Równanie NS Płyta: $\frac{\text{dV}}{\text{dt}} + \frac{\partial Vy}{\partial t} + Vx\frac{\partial Vx}{\partial x} + Vy\frac{\partial Vy}{\partial y} + Vz\frac{\partial Vz}{\partial z} = Y - \frac{1}{\rho}*\frac{\partial P}{\partial x} + \nu\Delta Vy + \frac{\nu}{3}*\frac{\partial}{\partial y}(div\overrightarrow{V})|$ $,\ Vx = Vz = 0\left| y \rightarrow 0 \right|Vy = u \rightarrow \frac{\partial Vu}{\partial y} = 0\left| \text{wektor}\overrightarrow{V} = \overrightarrow{i}Vx + \overrightarrow{j}Vy + \overrightarrow{k}Vz = \overrightarrow{i}V + \overrightarrow{j}u + \overrightarrow{k}\text{w\ } \right|Vy = u = \frac{\partial u}{\partial t} = \frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}u}{\partial y^{2}} + \frac{\partial^{2}u}{\partial z^{2}} \rightarrow \frac{\partial u}{\partial t} = v\frac{\partial^{2}u}{\partial y} \rightarrow kierunek\ x\ |u = f\left( y,t \right);\frac{\partial p}{\partial u} = 0 \rightarrow kierunek\ y|\ war.\ brzeg.:y = 0;t \geq 0 \rightarrow u\left( 0,t \right) = 0\ |$ S=$\frac{y}{\sqrt{\text{νt}}};\frac{\partial u}{\partial t} = \frac{\partial u}{\partial s}\frac{\partial s}{\partial t} = \frac{\partial u}{\partial s}\frac{1}{\sqrt{\nu}}x*\left( - \frac{1}{2}t^{\frac{3}{2}} \right) = - \frac{1}{2}\frac{xt^{- \frac{1}{2}}}{\sqrt{v}t}*\frac{\partial u}{\partial t} = - \frac{1}{2}\frac{s}{t}\frac{\partial u}{\partial t}$ |$\frac{\partial u}{\partial y} = \frac{\partial u}{\partial s}\frac{\partial s}{\partial y} = \frac{\partial u}{\partial s}\frac{1}{\sqrt{\text{νt}}} \rightarrow \frac{\partial^{2}u}{\partial y^{2}} = \frac{\partial}{\partial y}\left( \frac{\partial u}{\partial s}\frac{1}{\sqrt{\text{νt}}} \right) = \frac{1}{\sqrt{\text{νt}}}\frac{\partial}{\partial y}\left( \frac{\partial u}{\partial s} \right) = \frac{1}{\sqrt{\text{νt}}}\frac{\partial}{\partial s}\left( \frac{\partial u}{\partial y} \right) = \frac{1}{\sqrt{\text{νt}}}$ $\frac{\partial}{\partial s}(\frac{1}{\sqrt{\text{νt}}}$ $\frac{\partial u}{\partial s}) = \ \frac{1}{\text{vt}}\frac{\partial^{2}u}{\partial s^{2}}\ \left| - \frac{1}{2}\frac{s}{t}\frac{\text{du}}{\text{ds}} = v\frac{1}{\text{vt}}\frac{d^{2}u}{ds^{2}} \right|\frac{d^{2}u}{ds^{2}} + \frac{1}{2}\frac{\text{sdu}}{\text{ds}} = 0\backslash:\frac{\text{du}}{\text{ds}}\ |\frac{\frac{d}{\text{ds}}\left( \frac{\text{du}}{\text{ds}} \right)}{\frac{\text{du}}{\text{ds}}} = - \frac{1}{2}s\backslash*\int_{}^{}{|\ ln\frac{\text{du}}{\text{ds}}} = - \int_{}^{}{\frac{1}{2}sds = - \frac{1}{4}s^{2} + C_{1}|\ \frac{\text{du}}{\text{ds}} = e^{- \frac{1}{4}s^{2}} + C = Ce^{- \left( \frac{s}{2} \right)^{2}\ }*\int_{0}^{s}{|\ u(ss\hat{}s = ) - u(0) = C_{1}\int_{0}^{s}{e^{- \left( \frac{s}{2} \right)^{2}\ }\text{ds\ }\left| \text{zmiana\ z}\text{zmiana\ }miennych\ s = \xi \right|\text{schnn}\ }}}\text{\ \ }\frac{s}{2} = \xi \rightarrow ds = 2d\xi\ ;s = 2\xi|\ u\left( \xi \right) - u(0) = C\int_{0}^{2\xi}{e^{- \xi^{2}}\ d\xi}$ |$0 - u\left( o \right) = C\int_{0}^{\infty}{e^{- \xi^{2}}\ d\xi = c\frac{\sqrt{\pi}}{2}}|C = - \frac{2}{\sqrt{\pi}}u|$ u($\ \xi) - u\left( o \right) = u\frac{2}{\sqrt{\pi}}$ $\int_{0}^{2\xi}{e^{- \xi^{2}}d\xi} = u\frac{2}{\sqrt{\pi}}\int_{0}^{s}{e^{- \xi^{2}}d\xi}|\ u\left( s \right) = u(1 - \frac{2}{\sqrt{\pi}}$ ∫0se−ξ2dξ )|$s = \frac{y}{\sqrt{\text{νt}}}$ ; $g\left( s \right) = \frac{u(\frac{1y}{\sqrt{\text{νt}}}\ )}{u}$ = $1 - \frac{2}{\sqrt{\pi}}$ ∫0s e−ξ2dξ = ε