Boundary Conditions for Viscous Flows

Boundary conditions are the mechanism by which a solution of a differential equation is related to a specific physical problem. Whether or not a flow is viscous, there can be no velocity component normal to the solid surface of a body. If the body surface is not solid and there is suction or blowing through the surface, then the normal component of velocity is not zero at the wall but rather proportional to the mass flow through the surface. The focus here is on solid surfaces so that one boundary condition at our disposal is that there is no flow through the solid surface (i. e., the normal component of velocity at the surface is zero).

For a viscous flow, the physically observed fact is that the tangential velocity of the flow at the body surface is the same as that of the body. If the body is in motion, the fluid “sticks” to the surface and is carried along by the body. If the body is at rest (i. e., the usual point of view for problem solving), then the tangential velocity at the body surface is zero. This is the so-called no-slip boundary condition that must be imposed on the tangential velocity at a surface in a viscous flow.

The reason that the fluid sticks to the surface is explained at the molecular level in Chapter 2 in terms of the momentum transfer between the molecules and the molecular structure of the solid surface. In effect, all momentum parallel to the sur­face is destroyed by inelastic collisions on the rough interface.

Notice the fundamental difference in the boundary condition that is imposed on the tangential-velocity component at the surface of a body in inviscid and viscous flows. If the flow is inviscid, the velocity vector at the surface simply must be tan­gent to the body without any magnitude specified (i. e., a repetition of the previous “no-flow-through-the-surface” boundary condition). However, if the flow is viscous, then the tangential-velocity component at a solid surface at rest must be zero (i. e., the no-slip condition).

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