Question

Consider the Navier Stokes equations for a compressible Newtonian fluid (see page 9 of the CONSTITUTIVE EQUATIONS lecture), show that for incompressible flow of a Newtonian fluid with constant viscosity the right-hand side terms of the momentum equation reduce to the simple expression Op where ▽-▽ . ▽ is the Laplacian operator.

Answer 1

Rate of change of u - momentum within control volume = Delta x Delta y = Partial(pu)/partial t Rate at which u-momentum enters through Face 1: + su^2 Delta y Face 2: -su^2 Delta y Face 3: + suv Delta x Face 4: -suv Delta x pressure force acting on Face 1: + p Delta y pressure force acting on Face 2: - p Delta y Normal viscous force on Face 1: - T_x_p Delta y Normal viscous force on Face 2: + T_x_p Delta y Tangential viscous force on Face 3: - T_yx Delta x Tangential viscous force on Face 1: + T_yx Delta x summing up these ant divide with Delta_x Delta_y we get u - momentum partial differential (pu)/partial differential t + partial differential(su^2 + p)/partial differential x + partial differential (suv)/partial differential y = partial differential tau_xx /partial differential x + partial differential tau_xy/partial differential y v- momentum partial differential (sv)/partial differential t + partial differential (suv)/partial differential x + partial differential (sv^2 + p)/partial differential y = partial differential tau_xy/partial differential x + partial differential tau yy/partial differential y