Prove that pressure gradient in a static fluid is represented by a)dp/dz =-dg and b)dp/dx =0 , where z direction is vertically upwards, x axis is along horizontal, d is density of fluid and p is pressure.

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Solution

$W e k n o w t h a t p r e s s u r e d u e t o a t m o p h e r e a t a p l a c e d e p e n d s u p o n h e i g h t h f r o m t h e s u r f a c e o f s e a l e v e l, d e n s i t y d o f t h e s u b s \tan c e w h i c h r e c o r d s t h e p r e s s u r e a n d t h e v a l u e o f a c c e l e r a t i o n d u e t o g r a v i t y g a t t h a t p l a c e i . e . P = h d g A s w e m o v e u p f r o m t h e s e a l e v e l t h e v a l u e o f a t m o s p h e r i c p r e s s u r e d e c r e a s e s . T h e r e f o r e a t m o s p h e r i c p r e s s u r e P = - h d g D i f f e r e n t i a t i n g a b o v e e q n w . r . t . h e i g h t h, w e g e t, \frac{d P}{d h} = - d g a n d t h e d i f f e r e n t i a t i o n o f p r e s s u r e P w i t h r e s p e c t t o h o r i z o n t a l d i s \tan c e x i s g i v e n b y, \frac{d P}{d x} = 0 T h i s s h o w s t h a t p r e s s u r e d o e s n o t d e p e n d o n h o r i z o n t a l d i s \tan c e .$

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d

q = \frac{1}{2} n v^{2}

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