 Swiss Mathematician and Physicist  Born in the early 1700s  Made mathematical applications to fluid mechanics

 Pursued a business degree as there were no applications of a mathematics degree  Shifted to a medical profession and earned a PhD in botany and anatomy  Was close friends with Euler  Became a professor in 1724

 ½ ρ μ ^2+P=constant ρ = density of the fluid around a body μ =velocity of fluid P=Pressure

 French Engineer and Physicist  Contributions include a useable Theory of Elasticity and Navier-Stokes Equations

 Lived in the late 18 th century and early 19 th century  Attended Ecole Polytechnique  Choisy, Asnieres and Argenteuil in the Department of the Seine, and built a foot bridge to the Ill de la Cite in Paris

 Describes the motion of viscous fluid substances  As oppose to Euler’s equations for inviscid flow, Navier-Stokes equations are for a dissipative system  Navier-Stokes Equations are utilized in the explanations of fluid viscosity in boundary layer

v = kinematic viscosity u = velocity of the fluid parcel P = the pressure ρ = fluid density

 Layer in immediate vicinity of the bounding surface  Part of the flow where viscous forces distort the inviscid flow

 Boundary-Layer Theory began with Ludwig Prandtl’s paper on the motion of a fluid with very small velocity  Written in 1904  Understanding motion of real fluids

 Inviscid flow is the flow of an ideal fluid that is assumed to have no sheer stress  Viscosity is the state of being thick, sticky, and semi fluid in consistency, due to internal friction  Velocity Gradient is the difference in velocities over the distance between two layers (v/x)

 Boundary layer is dynamic  As the distance from initial contact increases on a wing, so does the height  As fluid passes over greater distance, more fluid is slowed by friction increasing height  Velocity of fluid on wing is 0 relative to the plane and increases the further away it is from the wing until it reaches free flow velocity

 Tau=M*u/y  Tau=M*du/dy M=viscosity coefficient tau=sheer stress u=velocity of fluid y=boundary layer

 Boundary Layer causes drag  Result of sheer force and skin friction  Sheer force causes two components to slide upon each other in opposite directions parallel to the contact plane  Drag Force increases as one moves over the wing

 Up to this point, it has been the subject described  As length across the wing increases, velocity gradient and sheer stress decrease and lead into stage two  In laminar, viscous sheer stress has held the fluid in constant motion within layers  When slowed, the fluids begin to rotate

 Laminar boundary layer soon becomes turbulent boundary layer

 Slow moving fluid moves to the faster moving region slowing it down. The net effect is an increase in momentum in the boundary layer. We call the part of the boundary layer the turbulent boundary layer

 Predicts the flow of a fluid Re=inertial forces/viscous forces= ρ *v*L/ μ =v*L/v v=mean velocity of object relative to fluid L=traveled length of fluid μ= dynamic viscosity of fluid Kinetic viscosity= μ / ρ ρ =density of fluid

 Laminar flow: Re < 2000  Transitional flow: 2000 < Re < 4000  Turbulent flow: Re > 4000

 At points very close to the boundary the velocity gradients become very large and the velocity gradients become very large with the viscous shear forces again becoming large enough to maintain the fluid in laminar motion. This region is known as the laminar sub-layer. This layer occurs within the turbulent zone and is next to the wall and very thin - a few hundredths of a mm.

 Height of Turbulent B.L. > thickness Laminar S.L. = Turbulence and loss of energy  Height of Turbulent B.L. < thickness Laminar S.L. = little effect on boundary layer

 Delta/l=5/sqrt(Re)

 Detachment of the boundary layer into a broader wake  Defined as the point between the forward and backward flow where sheer stress is zero  Laminar flow is too distorted by the chaotic turbulent flow to generated sufficient lift

 This separation can lead to the phenomenon known as stalling where lift cannot be generated because difference in pressure cannot be maintained

Three Conventional Ways to Allay Stall:  Arrange Engines so they draw air from the rear keeping a steady air flow  have holes on the greater pressure side of the wing so that the pressure on top decreases so the adverse pressure gradient reduces boundary layer separation  Install flaps to increase fluid velocity reducing boundary layer separation

 Wikipedia. Wikimedia Foundation. Web. 30 July  "Boundary Layer." Boundary Layer. Web. 30 July  "Boundary Layer Flow." Boundary Layer Flow. Web. 30 July  "History of Boundary Layer Theory." - Annual Review of Fluid Mechanics, 9(1):87. Web. 30 July 2015.