Blasius boundary layer
In physics and fluid mechanics, a Blasius boundary layer (named after Paul Richard Heinrich Blasius) describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow. Falkner and Skan later generalized Blasius' solution to wedge flow (Falkner–Skan boundary layer), i.e. flows in which the plate is not parallel to the flow. Prandtl's boundary layer equations. Using scaling arguments, Ludwig Prandtl has argued that about half of the terms in the Navier-Stokes equations are negligible in boundary layer flows (except in a  small region near the leading edge of the plate). This leads to a reduced set of equations known as the boundary layer equations. For steady incompressible flow with constant viscosity and density, these read: Mass Continuity: formula_1 formula_2-Momentum: formula_3 formula_4-Momentum: formula_5 Here the coordinate system is chosen with formula_2 pointing parallel to the plate in the direction of the flow and the formula_4 coordinate pointing normal to the plate, formula_8 and formula_9 are the formula_2 and formula_4 velocity components, formula_12 is the pressure, formula_13 is the density and formula_14 is the kinematic viscosity. A number of similarity solutions  to this set of equations have been found for various types of flow, including flow on a thin flat-plate. The term "similarity" refers to the property that the velocity profiles at different positions in the flow are the same apart from scaling factors. Similarity scaling factors reduce the set of partial differential equations to a relatively easily solved set of non-linear ordinary differential equations. Paul Richard Heinrich Blasius, one of Prandtl’s students, developed the similarity model corresponding to the flow for the case where the pressure gradient, formula_15/formula_16, along a thin flat-plate is negligible compared  to any pressure gradient in the boundary layer region. Blasius equation - first-order boundary layer. Blasius showed that for the case where formula_15/formula_16=0, the Prandtl formula_2-momentum equation has a self-similar solution. The self-similar solution exists because the equations and the boundary conditions are invariant under the transformation where formula_21 is any positive constant. He introduced the self-similar variables where formula_23 is the boundary layer thickness, formula_24 is the free stream velocity, and formula_25 is the stream function. The stream function is directly proportional to the normalized function, formula_26, which is only a function of the  similarity thickness variable. This leads directly to the velocity components: Where the prime denotes derivation with respect to formula_28. Substitution into the formula_2-momentum equation gives the Blasius equation The boundary conditions are the no-slip condition, the impermeability of the wall and the free stream velocity outside the boundary layer This is a third-order non-linear ordinary differential equation which can be solved numerically, e.g. with the shooting method. With the solution for formula_34 and its derivatives in hand, the Prandtl formula_4-momentum equation can be non-dimensionalized and rearanged to obtain the formula_4-pressure gradient, formula_15/formula_38 , as where formula_40 is the Blasius displacement thickness. The Blasius normal velocity formula_41 and the formula_4-pressure gradient asymptotes to a value of 0.86 and 0.43, respectively, at large formula_43-values whereas formula_44 asymptotes to the free stream velocity formula_24. As formula_43 goes to zero, the scaled formula_4-pressure gradient goes to 0.16603. The limiting form for small formula_48 is and the limiting form for large formula_50 is The characteristic parameters for boundary layers are the two sigma viscous boundary layer thickness, formula_52, the displacement thickness formula_40, the momentum thickness formula_54, the wall shear stress formula_55 and the drag force  formula_56 acting on a length formula_57 of the plate. For the Blasius solution, they are given by The factor formula_63 in the drag force formula is to account both sides of the plate. The Von Kármán Momentum integral and the energy integral for Blasius profile reduce to where formula_55 is the wall shear stress, formula_67 is the wall injection/suction velocity, formula_68 is the energy dissipation rate, formula_69 is the momentum thickness and formula_70 is the energy thickness. Uniqueness of Blasius solution. The Blasius solution is not unique from a mathematical perspective, as Ludwig Prandtl himself noted it in his  transposition theorem and analyzed by series of researchers such as Keith Stewartson, Paul A. Libby. To this solution, any one of the infinite discrete set of eigenfunctions can be added, each of which satisfies the linearly perturbed equation with homogeneous conditions and exponential decay at infinity. The first of these eigenfunctions turns out to be the formula_2 derivative of the first order Blasius solution, which represents the uncertainty in the effective location of the origin. Second-order boundary layer. This boundary layer approximation predicts a non-zero vertical velocity far away from the wall, which needs to be accounted in  next order outer inviscid layer and the corresponding inner boundary layer solution, which in turn will predict a new vertical velocity and so on. The vertical velocity at infinity for the first order boundary layer problem from the Blasius equation is The solution for second order boundary layer is zero. The solution for outer inviscid and inner boundary layer are Again as in the first order boundary problem, any one of the infinite set of eigensolution can be added to this solution. In all the solution formula_74 can be considered as a Reynolds number. Third-order boundary layer. Since the  second order inner problem is zero, the corresponding corrections to third order problem is null i.e., the third order outer problem is same as second order outer problem. The solution for third-order correction don't have an exact expression, but inner boundary layer expansion have the form, where formula_76 is the first eigensolution of the first order boundary layer solution (which is formula_2 derivative of the first order Blasius solution) and solution for formula_78 is nonunique and the problem is left with an undetermined constant. Blasius boundary layer with suction. Suction is one of the common methods to postpone  the boundary layer separation. Consider a uniform suction velocity at the wall formula_79. Bryan Thwaites showed that the solution for this problem is same as the Blasius solution without suction for distances very close to the leading edge. Introducing the transformation into the boundary layer equations leads to with boundary conditions, Von Mises transformation. Iglisch obtained the complete numerical solution in 1944. If further von Mises transformation is introduced then the equations become with boundary conditions, This parabolic partial differential equation can be marched starting from formula_87 numerically. Asymptotic suction profile. Since the convection due to suction and the diffusion  due to the solid wall are acting in the opposite direction, the profile will reach steady solution at large distance, unlike the Blasius profile where boundary layer grows indefinitely. The solution was first obtained by Griffith and F.W. Meredith. For distances from the leading edge of the plate formula_88, both the boundary layer thickness and the solution are independent of formula_2 given by Stewartson studied matching of full solution to the asymptotic suction profile. Compressible Blasius boundary layer. Here Blasius boundary layer with a specified specific enthalpy formula_91 at the wall is studied. The density formula_13, viscosity formula_93 and thermal  conductivity formula_94 are no longer constant here. The equation for conservation of mass, momentum and energy become where formula_96 is the Prandtl number with suffix formula_97 representing properties evaluated at infinity. The boundary conditions become Unlike the incompressible boundary layer, similarity solution exists only if the transformation holds and this is possible only if formula_101. Howarth transformation. Introducing the self-similar variables using Howarth–Dorodnitsyn transformation the equations reduce to where formula_104 is the specific heat ratio and formula_105 is the Mach number, where formula_106 is the speed of sound. The equation can be solved once formula_107 are specified. The  boundary conditions are The commonly used expressions for air are formula_109. If formula_110 is constant, then formula_111. The temperature inside the boundary layer will increase even though the plate temperature is maintained at the same temperature as ambient, due to dissipative heating and of course, these dissipation effects are only pronounced when the Mach number formula_112 is large. First-order Blasius boundary layer in parabolic coordinates. Since the boundary layer equations are Parabolic partial differential equation, the natural coordinates for the problem is parabolic coordinates. The transformation from Cartesian coordinates formula_113 to parabolic coordinates formula_114 is given by Blasius boundary