strong in the very common case of ( i ) low viscosity fluids that ( ii ) wet partially the solid with ( iii ) no precursor film on it . in this case the bulk dissipation is particularly small with respect to the large clr dissipation @xcite . the contact line motion is very slow and the liquid surface can be described in the quasi - static approximation @xcite . the dissipation in the liquid ( energy per unit time ) can then be approximated by the dissipation in the clr as @xcite @xmath0 where @xmath1 is the normal component of the contact line velocity and the integration is performed along the contact line . the generalized dissipation coefficient @xmath2 is a constant that is assumed to be much larger that the liquid shear viscosity . the expression ( [ diss ] ) assumes that the contribution of a piece of the clr to the dissipation is proportional to the contact line length . then the @xmath3 term is leading for small @xmath1 . the dissipation coefficient @xmath2 can be obtained e.g. from measurements of the kinetics of relaxation of an oval sessile drop toward its equilibrium shape ( see @xcite , where it was found @xmath4 times larger than the shear viscosity ) . it can also be obtained from the measurements of @xmath1 and the dynamic contact angle @xmath5 by using the expression @xmath6 where @xmath7 is the equilibrium value of the contact angle and @xmath8 is the surface tension . [ cos ] is common for many contact line motion models . it can be shown @xcite that eq . [ cos ] follows from the expression ( [ diss ] ) for the cases where @xmath1 does not vary along the contact line ( i.e. for the contact line of constant curvature ) .    usually , one is interested to know the displacement of the contact line ( or its statistical properties in the case of the irregular solid ) because its position serves as a boundary condition for the determination of the shape of the liquid surface . the recent articles on this subject show that no general approach to this problem is accepted . it is recognized generally @xcite that the static contact line equation should be non - local because the contact line displacement at one point influences its position at other points through the surface tension . in dynamics , the local relations similar to eq . [ cos ] were considered universal for a long time . in our previous work @xcite we developed a non - local dynamic approach . we showed that if non - locality is taken into account , eq .  [ cos ] is not valid in the general case where @xmath1 varies along the contact line . this variation can be due either to the initially inhomogeneous contact line curvature ( like in @xcite ) or to a inhomogeneous substrate . in this letter we analyze this latter case . we use this non - local approach to derive an equation of motion for the very common case of the capillary rise of the liquid along a vertical solid wall with the account of the surface defects . a cartesian reference system @xmath9 is chosen in such a way that the liquid surface far from the wall ( defined by the @xmath10 plane ) coincides with the @xmath11 plane , see fig . [ wilh ] . the liquid surface described by the equation @xmath12 is assumed to be weakly deformed so that @xmath13 the contact line can be described by the equation @xmath14 . a piece of the liquid surface of length @xmath15 in the @xmath16-direction ( see fig .  [ wilh ] ) is considered so that the final form for the equation of motion is found in the limit @xmath17 . first , following the approach @xcite , we find the energy @xmath18 of the liquid ( per contact line length ) as a functional of @xmath19 . it is convenient to break @xmath18 into two terms , @xmath20 . in the approximation ( [ weak ] ) , the first term reads @xmath21,\ ] ] where @xmath22^{1/2}$ ] is the capillary length , @xmath23 is the liquid density and @xmath24 is the gravity acceleration . the defects are accounted for in the second term @xcite @xmath25 by the fluctuations of the function @xmath26 which is the difference of the surface energies ( in @xmath8 units ) of the gas - solid and liquid - solid interfaces at the point @xmath27 of the solid surface . according to the young expression , @xmath28 $ ] , where @xmath7 is the local value of the equilibrium contact angle . when the surface defects exist , @xmath7 is an arbitrary function of @xmath29 .    in the quasi - equilibrium approximation , the surface shape @xmath30 is found by minimization of the functional @xmath18 resulting in the equation @xmath31 which can be solved by separating the variables by using the boundary condition @xmath32 and assuming that the function @xmath33 is bounded at @xmath34 . the solution reads @xmath35\exp\left(-x\sqrt{l_c^{-2}+\pi^2 n^2/l^2}\right),\ ] ] where the coefficients @xmath36 , @xmath37 , @xmath38 are the coefficients for the fourier series @xmath39.\ ] ] they can thus be related to @xmath40 by the expressions @xmath41 one can notice that @xmath36 is simply the horizontally averaged displacement of the contact line . the back substitution of eq . ( [ f ] ) with @xmath36 , @xmath37 , @xmath38 replaced by their values ( [ abh ] ) into eq . ( [ u1 ] ) results in the functional @xmath42.\ ] ] eq . ( [ u2 ] ) together with eq . ( [ funcu1 ] ) define the functional @xmath43 $ ] . now , one needs to calculate the dissipation function @xmath44 ( per unit length ) . in the approximation ( [ weak ] ) , @xmath45 in eq . ( [ diss ] ) and @xmath46 where the dot means a time derivative . the equation of motion can be written in the quasi - static approximation as @xmath47\over \delta h}=-{\delta t[\dot h]\over \delta \dot h},\ ] ] where @xmath48 means variational derivative . by substituting eqs . ( [ u2],[funcu1],[t ] ) into eq . ( [ var ] ) and taking the variational derivatives , one obtains the dynamic equation of the contact line motion : @xmath49,\ ] ] where @xmath19 and @xmath36 are assumed to be time dependent . the final form for the integral equation for the contact line motion can be obtained by taking the limit @xmath17 : @xmath50\sqrt{l_c^{-2}+p^2}\biggr\}.