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This paper focuses on the laminar boundary layer startup process (momentum and thermal) in incompressible flows. The unsteady boundary layer equations can be solved via similarity analysis by normalizing the stream-wise (x), wall-normal (y) and time (t) coordinates by a variable η and τ, respectively. The resulting ODEs are solved by a finite difference explicit algorithm. This can be done for two cases: flat plate flow where the change in pressure are zero (Blasius solution) and wedge or Falkner-Skan flow where the changes in pressure can be favorable (FPG) or adverse (APG). In addition, transient passive scalar transport is examined by setting several Prandtl numbers in the governing equation at two different wall thermal conditions: isothermal and isoflux. Numerical solutions for the transient evolution of the momentum and thermal boundary layer profiles are compared with analytical approximations for both small times (unsteady flow) and large (steady-state flow) times. 

The laminar boundary layer of a viscous incompressible fluid subject to a two-dimensional wall curvature is evaluated. It is well known that a curved surface induces streamwise pressure gradient as well as wall curvature driven pressure gradient. Under certain assumptions, a family of similarity solutions can be obtained under the influence of flow acceleration/deceleration, which is known as the Falkner-Skan similarity solutions. In this study, the effect of the wall normal pressure gradient is taken into consideration, and the freestream flow parameters are adjusted for flow over a curved surface. Present results are obtained by numerical solution of a generalized Falkner-Skan equation governing similar solutions for flows over curved surfaces. The Falkner-Skan equations are solved by an RK4 shooting algorithm. Additionally, the transport of a passive scalar is incorporated in the present analysis at different Prandtl numbers. The objective of this paper is to use the curvilinear or axisymmetric boundary layer and energy equations to assess the effect of Favorable, Adverse and Zero pressure gradient on the laminar momentum and thermal boundary layer development. Major conclusions are summarized as follows: (i) as the pressure gradient β increases from negative values (APG) towards positive (FPG) values, the displacement (Δ∗) and momentum (θ∗) thickness tend to decrease no matter the curvature type, and, (ii) the normalized wall shear stress (i.e., f′′) exhibits a linear decreasing behavior as the wall curvature switches from concave (negative) to convex (positive) at a constant pressure gradient.