Let \((\vec{V} \Delta t).\vec{ds}\) be the change in volume of elemental control volume in time Δt. Over the same time Δt, what is the change in volume of the whole control volume V with control surface S?(a) \(\int(\vec{V}\Delta t).\vec{ds}\)(b) \(\vec{V}\Delta t\)(c) \(\sum(\vec{V}\Delta t).\vec{ds}\)(d) \(\iint_s(\vec{V}\Delta t).\vec{ds}\)I had been asked this question in exam.My doubt is from Governing Equations in portion Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Let \((\vec{V} \Delta t).\vec{ds}\) be the change in volume of element

1.	Let \((\vec{V} \Delta t).\vec{ds}\) be the change in volume of elemental control volume in time Δt. Over the same time Δt, what is the change in volume of the whole control volume V with control surface S?(a) \(\int(\vec{V}\Delta t).\vec{ds}\)(b) \(\vec{V}\Delta t\)(c) \(\sum(\vec{V}\Delta t).\vec{ds}\)(d) \(\iint_s(\vec{V}\Delta t).\vec{ds}\)I had been asked this question in exam.My doubt is from Governing Equations in portion Governing Equations of Fluid Dynamics of Computational Fluid Dynamics
Answer» The CORRECT answer is (d) \(\iint_s(\VEC{V}\Delta t).\vec{ds}\) To elaborate: The CHANGE in volume of the WHOLE control volume is the summation of \((\vec{V} \Delta t).\vec{ds}\) over the total control surface S. This summation becomes integral as \(\vec{ds}\) is elemental. Therefore, \(\iint_s(\vec{V}\Delta t).\vec{ds}\) is the total change.

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