\ ] ]    to treat the contact line equation , it is convenient to introduce the spatial fluctuation @xmath51 and solve the equations for @xmath52 and @xmath36 separately . first , we derive the dynamic equation for @xmath36 by integrating eq . ( [ hl ] ) over @xmath16 from @xmath53 to @xmath54 , changing the integration order and dividing by @xmath15 . the integrals of all terms in the sum are zero and one obtains the following equation for @xmath36 @xmath55 where @xmath56 is the ( time dependent ) horizontal average of @xmath57 . by subtracting eq . ( [ h0 ] ) from eq . ( [ hh ] ) one obtains the equation for @xmath52 which has the same form as eq . ( [ hh ] ) where @xmath57 should be replaced by its fluctuation @xmath58 . equation for @xmath52 simplifies in the fourier space . by making use of the convolution theorem @xcite , one obtains @xmath59 where the tilde over a variable denotes its fourier transform with the parameter @xmath60 , e.g. @xmath61 this concludes the derivation of the equation of contact line motion . the stationary version of eq . ( [ h1f ] ) was derived first by pomeau and vannimenus @xcite . the static contact line position is defined by the minimum of the energy @xmath43 $ ] . however , it can exhibit multiple minima that correspond to the metastable states . in this case one needs to choose between them @xcite to determine correctly the contact line position and corresponding local contact angle ( which are both different in advancing and receding cases ) . in dynamics , eq . ( [ hh ] ) provides us with an unique solution because the system  knows " the direction of the contact line motion and its history . to show the relevance of this approach , we solve rigorously the contact line dynamics for a simple case of a single stripe - shaped defect at the wall , @xmath62 where @xmath63 and @xmath64 are constants . consider first the contact line displacement at equilibrium described by the equations @xmath65 notice that @xmath56 should be small enough so that @xmath30 defined by eq .  ( [ f ] ) satisfies conditions ( [ weak ] ) . obviously , eq .  ( [ h0 ] ) results in @xmath66 and @xmath67 where @xmath68 . to show how this result relates to earlier approaches to the single - defect problem , we analyze first the limit @xmath69 . the inverse transform can easily be taken using the tables @xcite and results in @xmath70     of the contact line from its horizontally averaged value . the defect area ( half - width of which is @xmath71 ) is shadowed . the time values in the units @xmath72 are shown above the corresponding curve.,height=264 ]    where @xmath73 is the modified bessel function of zeroth order . one recognizes the result @xcite . it leads to an unphysical divergence when @xmath74 : @xmath75 and a cut - off at small @xmath16 is needed @xcite .    using the present approach , one can understand the origin of this divergence . the limit @xmath69 in the rigorous result ( [ h1eq ] ) is equivalent to the limit @xmath76 since their product enters eq . ( [ h1eq ] ) . the asymptotics at @xmath76 in the fourier transform corresponds @xcite to the limit @xmath77 in the original function , which means that eq . ( [ hd ] ) is correct only at @xmath77 . however , it is not guaranteed that its asymptotics at @xmath74 is correct . let us return now to the rigorous expression ( [ h1eq ] ) . the fourier transform ( [ h1eq ] ) can be inverted analytically , @xmath78/\pi,\ ] ] where @xmath79 . it is clear now that @xmath80 remains finite , see the dotted curve in fig . [ stripe ] . the dynamic solution can be obtained the same way by using the initial condition @xmath81 : @xmath82\ } , \\ h_1(y , t)=\delta c\left[f(\delta+y)+f(\delta - y)\right.\left .- g(\delta+y , t)-g(\delta - y , t)\right]/\pi ,    \end{array}\ ] ] where @xmath83 the time evolution of @xmath84 is shown in fig . [ stripe ] . these curves can be compared to those obtained experimentally in @xcite where the motion of the contact line over a single defect was studied . the comparison shows a good qualitative agreement . the results can not be compared quantitatively since in each of these articles some parameters that enter eq . [ h1d ] ( @xmath85 in particular ) are missing . ( [ hh ] ) describes the spontaneous contact line motion for _ arbitrary _ distribution of surface energy given by the function @xmath26 ( provided @xmath86 is small enough so that the conditions ( [ weak ] ) are satisfied ) . this function can be considered random and the equation becomes stochastic . it can be used to establish any statistical parameter of the contact line in dynamics . in particular , the collective effect of defects on the contact line motion can be studied .    in this work we solve a simple example where the defect properties do not vary along the average contact line velocity vector . in the general case , where the local value of the equilibrium contact angle ( or of the function @xmath57 ) varies along this direction , the equation of the contact line motion becomes non - linear . more sophisticated methods are then needed to solve it . g. d. nadkarni & s. garoff , europhys . lett . * 20 * , 523 ( 1992 ) . j. m. marsh & a .- m . cazabat , phys . rev . lett . * 71 * , 2433 ( 1993 ) ; europhys . lett . * 23 * , 45 ( 1993 ) . a. paterson , m. fermigier , p. jenffer , & l. limat , phys . e * 51 * , 1291 ( 1995 ) .
the wetting flows are controlled by the contact line motion . 
 we derive an equation that describes the slow time evolution of the triple solid - liquid - fluid contact line for an arbitrary distribution of defects on a solid surface . 
 the capillary rise along a partially wetted infinite vertical wall is considered . 
 the contact line is assumed to be only slightly deformed by the defects . 
 the derived equation is solved exactly for a simple example of a single defect .