Nonlinear motion equations for a Non-Newtonian incompressible fluid in an orthogonal coordinate system

Summary Starting with an assumed relationship between the stress tensor, the non-Newtonian viscosity, and the strain rate tensor, the nonlinear equations of motion are developed for use in any orthogonal coordinate system. The resulting equations are written in terms of the scalar velocities, the non-Newtonian viscosity, the metric coefficients, and their derivatives.The non-Newtonian viscosity is assumed to be a scalar function of the strain rate tensor, and so depends upon the invariants of the strain rate tensor. For convenience, the necessary invariants are written out in complete form for use in any orthogonal coordinate system, in terms of the scalar velocities, the metric coefficients, and their derivatives.Using the resulting motion equations and a model of this type of viscosity, theOstwald-de Waele model, an example of time dependent flow is solved using a continuous time-discrete space method programmed on an analog computer.e_ij strain rate tensor - body force density, dynes/cm³ - F₁,F₂,F₃ components of body force density, dynes/cm³ - g acceleration of gravity - H function of time - h₁,h₂,h₃ metric coefficients - I₁,I₂,I₃ invariants - m constant - P pressure, dynes/cm² - r radius, cm - t time, sec - velocity vector, cm/sec - v₁,v₂,v₃ velocities in thex₁,x₂ andx₃ directions, respectively, cm/sec - v_n(t) velocity of thenth node, cm/sec - x₁,x₂,x₃ coordinate directions - z coordinate, cm - unit tensor - _ij Kronecker delta - _ij 2e_ij - nabla - _ijk alternating unit tensor - non-Newtonian viscosity, dynes/cm² - ₀,₁ constant viscosities, dynes sec/cm², dynes sec^m/cm² - angle, radians - v₀,v₁ constant kinematic viscosities, cm²/sec, cm² sec^m-2 - density, g/cm³ - _ij stress tensor - fluid dilationWith 3 figures