ME6204 Convective heat transfer Ch1: Equations of continuity, motion and energy Chapter 1: Equations of continuity, motion and energy Equation of continuity Conservation of mass One conservation law that is pertinent to the flow of a viscous fluid is that matter may neither be created nor destroyed. (x, y) () dx x u u + x, u z y, v v u () dy y v v + Mass enters and leaves the control volume exclusively through gross fluid motion. Transport due to such motion is often referred to as advection. Referring to the figure above, this law requires that, for steady flow, the net rate at which mass enters the control volume (inflow outflow) must equal zero. Hence, ()() ()() 0 y v x u 01dydx y v 1dxdy x u = + = + (1) Equation (1), the continuity equation, is a general expression of the overall mass conservation requirement, and it must be satisfied at every point in the fluid. The equation applies for a single species fluid, as well as for mixtures in which species diffusion and chemical reactions may be occurring. Dr. Chua Hui Tong, engchtnus.edu.sg 1.1 ME6204 Convective heat transfer Ch1: Equations of continuity, motion and energy If the fluid is incompressible, the density is a constant, and the continuity equation reduces to 0 y v x u = + (2) We shall now derive the continuity equation in a more general manner The rate of change of the mass within a given control volume V is = VV dV t dV dt d (3) where is the density (mass per unit volume) of matter within the control volume. This quantity is equal to the material flow of matter into the volume V through its surface , so that = v ddV t V (4) where d is an outward pointing elemental vector with magnitude d normal to the surface . v is the velocity of the matter. The quantities and v are all functions of time and space. Applying the Gauss theorem, which states that () =vvddV V , to equation (4), we obtain () = = V V dV ddV t v v and since equation (2) is valid for any arbitrary volume V, we have (v= t ) (5) In the rectangular coordinates, for which , we have the following form for the continuity equation kjiv w v u+= ()()() 0 z w y v x u t = + + + (6) In steady state, 0 t = ; and for a constant-density fluid, = constant. We therefore have the following simplified continuity equation Dr. Chua Hui Tong, engchtnus.edu.sg 1.2 ME6204 Convective heat transfer Ch1: Equations of continuity, motion and energy 0 z w y v x u = + + (7) Substantial derivative Now, for any quantity, that varies both in time and in space, i.e. = (x, y, z, t), its total (or substantial) time derivative is expressed as ()+ = + = + + + = v v t t dt dz zdt dy ydt dx xtdt d Hence, the total (or substantial) time derivative operator can be written as + =v tdt d (8) In the rectangular coordinates, this substantial time derivative operator could be expressed as z w y v x u tdt d + + + = (9) Hence, equation (5) could also be expressed as () () v vvv vvv v = = =+ = dt d dt d t t (10) Equation of motion Newtons second law of motion is pertinent to the flow of a viscous fluid. For a differential control volume in the fluid, this requirement states that the sum of all forces acting on the control volume must equal the net rate at which momentum leaves the control volume (outflow inflow). Two kinds of forces may act on the fluid: body forces, which are proportional to the volume, and surface forces, which are proportional to area. Gravitational, centrifugal, magnetic, and/or electric fields may contribute to the total body force, and we designate the x and y components of this force per unit volume of fluid as X and Y, respectively. The surface Dr. Chua Hui Tong, engchtnus.edu.sg 1.3 ME6204 Convective heat transfer Ch1: Equations of continuity, motion and energy forces Fs are due to the fluid static pressure as well as to viscous stresses. At any point in the fluid, the viscous stress (a force per unit area) may be resolved into two perpendicular components, which include a normal stress ii and a shear stress ij as shown below. (x, y) yy yx xx xy dx x xx xx + dx x xy xy + dy y yy yy + dy y yx yx + x z y -p dy y p p dx x p p -p A double subscript notation is used to specify the stress components. The first subscript indicates the surface orientation by providing the direction of its outward normal, and the second subscript indicates the direction of the force component. Accordingly, for the x surface in the figure above, the normal stress xx corresponds to a force component normal to the surface, and the shear stress xy corresponds to a force in the y direction along the surface. All the stress components are shown positive in the sense that both the surface normal and the force co