We consider the stationary Stokes and Navier-Stokes Equations for viscous, incompressible flow in parameter dependent bounded domains D T , subject to homogeneous Dirichlet ("noslip") boundary conditions on ∂D T . Here, D T is the image of a given fixed nominal LipschitzWe establish shape holomorphy of Leray solutions which is to say, holomorphy of the mapdenotes the pullback of the corresponding weak solutions and T varies in W k,∞ with k ∈ {1, 2}, depending on the type of pullback. We consider in particular parametrized families {T y : y ∈ U } ⊆ W 1,∞ of domain mappings, with parameter domain U = [−1, 1] N and with affine dependence of T y on y. The presently obtained shape holomorphy implies summability results and n-term approximation rate bounds for gpc ("generalized polynomial chaos") expansions for the corresponding parametric solution map y → (û y ,p