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Hello and welcome back in video 43 we have derived integral form of continuity equation using the Reynolds transport equation in this video we will derive a mathematical equation for the same principle using a different approach in previous videos we derived integral equations of conservation of mass momentum and energy of a fluid flowing steadily through a control volume such analysis will be useful for the calculation of total mass flow rate total thrust and total power output from simple steady flow thermal fluid systems note that the integral approach study global effects of system flow however more complex fluid dynamic analysis requires calculation of flow properties at local points inside the flow for example if we are interested in calculating the velocity vector of a fluid particle at a given location within the system or calculating the value of local density a differential approach may be required in this case a differential control volume is considered around the point of interest and motion of a differential fluid elemental system through this control volume is analyzed if the mass of differential fluid system is of interest that would lead to differential form of continuity equation similarly consideration of momentum will lead to differential form of linear momentum equation and total energy lead to differential form of energy equation consider integral form of continuity equation we have derived in video 43 Let It Be equation a by converting surface integral to a volume integral and using divergence theorem along with some mathematical manipulations we get equation C which is called the differential form of continuity in this equation vector V represents velocity field of the flow which has three scalar component you V and W note that u v w are functions of space variables X Y Z and time T in a Cartesian coordinate system the Del operator can be expressed as a vector function finally this equation can be expanded and simplified for a steady incompressible flow as equation d for two-dimensional steady incompressible flows this equation can be further simplified note that this is a two-dimensional partial differential equation with two unknown function Ill aria Bull's U and V solutions of this partial differential equation using appropriate boundary conditions will result in velocity field of a given two-dimensional flow system thank you for listening
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